2023-07-04T12:28:46Z

Fulvio: /* Tutorial 3 */

This tutorial will show you how to visualise wave-function and band-structure related information following a DFT Quantum Espresso calculation using the "qepy" module of Yambopy.

'''Concerning the ICTP school: recall that for the yambopy tutorials you have to load the yambo, quantum-espresso and anaconda3 modules:'''
spack load yambo
spack load quantum-espresso
spack load anaconda3
'''You can copy the tutorial databases as explained in the virtual machine setup page: '''
cp -r /media/ictpuser/smr3694/ictptutor/YAMBOPY_TUTORIALS ~/

The full tutorial, including the Quantum espresso and Yambo databases that we will read, can be downloaded and extracted from the yambo website:
$wget https://media.yambo-code.eu/educational/tutorials/files/databases_qepy.tar
$tar -xvf databases_qepy.tar
$cd databases_qepy

==Tutorial 1. BN (semiconductor). Band structure==

First enter in the folder
cd databases_qepy/bn-semiconductor
and have a look to the
vim plot-qe-bands.py
The qepy classes are useful both to execute Quantum Espresso and to analyze the results. Enter in the python environment, by typing <code>python</code>, then the full qepy library is imported by simply doing:

from qepy import *

===Plot Band structure===

The qepy class '''PwXML''' reads the data file generated by Quantum Espresso and post-processes the data. The class is instanced by doing:

xml = PwXML(prefix='bn', path='bands')

The variable prefix corresponds to the same variable of the QE input. The folder location is indicated by variable path. In order to plot the bands, we also define the k-points path (in crystal coordinates) using the function Path:

npoints = 50
path_kpoints = Path([ [[0.0, 0.0, 0.0],'$\Gamma$'],
[[0.5, 0.0, 0.0],'M'],
[[1./3,1./3,0.0],'K'],
[[0.0, 0.0, 0.0],'$\Gamma$']], [int(npoints*2),int(npoints),int(sqrt(5)*npoints)])

It is worth to note that the path should coincide with the selected path for the QE band calculations.

In order to show the plot we call the '''plot_eigen''' method of the '''PwXML''' class:

xml.plot_eigen(path_kpoints)

This function will automatically plot the bands as shown below running the script:

python plot-qe-bands.py

[[File:Bands BN 1.png| 400 px | center |Band structure of BN calculated at the level of DFT-LDA]]

Alternatively, we can use the function '''plot_eigen_ax'''. This functions requires as input a matplotlib '''figure''' object with given axes, as you will see in the next example.

===Plot atomic orbital projected Band structure===

In addition to the band structure, useful information regarding the atomic orbital nature of the electronic wave functions can be displayed using the class '''ProjwfcXML'''.
In order to make quantum espresso calculate the relevant data, we need to use the QE executable '''projwfc.x''', which will create the file '''atomic_proj.xml'''. This executable projects the Kohn-Sham wavefunctions onto orthogonalized atomic orbitals, among others functionalities. The orbital indexing and ordering are explained in the input documentation of the projwfc.x function which you are invited to check (https://www.quantum-espresso.org/Doc/INPUT_PROJWFC.html#idm94). We can run '''projwf.x''' directly from the python script using the qepy class '''ProjwfcIn''':

proj = ProjwfcIn(prefix='bn')
proj.run(folder='bands')

Be aware that this can take a while in a large system with many k-points. As in the class '''PwXML''', we then define the path_kpoints and also the selected atomic orbitals to project our bands. We have chosen to represent the projection weight onto the nitrogen (1) and boron (2) orbitals, which according to the projection orderings is given by

atom_1 = list(range(16))
atom_2 = list(range(16,32))

The full list of orbitals is written in the file '''bands/prowfc.log'''. We have also defined the figure box

import matplotlib.pyplot as plt
fig = plt.figure(figsize=(5,7))
ax = fig.add_axes( [ 0.12, 0.10, 0.70, 0.80 ])

The class '''ProjwfcXML''' then runs as in this example:

band = ProjwfcXML(prefix='bn',path='bands',qe_version='7.0')
band.plot_eigen(ax,path_kpoints=path_kpoints,cmap='viridis',selected_orbitals=atom_1,selected_orbitals_2=atom_2)

We can run now the file:

python plot-qe-orbitals.py

[[File:Bands AOP BN 1.png| 400 px | center | Atomic orbital projected band structure of monolayer BN]]

We have chosen the colormap 'viridis' (variable cmap). You see that the colormap goes from maximum '''selected_orbitals''' content (in this case nitrogen) to the maximum '''selected_orbitals_2''' content (in this case boron).
The colormap can be represented in many ways and qepy does not include a particular function for this. An example is:

import matplotlib as mpl
cmap=plt.get_cmap('viridis')
bx = fig.add_axes( [ 0.88, 0.10, 0.05, 0.80 ])
norm = mpl.colors.Normalize(vmin=0.,vmax=1.)
cb1 = mpl.colorbar.ColorbarBase(bx, cmap=cmap, norm=norm,orientation='vertical',ticks=[0,1])
cb1.set_ticklabels(['B', 'N'])

We can also plot the band orbital charater with size variation instead of a color scale. In this case we have to pass only the variable selected_orbitals (see the next tutorial).

==Tutorial 2. Iron. Ferromagnetic metalic material==

In the case of spin-polarized calculations we can plot the spin up and down band structures. We have included in the tutorial a small workflow example to run quantum espresso calculations from scratch. This is done using the classes Tasks and Flows developed in yambopy. You can check the file flow-iron.py for an example.
Yambopy includes basic predefined workflows to run the common QE and Yambo calculations. In this case we are using the class '''PwBandsTasks'''.

python flow-iron.py

In order to plot the spin-polarized bands. After doing all the calculations from scratch with the flows (flow-iron.py), we can make the band plot by running the script:

python plot-qe-bands.py

The class PwXML automatically detects the spin polarized case (nspin=2 in the QE input file). The spin up channel is displayed with red and the spin down channel with blue. In the case of iron we have selected this k-point path:

npoints = 50
path_kpoints = Path([ [[0.0, 0.0, 0.0 ],'G'],
[[0.0, 0.0, 1.0 ],'H'],
[[1./2,0.0,1./2.],'N'],
[[0.0, 0.0, 0.0 ],'G'],
[[1./2, 1./2, 1./2 ],'P'],
[[1./2,0.0,1./2. ],'N']], [npoints,npoints,npoints,npoints,npoints])

xml = PwXML(prefix='pw',path='bands/t0')
xml.plot_eigen(path_kpoints)

[[File:Figure 3-iron-bands.png| 600 px | center |Spin polarized band structure of iron calculated by DFT]]

The analysis of the projected atomic orbitals is also implemented. In this case the results are more cumbersome because the projection is separated in spin up and down channels. 
Let us look first at the file '''plot-qe-orbitals-size'''. 

'''NB: If you generated the quantum espresso databases using the flows instead of relying on the precomputed databases, you also need to rerun the quantum espresso executable projwfc.x to recompute the orbital projections. In this case, please uncomment the following lines in the script''' (plot-qe-orbitals-size.py) :
#proj = ProjwfcIn(prefix='pw')
#proj.run(folder='bands/t0')

Now, we can use the dot size as a function of the weight of the orbitals
atom_s = [8]
atom_p = [0,1,2]
atom_d = [3,4,5,6,7]

and the plots are done with
band = ProjwfcXML(prefix='pw',path='bands/t0',qe_version='7.0')
band.plot_eigen(ax,path_kpoints=path_kpoints,selected_orbitals=atom_s,color='pink',color_2='black')
band.plot_eigen(ax,path_kpoints=path_kpoints,selected_orbitals=atom_p,color='green',color_2='orange')
band.plot_eigen(ax,path_kpoints=path_kpoints,selected_orbitals=atom_d,color='red',color_2='blue')

As an example, we can select just the ''d'' orbitals by commenting the first two plots and then running the file:

plot-qe-orbitals-size.py

[[File:Figure 4-iron-bands-size-d-orbitals.png|400px|center|Iron band structure. Size is proportional to the weights of the projection on atomic d-orbitals. Red (blue) is up (down) spin polarization.]]

From the plot is clear that ''d'' orbitals are mainly localized around the Fermi energy. The plot above is in red and blue, while the default choice in your script should be pink and black. You can experiment with the colors and other ''matplotlib'' plot options and also plot the other orbital types.

Another option is to plot the orbital composition as a colormap running the file:

plot-qe-orbitals-colormap.py

[[File:Figure 5-iron-bands-colormap.png|400px|center]]

Here we have added the p and d orbitals in the analysis:

atom_p = [0,1,2]
atom_d = [3,4,5,6,7]
band = ProjwfcXML(prefix='pw',path='bands/t0',qe_version='7.0')
band.plot_eigen(ax,path_kpoints=path_kpoints,cmap='viridis',cmap2='rainbow',selected_orbitals=atom_p,selected_orbitals_2=atom_d)

The colormap bar is added in the same way as in Tutorial 1 (see script), but this time we have a different colormap for each spin polarisation.

==Tutorial 3: GW bands==

Yambopy can be used either to run Yambo and QE calculations, or to analyse the results of QE and Yambo by dealing with their generated databases. This is done with a variety of classes included in the qepy (for QE) or yambopy (for Yambo) modules.
In the case of Yambo GW quasi-particle calculations, we can use the yambopy class '''YamboQPDB''' to read the database produced by the simulation.
Enter in the folder
cd ../gw-bands
We can use this to find the scissor operator, plot the GW bands and to interpolate the GW bands on a smoother k-path. The example runs by typing:

python plot-qp.py

See below for an explanation of the tutorial. As usual, we can import the qepy and yambopy libraries:

from qepy import *
from yambopy import *
import matplotlib.pyplot as plt

We define the k-points path.
npoints = 10
path = Path([ [[ 0.0, 0.0, 0.0],'$\Gamma$'],
[[ 0.5, 0.0, 0.0],'M'],
[[1./3.,1./3., 0.0],'K'],
[[ 0.0, 0.0, 0.0],'$\Gamma$']], [int(npoints*2),int(npoints),int(sqrt(5)*npoints)] )

Importantly, the number of points is a free choice. We can increase the variable npoints as we wish, it just means that the interpolation step will take more time. In order to analyse GW results we need to have the file related to the basic data of our Yambo calculation ('''SAVE/ns.db1''') and the netcdf file with the quasi-particle results ('''ndb.QP'''). We load the data calling the respective classes:

# Read Lattice information from SAVE
lat = YamboSaveDB.from_db_file(folder='SAVE',filename='ns.db1')
# Read QP database
ydb = YamboQPDB.from_db(filename='ndb.QP',folder='qp-gw')
(in the yambopy module, each class is specialised to read a specific Yambo database)

The first option is to plot the energies and calculate the ideal Kohn-Sham to GW scissor operator. We need to select the index of the top valence band:

n_top_vb = 4
ydb.plot_scissor_ax(ax,n_top_vb)

Yambopy displays the fitting and also the data of the slope of each fitting. Notice that this is also a test if the GW calculations are running well. '''If the dependence is not linear you should double-check your results!'''

[[File:Figure 6-slope-scissor.png|400px|center]]

In this case the slope is:

valence bands:
slope: 1.0515886598785766
conduction bands:
slope: 1.026524081134514
scissor list (shift,c,v) [eV,adim,adim]: [1.8985204833551723, 1.026524081134514, 1.0515886598785766]

In addition to the scissor operator, we can plot the GW (and DFT) band structure along the path. The first choice would be to plot the actual GW calculations, without interpolation, to check that the results are meaningful (or not). This plot is independent of the number of k-points ('''npoints''') that we want to put in the interpolation. The class YamboQPDB finds the calculated points that belong to the path and plots them. Be aware that if we use coarse grids the class would not find any point and the function will not work.

ks_bs_0, qp_bs_0 = ydb.get_bs_path(lat,path)
ks_bs_0.plot_ax(ax,legend=True,color_bands='r',label='KS')
qp_bs_0.plot_ax(ax,legend=True,color_bands='b',label='QP-GW')

[[File:Figure 7-GW-band-structure-non-interpolated.png|400px|center]]

The interpolation of the DFT and GW band structures looks similar:

ks_bs, qp_bs = ydb.interpolate(lat,path,what='QP+KS',lpratio=20)
ks_bs.plot_ax(ax,legend=True,color_bands='r',label='KS')
qp_bs.plot_ax(ax,legend=True,color_bands='b',label='QP-GW')

The '''lpratio''' can be increased if the interpolation does not work as well as intended. The interpolation is the same one implemented in abipy and currently requires abipy installed in order to work.

[[File:Figure 8-GW-band-structure-interpolated.png|400px|center]]

Finally, we can compare the calculated GW eigenvalues with the interpolation.

[[File:Figure 8-GW-band-structure-comparison.png|400px|center]]

==Links==
* Back to [[Rome 2023#Tutorials]]
* Back to [[ICTP 2022#Tutorials]]

Tutorials

2023-06-15T14:34:24Z

Fulvio: /* Post Processing */

If you are starting out with Yambo, or even an experienced user, we recommend that you complete the following tutorials before trying to use Yambo for your system.

The tutorials are meant to give some introductory background to the key concepts behind Yambo. Practical topics such as convergence are also discussed.
Nonetheless, users are invited to first read and study the [[lectures|background material]] in order to get familiar with the fundamental physical quantities.

Two kinds of tutorials are provided: '''stand-alone''' and '''modular'''. In addition you must have a working environment where both Yambo (and eventually QE) are installed.

== Setting up Yambo (and eventually QE) ==
To be able to follow the school you need a running version of the yambo/QE code.

There are several different ways to prepare a working environment.

=== Virtual Machine(s) ===
The easiest way is to access to a virtual machine which contains both (i) yambo/QE and (ii) the tutorials.

You can do it in one of two ways:
* Virtual machine via [[ICTP cloud|ICTP cloud]] If the schools you are attending provided an ICTP virtual machine this is the preferred option. It works through internet connection inside a browser.
* Install the [[Yambo_Virtual_Machine|yambo virtual machine]] on your laptop / desktop. This requires Oracle virtual box. Pre-download of the Virtual machine. No internet connection needed.


=== User installation ===
You can also setup the yambo code on your on laptop / desktop using different methods.

As far as the Yambo source is concerned you can:
* Install [[Yambo via Docker|Yambo via Docker]]
* [[download|Download]] and [[Installation|install]] yambo on your laptop / desktop (requires a linux machine).
* Install yambo on your laptop/desktop/cluster [https://github.com/nicspalla/my-repo via Spack].
* Install using Anaconda.

=== Yambo User Installation with Anaconda ===
It is possible to install Yambo (up to v5.0.4) and Quantum-ESPRESSO via conda-forge (a conda channel/repository):
To setup Anaconda, please start from installing [https://www.anaconda.com/products/distribution#Downloads Anaconda] or [https://docs.conda.io/en/latest/miniconda.html Miniconda].

Then we suggest to create a conda environment and activate it:
conda create --name yambopy -c conda-forge
conda activate yambopy
Then you can install the prerequisites and the two codes:
conda install numpy scipy netcdf4 matplotlib pyyaml lxml pandas
conda install yambo
conda install qe

==Setting up Yambopy==

===Quick installation===

A quick way to start using Yambopy is described here.

* Make sure that you are using Python 3 and that you have the following python packages: <code>numpy</code>, <code>scipy</code>, <code>matplotlib</code>, <code>netCDF4</code>, <code>lxml</code>, <code>pyyaml</code>. Optionally, you may want to have abipy [[https://abinit.github.io/abipy/index.html]] installed for band structure interpolations.

* Go to a directory of your choice and clone yambopy from the git repository

git clone https://github.com/yambo-code/yambopy.git

If you don't want to use git, you may download a tarball from the git repository instead.

* Enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Installing dependencies with Anaconda===
We suggest installing yambopy using Anaconda [[https://www.anaconda.com/products/distribution]] to manage the various python packages.

In this case, you can follow these steps.

First, install the required dependencies:
conda install numpy scipy netcdf4 lxml pyyaml

Then we create a conda environment based on python 3.6 (this is to ensure compatibility with abipy if we want to install it later on):
conda create --name NAME_ENV python=3.6
Here choose <code>NAME_ENV</code> as you want, e.g. <code>yenv</code>.

Now, we install abipy and its dependency pymatgen using <code>pip</code>. Here make sure that you are using the <code>pip</code> version provided by Anaconda and not your system version.

pip install pymatgen
pip install abipy

Finally, we are ready to install yambopy:

git clone https://github.com/yambo-code/yambopy.git

(or download and extract tarball) and follow the steps outlined in the quick installation section.

Now enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Frequent issues===
When running the installation you may get a <code>SyntaxError</code> related to utf-8 encoding or it may complain that module <code>setuptools</code> is not installed even though it is. In this case, it means that the <code>sudo</code> command is not preserving the correct <code>PATH</code> for your python executable.

Solve the problem by running the installation step as

sudo /your/path/to/python setup.py install
or
sudo env PATH=$PATH python setup.py install

This applies only to the installation step and not to subsequent yambopy use.

== Tutorial files ==
The tutorial CORE databases can be obtained

* from the [[Yambo_Virtual_Machine|Yambo Virtual Machine]]
* from the Yambo web-page
* from the Yambo GIT tutorial repository

=== From the Yambo Virtual Machine (VM) ===
If you are using the VM, a recent version of the tutorial files is provided.Follow these [[Yambo_Virtual_Machine#Updating_the_Yambo_tutorial_files| instructions]] to update the tutorial files to the most recent version.

=== From the Yambo website ===
If you are using your own installation or the docker, the files needed to run the tutorials can be downloaded from the lists below.

After downloading the tar.gz files just unpack them in '''the YAMBO_TUTORIALS''' folder. For example
$ mkdir YAMBO_TUTORIALS
$ mv hBN.tar.gz YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ tar -xvfz hBN.tar.gz
$ ls YAMBO_TUTORIALS
hBN

====Files needed for modular tutorials====
All of the following should be downloaded prior to following the modular tutorials:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="4"| hBN || [https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz hBN.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-convergence-kpoints.tar.gz hBN-convergence-kpoints.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-para.tar.gz hBN-2D-para.tar.gz]
|-
|}

====Files needed for stand-alone tutorials====
At the start of each tutorial you will be told which specific file needs to be downloaded:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="2"| Silicon || [https://media.yambo-code.eu/educational/tutorials/files/Silicon.tar.gz Silicon.tar.gz]
|-
|[https://media.yambo-code.eu/educational/tutorials/files/Silicon_Electron-Phonon.tar.gz Silicon_Electron-Phonon.tar.gz]
|-
| LiF || [https://media.yambo-code.eu/educational/tutorials/files/LiF.tar.gz LiF.tar.gz]
|-
| Aluminum || [https://media.yambo-code.eu/educational/tutorials/files/Aluminum.tar.gz Aluminum.tar.gz]
|-
| GaSb || [https://media.yambo-code.eu/educational/tutorials/files/GaSb.tar.gz GaSb.tar.gz]
|-
| AlAs || [https://media.yambo-code.eu/educational/tutorials/files/AlAs.tar.gz AlAs.tar.gz]
|-
| Hydrogen_Chain || [https://media.yambo-code.eu/educational/tutorials/files/Hydrogen_Chain.tar.gz Hydrogen_Chain.tar.gz]
|-
| MoS2 for HPC || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_HPC_tutorial.tar.gz MoS2_HPC_tutorial.tar.gz]
|-
| MoS2 for HPC shorter version || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_2Dquasiparticle_tutorial.tar.gz MoS2_2Dquasiparticle_tutorial.tar.gz]
|-
| Yambopy for QE || [https://media.yambo-code.eu/educational/tutorials/files/databases_qepy.tar databases_qepy]
|-
| Yambopy for YAMBO || [https://media.yambo-code.eu/educational/tutorials/files/databases_yambopy.tar databases_yambopy]
|-
|}

=== From the Git Tutorial Repository (advanced users) ===
If you are using your own installation or the docker, the [https://github.com/yambo-code/tutorials tutorials repository] contains the updated tutorials CORE databases. To use it
$ git clone https://github.com/yambo-code/tutorials.git YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ ./setup.pl -install

== Stand-alone tutorials ==
These tutorials are self-contained and cover a variety of mixed topics, both physical and methodological. They are designed to be followed from start to finish in one page and do not require previous knowledge of yambo. Each tutorial requires download of a specific core database, and typically they cover a specific physical system (like bulk GaSb or a hydrogen chain). Ground state input files and pseudopotentials are provided. Output files are also provided for reference.

These tutorials can be accessed directly from this page of from the side bar. They include different kind of subjects:

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

=== Basic ===
* [[LiF|Linear Response in 3D. Excitons at work]]
* [[Silicon|GW convergence]]

=== Post Processing ===
* [[Yambo Post Processing (ypp)]]
* [[First steps in Yambopy]]
* [[Yambopy tutorial: band structures]]
* [[Yambopy tutorial: Yambo databases]]
* [[GW tutorial. Convergence and approximations (BN)]] [yambopy tutorial]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]] [yambopy tutorial]

=== Advanced ===
* [[Hydrogen chain|TDDFT Failure and long range correlations]]
* [[Linear response from real time simulations]]

==== GW and Quasi-particles ====
* [[Real_Axis_and_Lifetimes|Real Axis and Lifetimes]]
* [[Self-consistent GW on eigenvalues only]]
* [[GW tutorial on HPC]]

==== Electron phonon coupling ====
* [[Electron Phonon Coupling|Electron Phonon Coupling]]
* [[Optical properties at finite temperature]]
* [[Phonon-assisted luminescence by finite atomic displacements]]
* [[Exciton-phonon coupling and luminescence]]

==== Non linear response ====
* [http://www.attaccalite.com/lumen/linear_response.html Linear response using Dynamical Berry phase]
* [[Real time approach to non-linear response]]
* [[Correlation effects in the non-linear response]]
* [http://www.attaccalite.com/lumen/thg_in_silicon.html Third Harmonic Generation]
* [http://www.attaccalite.com/lumen/spin_orbit.html Spin-orbit coupling and non-linear response]
* [[Two-photon absorption]]
* [[Pump and Probe]]
* [[Parallelization for non-linear response calculations]]

==== Developing Yambo ====
* [[How to create a new project in Yambo]]
* [[How to create a new ypp interface]]
* [[Some hints on github]]




== Modular tutorials ==
These tutorials are designed to provide a deeper understanding of specific yambo tasks and runlevels. They are designed to avoid repetition of common procedures and physical concepts. As such, they make use of the same physical systems: bulk hexagonal boron nitride ''hBN'' and a hBN sheet ''hBN-2D''.

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

====Introduction====
* [[First steps: a walk through from DFT to optical properties]]
====Quasiparticles in the GW approximation====
* [[How to obtain the quasi-particle band structure of a bulk material: h-BN]]
====Using Yambo in Parallel====
This modules contains very general discussions of the parallel environment of Yambo. Still the actual run of the code is specific to the CECAM cluster. If you want to run these modules just replace the parallel queue instructions with simple MPI commands.

* [[GW_parallel_strategies|Parallel GW (CECAM specific)]]: strategies for running Yambo in parallel
[[GW_parallel_strategies_CECAM]]
* [[Pushing_convergence_in_parallel|GW convergence (CECAM specific)]]: use Yambo in parallel to converge a GW calculation for a layer of hBN (hBN-2D)

====Excitons and the Bethe-Salpeter Equation====
* [[How to obtain an optical spectrum|Calculating optical spectra including excitonic effects: a step-by-step guide]]
* [[How to choose the input parameters|Obtaining a converged optical spectrum]]
* [[How to treat low dimensional systems|Many-body effects in low-dimensional systems: numerical issues and remedies]]
* [[How to analyse excitons|Analysis of excitonic spectra in a 2D material]]


====Yambopy====
* [[First steps in Yambopy]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]
* [[Yambopy tutorial: band structures | Database and plotting tutorial for quantum espresso: qepy]]
* [[Yambopy tutorial: Yambo databases | Database and plotting tutorial for yambo: yambopy ]]


=== Modules ===
Alternatively, users can learn more about a specific runlevel or task by looking at the individual '''[[Modules|documentation modules]]'''. These provide a focus on the input parameters, run time behaviour, and underlying physics. Although they can be followed separately, non-experts are urged to follow them as part of the more structured tutorials given above.

Tutorials

2023-06-15T14:34:02Z

Fulvio: /* Post Processing */

If you are starting out with Yambo, or even an experienced user, we recommend that you complete the following tutorials before trying to use Yambo for your system.

The tutorials are meant to give some introductory background to the key concepts behind Yambo. Practical topics such as convergence are also discussed.
Nonetheless, users are invited to first read and study the [[lectures|background material]] in order to get familiar with the fundamental physical quantities.

Two kinds of tutorials are provided: '''stand-alone''' and '''modular'''. In addition you must have a working environment where both Yambo (and eventually QE) are installed.

== Setting up Yambo (and eventually QE) ==
To be able to follow the school you need a running version of the yambo/QE code.

There are several different ways to prepare a working environment.

=== Virtual Machine(s) ===
The easiest way is to access to a virtual machine which contains both (i) yambo/QE and (ii) the tutorials.

You can do it in one of two ways:
* Virtual machine via [[ICTP cloud|ICTP cloud]] If the schools you are attending provided an ICTP virtual machine this is the preferred option. It works through internet connection inside a browser.
* Install the [[Yambo_Virtual_Machine|yambo virtual machine]] on your laptop / desktop. This requires Oracle virtual box. Pre-download of the Virtual machine. No internet connection needed.


=== User installation ===
You can also setup the yambo code on your on laptop / desktop using different methods.

As far as the Yambo source is concerned you can:
* Install [[Yambo via Docker|Yambo via Docker]]
* [[download|Download]] and [[Installation|install]] yambo on your laptop / desktop (requires a linux machine).
* Install yambo on your laptop/desktop/cluster [https://github.com/nicspalla/my-repo via Spack].
* Install using Anaconda.

=== Yambo User Installation with Anaconda ===
It is possible to install Yambo (up to v5.0.4) and Quantum-ESPRESSO via conda-forge (a conda channel/repository):
To setup Anaconda, please start from installing [https://www.anaconda.com/products/distribution#Downloads Anaconda] or [https://docs.conda.io/en/latest/miniconda.html Miniconda].

Then we suggest to create a conda environment and activate it:
conda create --name yambopy -c conda-forge
conda activate yambopy
Then you can install the prerequisites and the two codes:
conda install numpy scipy netcdf4 matplotlib pyyaml lxml pandas
conda install yambo
conda install qe

==Setting up Yambopy==

===Quick installation===

A quick way to start using Yambopy is described here.

* Make sure that you are using Python 3 and that you have the following python packages: <code>numpy</code>, <code>scipy</code>, <code>matplotlib</code>, <code>netCDF4</code>, <code>lxml</code>, <code>pyyaml</code>. Optionally, you may want to have abipy [[https://abinit.github.io/abipy/index.html]] installed for band structure interpolations.

* Go to a directory of your choice and clone yambopy from the git repository

git clone https://github.com/yambo-code/yambopy.git

If you don't want to use git, you may download a tarball from the git repository instead.

* Enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Installing dependencies with Anaconda===
We suggest installing yambopy using Anaconda [[https://www.anaconda.com/products/distribution]] to manage the various python packages.

In this case, you can follow these steps.

First, install the required dependencies:
conda install numpy scipy netcdf4 lxml pyyaml

Then we create a conda environment based on python 3.6 (this is to ensure compatibility with abipy if we want to install it later on):
conda create --name NAME_ENV python=3.6
Here choose <code>NAME_ENV</code> as you want, e.g. <code>yenv</code>.

Now, we install abipy and its dependency pymatgen using <code>pip</code>. Here make sure that you are using the <code>pip</code> version provided by Anaconda and not your system version.

pip install pymatgen
pip install abipy

Finally, we are ready to install yambopy:

git clone https://github.com/yambo-code/yambopy.git

(or download and extract tarball) and follow the steps outlined in the quick installation section.

Now enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Frequent issues===
When running the installation you may get a <code>SyntaxError</code> related to utf-8 encoding or it may complain that module <code>setuptools</code> is not installed even though it is. In this case, it means that the <code>sudo</code> command is not preserving the correct <code>PATH</code> for your python executable.

Solve the problem by running the installation step as

sudo /your/path/to/python setup.py install
or
sudo env PATH=$PATH python setup.py install

This applies only to the installation step and not to subsequent yambopy use.

== Tutorial files ==
The tutorial CORE databases can be obtained

* from the [[Yambo_Virtual_Machine|Yambo Virtual Machine]]
* from the Yambo web-page
* from the Yambo GIT tutorial repository

=== From the Yambo Virtual Machine (VM) ===
If you are using the VM, a recent version of the tutorial files is provided.Follow these [[Yambo_Virtual_Machine#Updating_the_Yambo_tutorial_files| instructions]] to update the tutorial files to the most recent version.

=== From the Yambo website ===
If you are using your own installation or the docker, the files needed to run the tutorials can be downloaded from the lists below.

After downloading the tar.gz files just unpack them in '''the YAMBO_TUTORIALS''' folder. For example
$ mkdir YAMBO_TUTORIALS
$ mv hBN.tar.gz YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ tar -xvfz hBN.tar.gz
$ ls YAMBO_TUTORIALS
hBN

====Files needed for modular tutorials====
All of the following should be downloaded prior to following the modular tutorials:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="4"| hBN || [https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz hBN.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-convergence-kpoints.tar.gz hBN-convergence-kpoints.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-para.tar.gz hBN-2D-para.tar.gz]
|-
|}

====Files needed for stand-alone tutorials====
At the start of each tutorial you will be told which specific file needs to be downloaded:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="2"| Silicon || [https://media.yambo-code.eu/educational/tutorials/files/Silicon.tar.gz Silicon.tar.gz]
|-
|[https://media.yambo-code.eu/educational/tutorials/files/Silicon_Electron-Phonon.tar.gz Silicon_Electron-Phonon.tar.gz]
|-
| LiF || [https://media.yambo-code.eu/educational/tutorials/files/LiF.tar.gz LiF.tar.gz]
|-
| Aluminum || [https://media.yambo-code.eu/educational/tutorials/files/Aluminum.tar.gz Aluminum.tar.gz]
|-
| GaSb || [https://media.yambo-code.eu/educational/tutorials/files/GaSb.tar.gz GaSb.tar.gz]
|-
| AlAs || [https://media.yambo-code.eu/educational/tutorials/files/AlAs.tar.gz AlAs.tar.gz]
|-
| Hydrogen_Chain || [https://media.yambo-code.eu/educational/tutorials/files/Hydrogen_Chain.tar.gz Hydrogen_Chain.tar.gz]
|-
| MoS2 for HPC || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_HPC_tutorial.tar.gz MoS2_HPC_tutorial.tar.gz]
|-
| MoS2 for HPC shorter version || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_2Dquasiparticle_tutorial.tar.gz MoS2_2Dquasiparticle_tutorial.tar.gz]
|-
| Yambopy for QE || [https://media.yambo-code.eu/educational/tutorials/files/databases_qepy.tar databases_qepy]
|-
| Yambopy for YAMBO || [https://media.yambo-code.eu/educational/tutorials/files/databases_yambopy.tar databases_yambopy]
|-
|}

=== From the Git Tutorial Repository (advanced users) ===
If you are using your own installation or the docker, the [https://github.com/yambo-code/tutorials tutorials repository] contains the updated tutorials CORE databases. To use it
$ git clone https://github.com/yambo-code/tutorials.git YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ ./setup.pl -install

== Stand-alone tutorials ==
These tutorials are self-contained and cover a variety of mixed topics, both physical and methodological. They are designed to be followed from start to finish in one page and do not require previous knowledge of yambo. Each tutorial requires download of a specific core database, and typically they cover a specific physical system (like bulk GaSb or a hydrogen chain). Ground state input files and pseudopotentials are provided. Output files are also provided for reference.

These tutorials can be accessed directly from this page of from the side bar. They include different kind of subjects:

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

=== Basic ===
* [[LiF|Linear Response in 3D. Excitons at work]]
* [[Silicon|GW convergence]]

=== Post Processing ===
* [[Yambo Post Processing (ypp)]]
* [[First steps in Yambopy]]
* [[Yambopy tutorial: band structures]]
* [[Yambopy tutorial: Yambo databases]]
* [[GW tutorial. Convergence and approximations (BN)]] (yambopy tutorial)
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]] (yambopy tutorial)

=== Advanced ===
* [[Hydrogen chain|TDDFT Failure and long range correlations]]
* [[Linear response from real time simulations]]

==== GW and Quasi-particles ====
* [[Real_Axis_and_Lifetimes|Real Axis and Lifetimes]]
* [[Self-consistent GW on eigenvalues only]]
* [[GW tutorial on HPC]]

==== Electron phonon coupling ====
* [[Electron Phonon Coupling|Electron Phonon Coupling]]
* [[Optical properties at finite temperature]]
* [[Phonon-assisted luminescence by finite atomic displacements]]
* [[Exciton-phonon coupling and luminescence]]

==== Non linear response ====
* [http://www.attaccalite.com/lumen/linear_response.html Linear response using Dynamical Berry phase]
* [[Real time approach to non-linear response]]
* [[Correlation effects in the non-linear response]]
* [http://www.attaccalite.com/lumen/thg_in_silicon.html Third Harmonic Generation]
* [http://www.attaccalite.com/lumen/spin_orbit.html Spin-orbit coupling and non-linear response]
* [[Two-photon absorption]]
* [[Pump and Probe]]
* [[Parallelization for non-linear response calculations]]

==== Developing Yambo ====
* [[How to create a new project in Yambo]]
* [[How to create a new ypp interface]]
* [[Some hints on github]]




== Modular tutorials ==
These tutorials are designed to provide a deeper understanding of specific yambo tasks and runlevels. They are designed to avoid repetition of common procedures and physical concepts. As such, they make use of the same physical systems: bulk hexagonal boron nitride ''hBN'' and a hBN sheet ''hBN-2D''.

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

====Introduction====
* [[First steps: a walk through from DFT to optical properties]]
====Quasiparticles in the GW approximation====
* [[How to obtain the quasi-particle band structure of a bulk material: h-BN]]
====Using Yambo in Parallel====
This modules contains very general discussions of the parallel environment of Yambo. Still the actual run of the code is specific to the CECAM cluster. If you want to run these modules just replace the parallel queue instructions with simple MPI commands.

* [[GW_parallel_strategies|Parallel GW (CECAM specific)]]: strategies for running Yambo in parallel
[[GW_parallel_strategies_CECAM]]
* [[Pushing_convergence_in_parallel|GW convergence (CECAM specific)]]: use Yambo in parallel to converge a GW calculation for a layer of hBN (hBN-2D)

====Excitons and the Bethe-Salpeter Equation====
* [[How to obtain an optical spectrum|Calculating optical spectra including excitonic effects: a step-by-step guide]]
* [[How to choose the input parameters|Obtaining a converged optical spectrum]]
* [[How to treat low dimensional systems|Many-body effects in low-dimensional systems: numerical issues and remedies]]
* [[How to analyse excitons|Analysis of excitonic spectra in a 2D material]]


====Yambopy====
* [[First steps in Yambopy]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]
* [[Yambopy tutorial: band structures | Database and plotting tutorial for quantum espresso: qepy]]
* [[Yambopy tutorial: Yambo databases | Database and plotting tutorial for yambo: yambopy ]]


=== Modules ===
Alternatively, users can learn more about a specific runlevel or task by looking at the individual '''[[Modules|documentation modules]]'''. These provide a focus on the input parameters, run time behaviour, and underlying physics. Although they can be followed separately, non-experts are urged to follow them as part of the more structured tutorials given above.

Tutorials

2023-06-15T14:32:55Z

Fulvio: /* Post Processing */

If you are starting out with Yambo, or even an experienced user, we recommend that you complete the following tutorials before trying to use Yambo for your system.

The tutorials are meant to give some introductory background to the key concepts behind Yambo. Practical topics such as convergence are also discussed.
Nonetheless, users are invited to first read and study the [[lectures|background material]] in order to get familiar with the fundamental physical quantities.

Two kinds of tutorials are provided: '''stand-alone''' and '''modular'''. In addition you must have a working environment where both Yambo (and eventually QE) are installed.

== Setting up Yambo (and eventually QE) ==
To be able to follow the school you need a running version of the yambo/QE code.

There are several different ways to prepare a working environment.

=== Virtual Machine(s) ===
The easiest way is to access to a virtual machine which contains both (i) yambo/QE and (ii) the tutorials.

You can do it in one of two ways:
* Virtual machine via [[ICTP cloud|ICTP cloud]] If the schools you are attending provided an ICTP virtual machine this is the preferred option. It works through internet connection inside a browser.
* Install the [[Yambo_Virtual_Machine|yambo virtual machine]] on your laptop / desktop. This requires Oracle virtual box. Pre-download of the Virtual machine. No internet connection needed.


=== User installation ===
You can also setup the yambo code on your on laptop / desktop using different methods.

As far as the Yambo source is concerned you can:
* Install [[Yambo via Docker|Yambo via Docker]]
* [[download|Download]] and [[Installation|install]] yambo on your laptop / desktop (requires a linux machine).
* Install yambo on your laptop/desktop/cluster [https://github.com/nicspalla/my-repo via Spack].
* Install using Anaconda.

=== Yambo User Installation with Anaconda ===
It is possible to install Yambo (up to v5.0.4) and Quantum-ESPRESSO via conda-forge (a conda channel/repository):
To setup Anaconda, please start from installing [https://www.anaconda.com/products/distribution#Downloads Anaconda] or [https://docs.conda.io/en/latest/miniconda.html Miniconda].

Then we suggest to create a conda environment and activate it:
conda create --name yambopy -c conda-forge
conda activate yambopy
Then you can install the prerequisites and the two codes:
conda install numpy scipy netcdf4 matplotlib pyyaml lxml pandas
conda install yambo
conda install qe

==Setting up Yambopy==

===Quick installation===

A quick way to start using Yambopy is described here.

* Make sure that you are using Python 3 and that you have the following python packages: <code>numpy</code>, <code>scipy</code>, <code>matplotlib</code>, <code>netCDF4</code>, <code>lxml</code>, <code>pyyaml</code>. Optionally, you may want to have abipy [[https://abinit.github.io/abipy/index.html]] installed for band structure interpolations.

* Go to a directory of your choice and clone yambopy from the git repository

git clone https://github.com/yambo-code/yambopy.git

If you don't want to use git, you may download a tarball from the git repository instead.

* Enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Installing dependencies with Anaconda===
We suggest installing yambopy using Anaconda [[https://www.anaconda.com/products/distribution]] to manage the various python packages.

In this case, you can follow these steps.

First, install the required dependencies:
conda install numpy scipy netcdf4 lxml pyyaml

Then we create a conda environment based on python 3.6 (this is to ensure compatibility with abipy if we want to install it later on):
conda create --name NAME_ENV python=3.6
Here choose <code>NAME_ENV</code> as you want, e.g. <code>yenv</code>.

Now, we install abipy and its dependency pymatgen using <code>pip</code>. Here make sure that you are using the <code>pip</code> version provided by Anaconda and not your system version.

pip install pymatgen
pip install abipy

Finally, we are ready to install yambopy:

git clone https://github.com/yambo-code/yambopy.git

(or download and extract tarball) and follow the steps outlined in the quick installation section.

Now enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Frequent issues===
When running the installation you may get a <code>SyntaxError</code> related to utf-8 encoding or it may complain that module <code>setuptools</code> is not installed even though it is. In this case, it means that the <code>sudo</code> command is not preserving the correct <code>PATH</code> for your python executable.

Solve the problem by running the installation step as

sudo /your/path/to/python setup.py install
or
sudo env PATH=$PATH python setup.py install

This applies only to the installation step and not to subsequent yambopy use.

== Tutorial files ==
The tutorial CORE databases can be obtained

* from the [[Yambo_Virtual_Machine|Yambo Virtual Machine]]
* from the Yambo web-page
* from the Yambo GIT tutorial repository

=== From the Yambo Virtual Machine (VM) ===
If you are using the VM, a recent version of the tutorial files is provided.Follow these [[Yambo_Virtual_Machine#Updating_the_Yambo_tutorial_files| instructions]] to update the tutorial files to the most recent version.

=== From the Yambo website ===
If you are using your own installation or the docker, the files needed to run the tutorials can be downloaded from the lists below.

After downloading the tar.gz files just unpack them in '''the YAMBO_TUTORIALS''' folder. For example
$ mkdir YAMBO_TUTORIALS
$ mv hBN.tar.gz YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ tar -xvfz hBN.tar.gz
$ ls YAMBO_TUTORIALS
hBN

====Files needed for modular tutorials====
All of the following should be downloaded prior to following the modular tutorials:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="4"| hBN || [https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz hBN.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-convergence-kpoints.tar.gz hBN-convergence-kpoints.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-para.tar.gz hBN-2D-para.tar.gz]
|-
|}

====Files needed for stand-alone tutorials====
At the start of each tutorial you will be told which specific file needs to be downloaded:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="2"| Silicon || [https://media.yambo-code.eu/educational/tutorials/files/Silicon.tar.gz Silicon.tar.gz]
|-
|[https://media.yambo-code.eu/educational/tutorials/files/Silicon_Electron-Phonon.tar.gz Silicon_Electron-Phonon.tar.gz]
|-
| LiF || [https://media.yambo-code.eu/educational/tutorials/files/LiF.tar.gz LiF.tar.gz]
|-
| Aluminum || [https://media.yambo-code.eu/educational/tutorials/files/Aluminum.tar.gz Aluminum.tar.gz]
|-
| GaSb || [https://media.yambo-code.eu/educational/tutorials/files/GaSb.tar.gz GaSb.tar.gz]
|-
| AlAs || [https://media.yambo-code.eu/educational/tutorials/files/AlAs.tar.gz AlAs.tar.gz]
|-
| Hydrogen_Chain || [https://media.yambo-code.eu/educational/tutorials/files/Hydrogen_Chain.tar.gz Hydrogen_Chain.tar.gz]
|-
| MoS2 for HPC || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_HPC_tutorial.tar.gz MoS2_HPC_tutorial.tar.gz]
|-
| MoS2 for HPC shorter version || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_2Dquasiparticle_tutorial.tar.gz MoS2_2Dquasiparticle_tutorial.tar.gz]
|-
| Yambopy for QE || [https://media.yambo-code.eu/educational/tutorials/files/databases_qepy.tar databases_qepy]
|-
| Yambopy for YAMBO || [https://media.yambo-code.eu/educational/tutorials/files/databases_yambopy.tar databases_yambopy]
|-
|}

=== From the Git Tutorial Repository (advanced users) ===
If you are using your own installation or the docker, the [https://github.com/yambo-code/tutorials tutorials repository] contains the updated tutorials CORE databases. To use it
$ git clone https://github.com/yambo-code/tutorials.git YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ ./setup.pl -install

== Stand-alone tutorials ==
These tutorials are self-contained and cover a variety of mixed topics, both physical and methodological. They are designed to be followed from start to finish in one page and do not require previous knowledge of yambo. Each tutorial requires download of a specific core database, and typically they cover a specific physical system (like bulk GaSb or a hydrogen chain). Ground state input files and pseudopotentials are provided. Output files are also provided for reference.

These tutorials can be accessed directly from this page of from the side bar. They include different kind of subjects:

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

=== Basic ===
* [[LiF|Linear Response in 3D. Excitons at work]]
* [[Silicon|GW convergence]]

=== Post Processing ===
* [[Yambo Post Processing (ypp)]]
* [[First steps in Yambopy]]
* [[Yambopy tutorial: band structures]]
* [[Yambopy tutorial: Yambo databases]]
* [[Yambopy][GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]

=== Advanced ===
* [[Hydrogen chain|TDDFT Failure and long range correlations]]
* [[Linear response from real time simulations]]

==== GW and Quasi-particles ====
* [[Real_Axis_and_Lifetimes|Real Axis and Lifetimes]]
* [[Self-consistent GW on eigenvalues only]]
* [[GW tutorial on HPC]]

==== Electron phonon coupling ====
* [[Electron Phonon Coupling|Electron Phonon Coupling]]
* [[Optical properties at finite temperature]]
* [[Phonon-assisted luminescence by finite atomic displacements]]
* [[Exciton-phonon coupling and luminescence]]

==== Non linear response ====
* [http://www.attaccalite.com/lumen/linear_response.html Linear response using Dynamical Berry phase]
* [[Real time approach to non-linear response]]
* [[Correlation effects in the non-linear response]]
* [http://www.attaccalite.com/lumen/thg_in_silicon.html Third Harmonic Generation]
* [http://www.attaccalite.com/lumen/spin_orbit.html Spin-orbit coupling and non-linear response]
* [[Two-photon absorption]]
* [[Pump and Probe]]
* [[Parallelization for non-linear response calculations]]

==== Developing Yambo ====
* [[How to create a new project in Yambo]]
* [[How to create a new ypp interface]]
* [[Some hints on github]]




== Modular tutorials ==
These tutorials are designed to provide a deeper understanding of specific yambo tasks and runlevels. They are designed to avoid repetition of common procedures and physical concepts. As such, they make use of the same physical systems: bulk hexagonal boron nitride ''hBN'' and a hBN sheet ''hBN-2D''.

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

====Introduction====
* [[First steps: a walk through from DFT to optical properties]]
====Quasiparticles in the GW approximation====
* [[How to obtain the quasi-particle band structure of a bulk material: h-BN]]
====Using Yambo in Parallel====
This modules contains very general discussions of the parallel environment of Yambo. Still the actual run of the code is specific to the CECAM cluster. If you want to run these modules just replace the parallel queue instructions with simple MPI commands.

* [[GW_parallel_strategies|Parallel GW (CECAM specific)]]: strategies for running Yambo in parallel
[[GW_parallel_strategies_CECAM]]
* [[Pushing_convergence_in_parallel|GW convergence (CECAM specific)]]: use Yambo in parallel to converge a GW calculation for a layer of hBN (hBN-2D)

====Excitons and the Bethe-Salpeter Equation====
* [[How to obtain an optical spectrum|Calculating optical spectra including excitonic effects: a step-by-step guide]]
* [[How to choose the input parameters|Obtaining a converged optical spectrum]]
* [[How to treat low dimensional systems|Many-body effects in low-dimensional systems: numerical issues and remedies]]
* [[How to analyse excitons|Analysis of excitonic spectra in a 2D material]]


====Yambopy====
* [[First steps in Yambopy]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]
* [[Yambopy tutorial: band structures | Database and plotting tutorial for quantum espresso: qepy]]
* [[Yambopy tutorial: Yambo databases | Database and plotting tutorial for yambo: yambopy ]]


=== Modules ===
Alternatively, users can learn more about a specific runlevel or task by looking at the individual '''[[Modules|documentation modules]]'''. These provide a focus on the input parameters, run time behaviour, and underlying physics. Although they can be followed separately, non-experts are urged to follow them as part of the more structured tutorials given above.

Tutorials

2023-06-15T14:24:43Z

Fulvio: /* Post Processing */

If you are starting out with Yambo, or even an experienced user, we recommend that you complete the following tutorials before trying to use Yambo for your system.

The tutorials are meant to give some introductory background to the key concepts behind Yambo. Practical topics such as convergence are also discussed.
Nonetheless, users are invited to first read and study the [[lectures|background material]] in order to get familiar with the fundamental physical quantities.

Two kinds of tutorials are provided: '''stand-alone''' and '''modular'''. In addition you must have a working environment where both Yambo (and eventually QE) are installed.

== Setting up Yambo (and eventually QE) ==
To be able to follow the school you need a running version of the yambo/QE code.

There are several different ways to prepare a working environment.

=== Virtual Machine(s) ===
The easiest way is to access to a virtual machine which contains both (i) yambo/QE and (ii) the tutorials.

You can do it in one of two ways:
* Virtual machine via [[ICTP cloud|ICTP cloud]] If the schools you are attending provided an ICTP virtual machine this is the preferred option. It works through internet connection inside a browser.
* Install the [[Yambo_Virtual_Machine|yambo virtual machine]] on your laptop / desktop. This requires Oracle virtual box. Pre-download of the Virtual machine. No internet connection needed.


=== User installation ===
You can also setup the yambo code on your on laptop / desktop using different methods.

As far as the Yambo source is concerned you can:
* Install [[Yambo via Docker|Yambo via Docker]]
* [[download|Download]] and [[Installation|install]] yambo on your laptop / desktop (requires a linux machine).
* Install yambo on your laptop/desktop/cluster [https://github.com/nicspalla/my-repo via Spack].
* Install using Anaconda.

=== Yambo User Installation with Anaconda ===
It is possible to install Yambo (up to v5.0.4) and Quantum-ESPRESSO via conda-forge (a conda channel/repository):
To setup Anaconda, please start from installing [https://www.anaconda.com/products/distribution#Downloads Anaconda] or [https://docs.conda.io/en/latest/miniconda.html Miniconda].

Then we suggest to create a conda environment and activate it:
conda create --name yambopy -c conda-forge
conda activate yambopy
Then you can install the prerequisites and the two codes:
conda install numpy scipy netcdf4 matplotlib pyyaml lxml pandas
conda install yambo
conda install qe

==Setting up Yambopy==

===Quick installation===

A quick way to start using Yambopy is described here.

* Make sure that you are using Python 3 and that you have the following python packages: <code>numpy</code>, <code>scipy</code>, <code>matplotlib</code>, <code>netCDF4</code>, <code>lxml</code>, <code>pyyaml</code>. Optionally, you may want to have abipy [[https://abinit.github.io/abipy/index.html]] installed for band structure interpolations.

* Go to a directory of your choice and clone yambopy from the git repository

git clone https://github.com/yambo-code/yambopy.git

If you don't want to use git, you may download a tarball from the git repository instead.

* Enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Installing dependencies with Anaconda===
We suggest installing yambopy using Anaconda [[https://www.anaconda.com/products/distribution]] to manage the various python packages.

In this case, you can follow these steps.

First, install the required dependencies:
conda install numpy scipy netcdf4 lxml pyyaml

Then we create a conda environment based on python 3.6 (this is to ensure compatibility with abipy if we want to install it later on):
conda create --name NAME_ENV python=3.6
Here choose <code>NAME_ENV</code> as you want, e.g. <code>yenv</code>.

Now, we install abipy and its dependency pymatgen using <code>pip</code>. Here make sure that you are using the <code>pip</code> version provided by Anaconda and not your system version.

pip install pymatgen
pip install abipy

Finally, we are ready to install yambopy:

git clone https://github.com/yambo-code/yambopy.git

(or download and extract tarball) and follow the steps outlined in the quick installation section.

Now enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Frequent issues===
When running the installation you may get a <code>SyntaxError</code> related to utf-8 encoding or it may complain that module <code>setuptools</code> is not installed even though it is. In this case, it means that the <code>sudo</code> command is not preserving the correct <code>PATH</code> for your python executable.

Solve the problem by running the installation step as

sudo /your/path/to/python setup.py install
or
sudo env PATH=$PATH python setup.py install

This applies only to the installation step and not to subsequent yambopy use.

== Tutorial files ==
The tutorial CORE databases can be obtained

* from the [[Yambo_Virtual_Machine|Yambo Virtual Machine]]
* from the Yambo web-page
* from the Yambo GIT tutorial repository

=== From the Yambo Virtual Machine (VM) ===
If you are using the VM, a recent version of the tutorial files is provided.Follow these [[Yambo_Virtual_Machine#Updating_the_Yambo_tutorial_files| instructions]] to update the tutorial files to the most recent version.

=== From the Yambo website ===
If you are using your own installation or the docker, the files needed to run the tutorials can be downloaded from the lists below.

After downloading the tar.gz files just unpack them in '''the YAMBO_TUTORIALS''' folder. For example
$ mkdir YAMBO_TUTORIALS
$ mv hBN.tar.gz YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ tar -xvfz hBN.tar.gz
$ ls YAMBO_TUTORIALS
hBN

====Files needed for modular tutorials====
All of the following should be downloaded prior to following the modular tutorials:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="4"| hBN || [https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz hBN.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-convergence-kpoints.tar.gz hBN-convergence-kpoints.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-para.tar.gz hBN-2D-para.tar.gz]
|-
|}

====Files needed for stand-alone tutorials====
At the start of each tutorial you will be told which specific file needs to be downloaded:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="2"| Silicon || [https://media.yambo-code.eu/educational/tutorials/files/Silicon.tar.gz Silicon.tar.gz]
|-
|[https://media.yambo-code.eu/educational/tutorials/files/Silicon_Electron-Phonon.tar.gz Silicon_Electron-Phonon.tar.gz]
|-
| LiF || [https://media.yambo-code.eu/educational/tutorials/files/LiF.tar.gz LiF.tar.gz]
|-
| Aluminum || [https://media.yambo-code.eu/educational/tutorials/files/Aluminum.tar.gz Aluminum.tar.gz]
|-
| GaSb || [https://media.yambo-code.eu/educational/tutorials/files/GaSb.tar.gz GaSb.tar.gz]
|-
| AlAs || [https://media.yambo-code.eu/educational/tutorials/files/AlAs.tar.gz AlAs.tar.gz]
|-
| Hydrogen_Chain || [https://media.yambo-code.eu/educational/tutorials/files/Hydrogen_Chain.tar.gz Hydrogen_Chain.tar.gz]
|-
| MoS2 for HPC || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_HPC_tutorial.tar.gz MoS2_HPC_tutorial.tar.gz]
|-
| MoS2 for HPC shorter version || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_2Dquasiparticle_tutorial.tar.gz MoS2_2Dquasiparticle_tutorial.tar.gz]
|-
| Yambopy for QE || [https://media.yambo-code.eu/educational/tutorials/files/databases_qepy.tar databases_qepy]
|-
| Yambopy for YAMBO || [https://media.yambo-code.eu/educational/tutorials/files/databases_yambopy.tar databases_yambopy]
|-
|}

=== From the Git Tutorial Repository (advanced users) ===
If you are using your own installation or the docker, the [https://github.com/yambo-code/tutorials tutorials repository] contains the updated tutorials CORE databases. To use it
$ git clone https://github.com/yambo-code/tutorials.git YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ ./setup.pl -install

== Stand-alone tutorials ==
These tutorials are self-contained and cover a variety of mixed topics, both physical and methodological. They are designed to be followed from start to finish in one page and do not require previous knowledge of yambo. Each tutorial requires download of a specific core database, and typically they cover a specific physical system (like bulk GaSb or a hydrogen chain). Ground state input files and pseudopotentials are provided. Output files are also provided for reference.

These tutorials can be accessed directly from this page of from the side bar. They include different kind of subjects:

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

=== Basic ===
* [[LiF|Linear Response in 3D. Excitons at work]]
* [[Silicon|GW convergence]]

=== Post Processing ===
* [[Yambo Post Processing (ypp)]]
* [[First steps in Yambopy]]
* [[Yambopy tutorial: band structures]]
* [[Yambopy tutorial: Yambo databases]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]

=== Advanced ===
* [[Hydrogen chain|TDDFT Failure and long range correlations]]
* [[Linear response from real time simulations]]

==== GW and Quasi-particles ====
* [[Real_Axis_and_Lifetimes|Real Axis and Lifetimes]]
* [[Self-consistent GW on eigenvalues only]]
* [[GW tutorial on HPC]]

==== Electron phonon coupling ====
* [[Electron Phonon Coupling|Electron Phonon Coupling]]
* [[Optical properties at finite temperature]]
* [[Phonon-assisted luminescence by finite atomic displacements]]
* [[Exciton-phonon coupling and luminescence]]

==== Non linear response ====
* [http://www.attaccalite.com/lumen/linear_response.html Linear response using Dynamical Berry phase]
* [[Real time approach to non-linear response]]
* [[Correlation effects in the non-linear response]]
* [http://www.attaccalite.com/lumen/thg_in_silicon.html Third Harmonic Generation]
* [http://www.attaccalite.com/lumen/spin_orbit.html Spin-orbit coupling and non-linear response]
* [[Two-photon absorption]]
* [[Pump and Probe]]
* [[Parallelization for non-linear response calculations]]

==== Developing Yambo ====
* [[How to create a new project in Yambo]]
* [[How to create a new ypp interface]]
* [[Some hints on github]]




== Modular tutorials ==
These tutorials are designed to provide a deeper understanding of specific yambo tasks and runlevels. They are designed to avoid repetition of common procedures and physical concepts. As such, they make use of the same physical systems: bulk hexagonal boron nitride ''hBN'' and a hBN sheet ''hBN-2D''.

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

====Introduction====
* [[First steps: a walk through from DFT to optical properties]]
====Quasiparticles in the GW approximation====
* [[How to obtain the quasi-particle band structure of a bulk material: h-BN]]
====Using Yambo in Parallel====
This modules contains very general discussions of the parallel environment of Yambo. Still the actual run of the code is specific to the CECAM cluster. If you want to run these modules just replace the parallel queue instructions with simple MPI commands.

* [[GW_parallel_strategies|Parallel GW (CECAM specific)]]: strategies for running Yambo in parallel
[[GW_parallel_strategies_CECAM]]
* [[Pushing_convergence_in_parallel|GW convergence (CECAM specific)]]: use Yambo in parallel to converge a GW calculation for a layer of hBN (hBN-2D)

====Excitons and the Bethe-Salpeter Equation====
* [[How to obtain an optical spectrum|Calculating optical spectra including excitonic effects: a step-by-step guide]]
* [[How to choose the input parameters|Obtaining a converged optical spectrum]]
* [[How to treat low dimensional systems|Many-body effects in low-dimensional systems: numerical issues and remedies]]
* [[How to analyse excitons|Analysis of excitonic spectra in a 2D material]]


====Yambopy====
* [[First steps in Yambopy]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]
* [[Yambopy tutorial: band structures | Database and plotting tutorial for quantum espresso: qepy]]
* [[Yambopy tutorial: Yambo databases | Database and plotting tutorial for yambo: yambopy ]]


=== Modules ===
Alternatively, users can learn more about a specific runlevel or task by looking at the individual '''[[Modules|documentation modules]]'''. These provide a focus on the input parameters, run time behaviour, and underlying physics. Although they can be followed separately, non-experts are urged to follow them as part of the more structured tutorials given above.

Tutorials

2023-06-15T14:24:07Z

Fulvio: /* Post Processing */

If you are starting out with Yambo, or even an experienced user, we recommend that you complete the following tutorials before trying to use Yambo for your system.

The tutorials are meant to give some introductory background to the key concepts behind Yambo. Practical topics such as convergence are also discussed.
Nonetheless, users are invited to first read and study the [[lectures|background material]] in order to get familiar with the fundamental physical quantities.

Two kinds of tutorials are provided: '''stand-alone''' and '''modular'''. In addition you must have a working environment where both Yambo (and eventually QE) are installed.

== Setting up Yambo (and eventually QE) ==
To be able to follow the school you need a running version of the yambo/QE code.

There are several different ways to prepare a working environment.

=== Virtual Machine(s) ===
The easiest way is to access to a virtual machine which contains both (i) yambo/QE and (ii) the tutorials.

You can do it in one of two ways:
* Virtual machine via [[ICTP cloud|ICTP cloud]] If the schools you are attending provided an ICTP virtual machine this is the preferred option. It works through internet connection inside a browser.
* Install the [[Yambo_Virtual_Machine|yambo virtual machine]] on your laptop / desktop. This requires Oracle virtual box. Pre-download of the Virtual machine. No internet connection needed.


=== User installation ===
You can also setup the yambo code on your on laptop / desktop using different methods.

As far as the Yambo source is concerned you can:
* Install [[Yambo via Docker|Yambo via Docker]]
* [[download|Download]] and [[Installation|install]] yambo on your laptop / desktop (requires a linux machine).
* Install yambo on your laptop/desktop/cluster [https://github.com/nicspalla/my-repo via Spack].
* Install using Anaconda.

=== Yambo User Installation with Anaconda ===
It is possible to install Yambo (up to v5.0.4) and Quantum-ESPRESSO via conda-forge (a conda channel/repository):
To setup Anaconda, please start from installing [https://www.anaconda.com/products/distribution#Downloads Anaconda] or [https://docs.conda.io/en/latest/miniconda.html Miniconda].

Then we suggest to create a conda environment and activate it:
conda create --name yambopy -c conda-forge
conda activate yambopy
Then you can install the prerequisites and the two codes:
conda install numpy scipy netcdf4 matplotlib pyyaml lxml pandas
conda install yambo
conda install qe

==Setting up Yambopy==

===Quick installation===

A quick way to start using Yambopy is described here.

* Make sure that you are using Python 3 and that you have the following python packages: <code>numpy</code>, <code>scipy</code>, <code>matplotlib</code>, <code>netCDF4</code>, <code>lxml</code>, <code>pyyaml</code>. Optionally, you may want to have abipy [[https://abinit.github.io/abipy/index.html]] installed for band structure interpolations.

* Go to a directory of your choice and clone yambopy from the git repository

git clone https://github.com/yambo-code/yambopy.git

If you don't want to use git, you may download a tarball from the git repository instead.

* Enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Installing dependencies with Anaconda===
We suggest installing yambopy using Anaconda [[https://www.anaconda.com/products/distribution]] to manage the various python packages.

In this case, you can follow these steps.

First, install the required dependencies:
conda install numpy scipy netcdf4 lxml pyyaml

Then we create a conda environment based on python 3.6 (this is to ensure compatibility with abipy if we want to install it later on):
conda create --name NAME_ENV python=3.6
Here choose <code>NAME_ENV</code> as you want, e.g. <code>yenv</code>.

Now, we install abipy and its dependency pymatgen using <code>pip</code>. Here make sure that you are using the <code>pip</code> version provided by Anaconda and not your system version.

pip install pymatgen
pip install abipy

Finally, we are ready to install yambopy:

git clone https://github.com/yambo-code/yambopy.git

(or download and extract tarball) and follow the steps outlined in the quick installation section.

Now enter into the yambopy folder and install

cd yambopy
sudo python setup.py install

If you don't have administrative privileges (for example on a computing cluster), type instead

cd yambopy
python setup.py install --user

===Frequent issues===
When running the installation you may get a <code>SyntaxError</code> related to utf-8 encoding or it may complain that module <code>setuptools</code> is not installed even though it is. In this case, it means that the <code>sudo</code> command is not preserving the correct <code>PATH</code> for your python executable.

Solve the problem by running the installation step as

sudo /your/path/to/python setup.py install
or
sudo env PATH=$PATH python setup.py install

This applies only to the installation step and not to subsequent yambopy use.

== Tutorial files ==
The tutorial CORE databases can be obtained

* from the [[Yambo_Virtual_Machine|Yambo Virtual Machine]]
* from the Yambo web-page
* from the Yambo GIT tutorial repository

=== From the Yambo Virtual Machine (VM) ===
If you are using the VM, a recent version of the tutorial files is provided.Follow these [[Yambo_Virtual_Machine#Updating_the_Yambo_tutorial_files| instructions]] to update the tutorial files to the most recent version.

=== From the Yambo website ===
If you are using your own installation or the docker, the files needed to run the tutorials can be downloaded from the lists below.

After downloading the tar.gz files just unpack them in '''the YAMBO_TUTORIALS''' folder. For example
$ mkdir YAMBO_TUTORIALS
$ mv hBN.tar.gz YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ tar -xvfz hBN.tar.gz
$ ls YAMBO_TUTORIALS
hBN

====Files needed for modular tutorials====
All of the following should be downloaded prior to following the modular tutorials:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="4"| hBN || [https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz hBN.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-convergence-kpoints.tar.gz hBN-convergence-kpoints.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz]
|-
| [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-para.tar.gz hBN-2D-para.tar.gz]
|-
|}

====Files needed for stand-alone tutorials====
At the start of each tutorial you will be told which specific file needs to be downloaded:

{| class="wikitable"
|-
! Tutorial !! File(s)
|-
| rowspan="2"| Silicon || [https://media.yambo-code.eu/educational/tutorials/files/Silicon.tar.gz Silicon.tar.gz]
|-
|[https://media.yambo-code.eu/educational/tutorials/files/Silicon_Electron-Phonon.tar.gz Silicon_Electron-Phonon.tar.gz]
|-
| LiF || [https://media.yambo-code.eu/educational/tutorials/files/LiF.tar.gz LiF.tar.gz]
|-
| Aluminum || [https://media.yambo-code.eu/educational/tutorials/files/Aluminum.tar.gz Aluminum.tar.gz]
|-
| GaSb || [https://media.yambo-code.eu/educational/tutorials/files/GaSb.tar.gz GaSb.tar.gz]
|-
| AlAs || [https://media.yambo-code.eu/educational/tutorials/files/AlAs.tar.gz AlAs.tar.gz]
|-
| Hydrogen_Chain || [https://media.yambo-code.eu/educational/tutorials/files/Hydrogen_Chain.tar.gz Hydrogen_Chain.tar.gz]
|-
| MoS2 for HPC || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_HPC_tutorial.tar.gz MoS2_HPC_tutorial.tar.gz]
|-
| MoS2 for HPC shorter version || [https://media.yambo-code.eu/educational/tutorials/files/MoS2_2Dquasiparticle_tutorial.tar.gz MoS2_2Dquasiparticle_tutorial.tar.gz]
|-
| Yambopy for QE || [https://media.yambo-code.eu/educational/tutorials/files/databases_qepy.tar databases_qepy]
|-
| Yambopy for YAMBO || [https://media.yambo-code.eu/educational/tutorials/files/databases_yambopy.tar databases_yambopy]
|-
|}

=== From the Git Tutorial Repository (advanced users) ===
If you are using your own installation or the docker, the [https://github.com/yambo-code/tutorials tutorials repository] contains the updated tutorials CORE databases. To use it
$ git clone https://github.com/yambo-code/tutorials.git YAMBO_TUTORIALS
$ cd YAMBO_TUTORIALS
$ ./setup.pl -install

== Stand-alone tutorials ==
These tutorials are self-contained and cover a variety of mixed topics, both physical and methodological. They are designed to be followed from start to finish in one page and do not require previous knowledge of yambo. Each tutorial requires download of a specific core database, and typically they cover a specific physical system (like bulk GaSb or a hydrogen chain). Ground state input files and pseudopotentials are provided. Output files are also provided for reference.

These tutorials can be accessed directly from this page of from the side bar. They include different kind of subjects:

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

=== Basic ===
* [[LiF|Linear Response in 3D. Excitons at work]]
* [[Silicon|GW convergence]]

=== Post Processing ===
* [[Yambo Post Processing (ypp)]]
* [[First Steps in YamboPy]]
* [[Yambopy tutorial: band structures]]
* [[Yambopy tutorial: Yambo databases]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]

=== Advanced ===
* [[Hydrogen chain|TDDFT Failure and long range correlations]]
* [[Linear response from real time simulations]]

==== GW and Quasi-particles ====
* [[Real_Axis_and_Lifetimes|Real Axis and Lifetimes]]
* [[Self-consistent GW on eigenvalues only]]
* [[GW tutorial on HPC]]

==== Electron phonon coupling ====
* [[Electron Phonon Coupling|Electron Phonon Coupling]]
* [[Optical properties at finite temperature]]
* [[Phonon-assisted luminescence by finite atomic displacements]]
* [[Exciton-phonon coupling and luminescence]]

==== Non linear response ====
* [http://www.attaccalite.com/lumen/linear_response.html Linear response using Dynamical Berry phase]
* [[Real time approach to non-linear response]]
* [[Correlation effects in the non-linear response]]
* [http://www.attaccalite.com/lumen/thg_in_silicon.html Third Harmonic Generation]
* [http://www.attaccalite.com/lumen/spin_orbit.html Spin-orbit coupling and non-linear response]
* [[Two-photon absorption]]
* [[Pump and Probe]]
* [[Parallelization for non-linear response calculations]]

==== Developing Yambo ====
* [[How to create a new project in Yambo]]
* [[How to create a new ypp interface]]
* [[Some hints on github]]




== Modular tutorials ==
These tutorials are designed to provide a deeper understanding of specific yambo tasks and runlevels. They are designed to avoid repetition of common procedures and physical concepts. As such, they make use of the same physical systems: bulk hexagonal boron nitride ''hBN'' and a hBN sheet ''hBN-2D''.

'''Warning''': These tutorials were prepared using previous version of the Yambo code: some command lines, variables, reports and outputs can be slightly different from the last version of the code. Scripts for parsing output cannot work anymore and should be edited to work with the new outputs. New command lines can be accessed typing <code>yambo -h </code>

====Introduction====
* [[First steps: a walk through from DFT to optical properties]]
====Quasiparticles in the GW approximation====
* [[How to obtain the quasi-particle band structure of a bulk material: h-BN]]
====Using Yambo in Parallel====
This modules contains very general discussions of the parallel environment of Yambo. Still the actual run of the code is specific to the CECAM cluster. If you want to run these modules just replace the parallel queue instructions with simple MPI commands.

* [[GW_parallel_strategies|Parallel GW (CECAM specific)]]: strategies for running Yambo in parallel
[[GW_parallel_strategies_CECAM]]
* [[Pushing_convergence_in_parallel|GW convergence (CECAM specific)]]: use Yambo in parallel to converge a GW calculation for a layer of hBN (hBN-2D)

====Excitons and the Bethe-Salpeter Equation====
* [[How to obtain an optical spectrum|Calculating optical spectra including excitonic effects: a step-by-step guide]]
* [[How to choose the input parameters|Obtaining a converged optical spectrum]]
* [[How to treat low dimensional systems|Many-body effects in low-dimensional systems: numerical issues and remedies]]
* [[How to analyse excitons|Analysis of excitonic spectra in a 2D material]]


====Yambopy====
* [[First steps in Yambopy]]
* [[GW tutorial. Convergence and approximations (BN)]]
* [[Bethe-Salpeter equation tutorial. Optical absorption (BN)]]
* [[Yambopy tutorial: band structures | Database and plotting tutorial for quantum espresso: qepy]]
* [[Yambopy tutorial: Yambo databases | Database and plotting tutorial for yambo: yambopy ]]


=== Modules ===
Alternatively, users can learn more about a specific runlevel or task by looking at the individual '''[[Modules|documentation modules]]'''. These provide a focus on the input parameters, run time behaviour, and underlying physics. Although they can be followed separately, non-experts are urged to follow them as part of the more structured tutorials given above.

Second-harmonic generation of 2D-hBN

2023-05-25T14:23:14Z

Fulvio: /* Run the simulation */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample 2N+1 times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Phys. Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =
You should be in the folder <code>YAMBO_TUTORIALS/hBN-2D/YAMBO/</code>.
In this example, we will consider a single layer of hexagonal boron nitride (hBN).
Execute the commands (remember to ad -nompi if you run on marconi100)
yambo_nl
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
cd FixSymm
yambo_nl
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]
If you instead want to track the progress of the full simulation, you should check the log file(s). If you are doing a parallel run, you can type
tail -f 01_SHG_ip_files/LOG/l-01_SHG_ip_nloptics_CPU_1

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
Xorder= 4 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F 02_SHG_ip_pp.in -J 01_SHG_ip -C 01_SHG_ip_files

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
HARRLvcs= 1000 mHa # [HA] Hartree RL components
EXXRLvcs= 1000 mHa # [XX] Exchange RL components

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequencies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from the current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

Also in this case, we note differences between the two simulations. As discussed for the independent particle approximation, this is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are under converged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, to eliminate spurious interaction with the periodic images, one should use the Coulomb cut-off as it was used for this system in the [[Quasi-particles of a 2D system| tutorial on quasiparticle calculations]]. As a reference, a converged calculation for this system is in <ref name="Gruning2014">M. Grüning and C. Attaccalite, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081102 Phys. Rev. B '''89''', 081102]</ref>.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T13:31:44Z

Fulvio: /* Output post-processing: the dielectric function */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample 2N+1 times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Phys. Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =
You should be in the folder <code>YAMBO_TUTORIALS/hBN-2D/YAMBO/</code>.
In this example, we will consider a single layer of hexagonal boron nitride (hBN).
Execute the commands (remember to ad -nompi if you run on marconi100)
yambo_nl
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
cd FixSymm
yambo_nl
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]
If you instead want to track the progress of the full simulation, you should check the log file(s). If you are doing a parallel run, you can type
tail -f 01_SHG_ip_files/LOG/l-01_SHG_ip_nloptics_CPU_1

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
Xorder= 4 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F 02_SHG_ip_pp.in -J 01_SHG_ip -C 01_SHG_ip_files

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequencies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from the current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

Also in this case, we note differences between the two simulations. As discussed for the independent particle approximation, this is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are under converged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, to eliminate spurious interaction with the periodic images, one should use the Coulomb cut-off as it was used for this system in the [[Quasi-particles of a 2D system| tutorial on quasiparticle calculations]]. As a reference, a converged calculation for this system is in <ref name="Gruning2014">M. Grüning and C. Attaccalite, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081102 Phys. Rev. B '''89''', 081102]</ref>.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T13:23:43Z

Fulvio: /* Output post-processing: the dielectric function */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample 2N+1 times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Phys. Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =
You should be in the folder <code>YAMBO_TUTORIALS/hBN-2D/YAMBO/</code>.
In this example, we will consider a single layer of hexagonal boron nitride (hBN).
Execute the commands (remember to ad -nompi if you run on marconi100)
yambo_nl
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
cd FixSymm
yambo_nl
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]
If you instead want to track the progress of the full simulation, you should check the log file(s). If you are doing a parallel run, you can type
tail -f 01_SHG_ip_files/LOG/l-01_SHG_ip_nloptics_CPU_1

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
Xorder= 4 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F 02_SHG_ip_pp.in -J 01_SHG_ip -C 01_SHG_ip_files.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequencies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from the current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

Also in this case, we note differences between the two simulations. As discussed for the independent particle approximation, this is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are under converged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, to eliminate spurious interaction with the periodic images, one should use the Coulomb cut-off as it was used for this system in the [[Quasi-particles of a 2D system| tutorial on quasiparticle calculations]]. As a reference, a converged calculation for this system is in <ref name="Gruning2014">M. Grüning and C. Attaccalite, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081102 Phys. Rev. B '''89''', 081102]</ref>.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T13:21:52Z

Fulvio: /* Output post-processing: the dielectric function */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample 2N+1 times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Phys. Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =
You should be in the folder <code>YAMBO_TUTORIALS/hBN-2D/YAMBO/</code>.
In this example, we will consider a single layer of hexagonal boron nitride (hBN).
Execute the commands (remember to ad -nompi if you run on marconi100)
yambo_nl
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
cd FixSymm
yambo_nl
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]
If you instead want to track the progress of the full simulation, you should check the log file(s). If you are doing a parallel run, you can type
tail -f 01_SHG_ip_files/LOG/l-01_SHG_ip_nloptics_CPU_1

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
Xorder= 4 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequencies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from the current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

Also in this case, we note differences between the two simulations. As discussed for the independent particle approximation, this is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are under converged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, to eliminate spurious interaction with the periodic images, one should use the Coulomb cut-off as it was used for this system in the [[Quasi-particles of a 2D system| tutorial on quasiparticle calculations]]. As a reference, a converged calculation for this system is in <ref name="Gruning2014">M. Grüning and C. Attaccalite, [https://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.081102 Phys. Rev. B '''89''', 081102]</ref>.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T09:41:18Z

Fulvio: /* Step 2: Independent-particle approximation */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]
If you instead want to track the progress of the full simulation, you should check the log file(s). If you are doing a parallel run, you can type
tail -f 01_SHG_ip_files/LOG/l-01_SHG_ip_nloptics_CPU_1

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T09:05:54Z

Fulvio: /* Step 2: Independent-particle approximation */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]
If you instead want to track the progress of the full simulation, you should check the log file(s). If you are doing a parallel run, you can type
tail -f 01_SHG_ip_files/LOG/l-01_SHG_ip_nloptics_CPU_1

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T09:03:48Z

Fulvio: /* Step 2: Independent-particle approximation */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check the progress of one specific frequency:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F2
If you want to know how many fs the simulation will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T08:55:40Z

Fulvio: /* Step 2: Independent-particle approximation */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

'''[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:'''
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
'''Otherwise, you can submit the job with a slurm script as explained in the introductory page]'''

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

This run will take several minutes. While it is running, you can check one of the outputs to track completion:
tail -f 01_SHG_ip_files/o-01_SHG_ip.polarization_F1
If you want to know how many fs the run will last (since this was decided by the code), you can get this information in the report file:
$ grep Total 01_SHG_ip_files/r-01_SHG_ip_nloptics
Total simulation time : 41.23193 [fs]

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T08:51:48Z

Fulvio: /* Step 1: Prerequisites */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz hBN-2D.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
Otherwise, you can submit the job with a slurm script as explained in the introductory page]

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Rome 2023

2023-05-25T08:51:19Z

Fulvio: /* DAY 4 - Thursday, May 25 */

A general description of the goal(s) of the school can be found on the [https://www.yambo-code.eu/2023/02/18/yambo-school-2023/ Yambo main website]

== Use CINECA computational resources ==
Yambo tutorials will be run on the MARCONI100 (M100) accelerated cluster. You can find info about M100 [https://wiki.u-gov.it/confluence/display/SCAIUS/UG3.2%3A+MARCONI100+UserGuide#UG3.2:MARCONI100UserGuide here].
In order to access computational resources provided by CINECA you need your personal username and password that were sent you by the organizers.

=== Connect to the cluster using ssh ===

You can access M100 via <code>ssh</code> protocol in different ways.

''' - Connect using username and password '''

Use the following command replacing your username:
$ ssh username@login.m100.cineca.it

However, in this way you have to type your password each time you want to connect.

''' - Connect using ssh key '''

You can setup a ssh key pair to avoid typing the password each time you want to connect to M100. To do so, go to your <code>.ssh</code> directory (usually located in the <code>home</code> directory):
$ cd $HOME/.ssh

If you don't have this directory, you can create it with <code>mkdir $HOME/.ssh</code>.

Once you are in the <code>.ssh</code> directory, run the <code>ssh-keygen</code> command to generate a private/public key pair:
$ ssh-keygen
Generating public/private rsa key pair.
Enter file in which to save the key: m100_id_rsa
Enter passphrase (empty for no passphrase):
Enter same passphrase again:
Your identification has been saved in <your_.ssh_dir>/m100_id_rsa
Your public key has been saved in <your_.ssh_dir>/m100_id_rsa.pub
The key fingerprint is:
<...>
The key's randomart image is:
<...>

Now you need to copy the '''public''' key to M100. You can do that with the following command (for this step you need to type your password):
$ ssh-copy-id -i <your_.ssh_dir>/m100_id_rsa.pub <username>@login.m100.cineca.it

Once the public key has been copied, you can connect to M100 without having to type the password using the <code>-i</code> option:
$ ssh -i <your_.ssh_dir>/m100_id_rsa username@login.m100.cineca.it

To simplify even more, you can paste the following lines in a file named <code>config</code> located inside the <code>.ssh</code> directory adjusting username and path:
Host m100
HostName login.m100.cineca.it
User username
IdentityFile <your_.ssh_dir>/m100_id_rsa

With the <code>config</code> file setup you can connect simply with
$ ssh m100

=== General instructions to run tutorials ===

Before proceeding, it is useful to know the different workspaces you have available on M100, which can be accessed using environment variables. The main ones are:
* <code>$HOME</code>: it's the <code>home</code> directory associated to your username;
* <code>$WORK</code>: it's the <code>work</code> directory associated to the account where the computational resources dedicated to this school are allocated;
* <code>$CINECA_SCRATCH</code>: it's the <code>scratch</code> directory associated to your username.
You can find more details about storage and FileSystems [https://wiki.u-gov.it/confluence/display/SCAIUS/UG2.5%3A+Data+storage+and+FileSystems here].

Please don't forget to '''run all tutorials in your scratch directory''':
$ echo $CINECA_SCRATCH
/m100_scratch/userexternal/username
$ cd $CINECA_SCRATCH

Computational resources on M100 are managed by the job scheduling system [https://slurm.schedmd.com/overview.html Slurm]. Most part of Yambo tutorials during this school can be run in serial, except some that need to be executed on multiple processors. Generally, Slurm batch jobs are submitted using a script, but the tutorials here are better understood if run interactively. The two procedures that we will use to submit interactive and non interactive jobs are explained below.

''' - Run a job using a batch script '''


This procedure is suggested for the tutorials and examples that need to be run in parallel. In these cases you need to submit the job using a batch script <code>job.sh</code>. Please note that the instructions in the batch script must be compatible with the specific M100 [https://wiki.u-gov.it/confluence/display/SCAIUS/UG3.2%3A+MARCONI100+UserGuide#UG3.2:MARCONI100UserGuide-SystemArchitecture architecture] and [https://wiki.u-gov.it/confluence/display/SCAIUS/UG3.2%3A+MARCONI100+UserGuide#UG3.2:MARCONI100UserGuide-Accounting accounting] systems. The complete list of Slurm options can be found [https://slurm.schedmd.com/sbatch.html here]. However you will find '''ready-to-use''' batch scripts in locations specified during the tutorials.

To submit the job, use the <code>sbatch</code> command:
$ sbatch job.sh
Submitted batch job <JOBID>

To check the job status, use the <code>squeue</code> command:
$ squeue -u <username>
JOBID PARTITION NAME USER ST TIME NODES NODELIST(REASON)
<...> m100_... JOB username R 0:01 <N> <...>

If you need to cancel your job, do:
$ scancel <JOBID>

''' - Open an interactive session '''

This procedure is suggested for most of the tutorials, since the majority of these is meant to be run in serial (relatively to MPI parallelization) from the command line. Use the command below to open an interactive session of 1 hour (complete documentation [https://slurm.schedmd.com/salloc.html here]):
$ salloc -A tra23_Yambo -p m100_sys_test -q qos_test --reservation=s_tra_yambo --nodes=1 --ntasks-per-node=1 --cpus-per-task=4 -t 01:00:00
salloc: Granted job allocation 10164647
salloc: Waiting for resource configuration
salloc: Nodes r256n01 are ready for job

We ask for 4 cpus-per-task because we can exploit OpenMP parallelization with the available resources.

With <code>squeue</code> you can see that there is now a job running:
$ squeue -u username
JOBID PARTITION NAME USER ST TIME NODES NODELIST(REASON)
10164647 m100_usr_ interact username R 0:02 1 r256n01

To run the tutorial, <code>ssh</code> into the node specified by the job allocation and <code>cd</code> to your scratch directory:
username@'''login02'''$ ssh r256n01
...
username@'''r256n01'''$ cd $CINECA_SCRATCH

Then, you need to manually load <code>yambo</code> as in the batch script above. Please note that the serial version of the code is in a different directory and does not need <code>spectrum_mpi</code>:
$ module purge
$ module load hpc-sdk/2022--binary spectrum_mpi/10.4.0--binary
$ export PATH=/m100_work/tra23_Yambo/softwares/YAMBO/5.2-cpu/bin:$PATH

Finally, set the <code>OMP_NUM_THREADS</code> environment variable to 4 (as in the <code>--cpus-per-task</code> option):
$ export OMP_NUM_THREADS=4

To close the interactive session when you have finished, log out of the compute node with the <code>exit</code> command, and then cancel the job:
$ exit
$ scancel <JOBID>

''' - Plot results with gnuplot '''

During the tutorials you will often need to plot the results of the calculations. In order to do so on M100, '''open a new terminal window''' and connect to M100 enabling X11 forwarding with the <code>-X</code> option:
$ ssh -X m100

Please note that <code>gnuplot</code> can be used in this way only from the login nodes:
username@'''login01'''$ cd <directory_with_data>
username@'''login01'''$ gnuplot
...
Terminal type is now '...'
gnuplot> plot <...>

''' - Set up yambopy '''

In order to run yambopy on m100, you must first setup the conda environment (to be done only once):
$ cd
$ module load anaconda/2020.11
$ conda init bash
$ source .bashrc

After this, every time you want to use yambopy you need to load its module and environment:
$ module load anaconda/2020.11
$ conda activate /m100_work/tra23_Yambo/softwares/YAMBO/env_yambopy

== Tutorials ==

Quick recap.
Before every tutorial do the following steps

$ ssh m100
$ cd $CINECA_SCRATCH
$ mkdir YAMBO_TUTORIALS '''(Only if you didn't before)'''
$ cd YAMBO_TUTORIALS

=== DAY 1 - Monday, 22 May ===

'''16:15 - 18:30 From the DFT ground state to the complete setup of a Many Body calculation using Yambo'''

To get the tutorial files needed for the following tutorials, follow these steps:
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz
$ ls
hBN-2D.tar.gz hBN.tar.gz
$ tar -xvf hBN-2D.tar.gz
$ tar -xvf hBN.tar.gz
$ ls
'''hBN-2D''' '''hBN''' hBN-2D.tar.gz hBN.tar.gz

Now that you have all the files, you may open the interactive job session with <code>salloc</code> as explained above and proceed with the tutorials.

* [[First steps: walk through from DFT(standalone)|First steps: Initialization and more ]]
* [[Next steps: RPA calculations (standalone)|Next steps: RPA calculations ]]

At this point, you may learn about the python pre-postprocessing capabilities offered by yambopy, our python interface to yambo and QE. First of all, let's create a dedicated directory, download and extract the related files.

$ cd $CINECA_SCRATCH
$ mkdir YAMBOPY_TUTORIALS
$ cd YAMBOPY_TUTORIALS
$ wget https://media.yambo-code.eu/educational/tutorials/files/databases_yambopy.tar
$ tar -xvf databases_yambopy.tar
$ cd databases_yambopy

Then, follow '''the first three sections''' of this link, which are related to initialization and linear response.
* [[Yambopy tutorial: Yambo databases|Reading databases with yambopy]]

=== DAY 2 - Tuesday, 23 May ===

'''14:00 - 16:30 A tour through GW simulation in a complex material (from the blackboard to numerical computation: convergence, algorithms, parallel usage)'''

To get all the tutorial files needed for the following tutorials, follow these steps:

wget https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz
wget https://media.yambo-code.eu/educational/tutorials/files/MoS2_2Dquasiparticle_tutorial.tar.gz
tar -xvf hBN.tar.gz
tar -xvf MoS2_2Dquasiparticle_tutorial.tar.gz
cd hBN

Now you can start the first tutorial:

* [[GW tutorial Rome 2023 | GW computations on practice: how to obtain the quasi-particle band structure of a bulk material ]]

If you have gone through the first tutorial, pass now to the second one:

cd $CINECA_SCRATCH
cd YAMBO_TUTORIALS
cd MoS2_HPC_tutorial

* [[Quasi-particles of a 2D system | Quasi-particles of a 2D system ]]

To conclude, you can learn an other method to plot the band structure in Yambo

* [[Yambopy tutorial: band structures| Yambopy tutorial: band structures]]

=== DAY 3 - Wednesday, 24 May ===
'''14:00 - 16:30 Bethe-Salpeter equation (BSE)''' Fulvio Paleari (CNR-Nano, Italy), Davide Sangalli (CNR-ISM, Italy)

To get the tutorial files needed for the following tutorials, follow these steps:
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN.tar.gz # NOTE: YOU SHOULD ALREADY HAVE THIS FROM DAY 1
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN-convergence-kpoints.tar.gz
$ tar -xvf hBN-convergence-kpoints.tar.gz
$ tar -xvf hBN.tar.gz

Now, you may open the interactive job session with <code>salloc</code> and proceed with the following tutorials.

* [[Calculating optical spectra including excitonic effects: a step-by-step guide|Perform a BSE calculation from beginning to end ]]
* [[How to analyse excitons - ICTP 2022 school|Analyse your results (exciton wavefunctions in real and reciprocal space, etc.) ]]
* [[BSE solvers overview|Solve the BSE eigenvalue problem with different numerical methods]]
* [[How to choose the input parameters|Choose the input parameters for a meaningful converged calculation]]
Now, go into the yambopy tutorial directory to learn about python analysis tools for the BSE:
$ cd $CINECA_SCRATCH
$ cd YAMBOPY_TUTORIALS/databases_yambopy

* [[Yambopy_tutorial:_Yambo_databases#Exciton_intro_1:_read_and_sort_data|Visualization of excitonic properties with yambopy]]

'''17:00 - 18:30 Bethe-Salpeter equation in real time (TD-HSEX)''' Fulvio Paleari (CNR-Nano, Italy), Davide Sangalli (CNR-ISM, Italy)

The files needed for the following tutorials can be downloaded following these steps:
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz
$ tar -xvf hBN-2D-RT.tar.gz

* [[Introduction_to_Real_Time_propagation_in_Yambo#Time_Dependent_Equation_for_the_Reduced_One--Body_Density--Matrix|Read the introductive section to real-time propagation for the one-body density matrix]] (the part about time-dependent Schrödinger equation will be covered on DAY 4 and you can skip it for now)
* [[Prerequisites for Real Time propagation with Yambo|Perform the setup for a real-time calculation]]
* [[Linear response from real time simulations (density matrix only)|Calculate the linear response in real time]]
* [[Real time Bethe-Salpeter Equation (density matrix only)|Calculate the BSE in real time]]

=== DAY 4 - Thursday, May 25 ===

'''14:00 - 16:30 Real-time approach with the time dependent berry phase''' Myrta Gruning (Queen's University Belfast), Davide Sangalli (CNR-ISM, Italy)

'''16:15 - 18:30 From the DFT ground state to the complete setup of a Many Body calculation using Yambo'''

For the tutorials we will use first the <code>hBN-2D-RT</code> folder (k-sampling 10x10x1) and then the <code>hBN-2D</code> folder (k-sampling 6x6x1)
You may already have them in the <code>YAMBO_TUTORIALS</code> folder
$ ls
'''hBN-2D''' '''hBN-2D-RT''' hBN-2D.tar.gz hBN-2D-RT.tar.gz

If you need to downoload again the tutorial files, follow these steps:
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D.tar.gz
$ wget https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz
$ tar -xvf hBN-2D.tar.gz
$ tar -xvf hBN-2D-RT.tar.gz

* [[Linear response from Bloch-states dynamics]]
* [[Second-harmonic generation of 2D-hBN]]

* [[Real time approach to non-linear response]] (additional tutorial)
* [[Correlation effects in the non-linear response]] (additional tutorial)

=== DAY 5 - Friday, 26 May ===

== Lectures ==

=== DAY 1 - Monday, 22 May ===

* D. Varsano, [https://media.yambo-code.eu/educational/Schools/ROME2023/scuola_intro.pdf Description and goal of the school].
* G. Stefanucci, [https://media.yambo-code.eu/educational/Schools/ROME2023/Stefanucci.pdf The Many-Body Problem: Key concepts of the Many-Body Perturbation Theory]
* M. Marsili, [https://media.yambo-code.eu/educational/Schools/ROME2023/marghe_linear_response.pdf Beyond the independent particle scheme: The linear response theory]

=== DAY 2 - Tuesday, 23 May ===

* E. Perfetto, [https://media.yambo-code.eu/educational/Schools/ROME2023/Talk_Perfetto.pdf An overview on non-equilibrium Green Functions]
* R. Frisenda, [https://media.yambo-code.eu/educational/Schools/ROME2023/FRISENDA%20-%20ARPES%20spectroscopy,%20an%20experimental%20overview.pdf ARPES spectroscopy, an experimental overview]
* A. Marini, [https://media.yambo-code.eu/educational/Schools/ROME2023/GW_marini.pdf The Quasi Particle concept and the GW method]
* A. Guandalini, [https://media.yambo-code.eu/educational/Schools/ROME2023/alberto_guandalini.pdf The GW method: approximations and algorithms]
* D.A. Leon, C. Cardoso, [https://media.yambo-code.eu/educational/Schools/ROME2023/Cardoso_YamboSchool2023_Rome.pdf Frequency dependence in GW: origin, modelling and practical implementations]

=== DAY 3 - Wednesday, 24 May ===

* A. Molina-Sánchez, Modelling excitons: from 2D materials to Pump and Probe experiments
* M. Palummo, The Bethe-Salpeter equation: derivations and main physical concepts
* F. Paleari, Real time approach to the Bethe-Salpeter equation
* D. Sangalli, TD-HSEX and real-time dynamics

=== DAY 4 - Thursday, 25 May ===

* S. Mor, Time resolved spectroscopy: an experimental overview
* M. Grüning, Nonlinear optics within Many-Body Perturbation Theory
* N. Tancogne-Dejean, Theory and simulation of High Harmonics Generation
* Y. Pavlyukh, Coherent electron-phonon dynamics within a time-linear GKBA scheme

Second-harmonic generation of 2D-hBN

2023-05-25T08:50:15Z

Fulvio: /* Step 2: Independent-particle approximation */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz hBN-2D-RT.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores.

[In order to get 4 cores while in an interacting node, exit your current allocation and rerun <code>salloc</code> asking for the proper number of tasks:
salloc -A tra23_Yambo -p m100_usr_prod -q m100_qos_dbg --nodes=1 --ntasks-per-node=4 --cpus-per-task=4 -t 02:00:00
Otherwise, you can submit the job with a slurm script as explained in the introductory page]

Now execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T08:35:06Z

Fulvio: /* Step 1: Prerequisites */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz hBN-2D-RT.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again (<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores, execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T08:34:51Z

Fulvio: /* Step 1: Prerequisites */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz hBN-2D-RT.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field. Now run the setup again ((<code>yambo_nl -nompi</code>).

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores, execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Dielectric function from Bloch-states dynamics

2023-05-25T08:26:40Z

Fulvio: /* Step 1: Prerequisites */

= Step 0: Theoretical background =

The dynamics of Bloch-states is found by integrating a set of Time Dependent Effective Schrödinger Equations (TD-ESEs). Effective means that an effective Hamiltonian is considered. We consider the many-body effective Hamiltonians seen in the [[Linear response from real time simulations]]. In addition, one can also consider effective Hamiltonians based on density-functional theory. The latter choice gives time-dependent density-functional theory.

At difference with what seen in [[Linear response from real time simulations]], the coupling between electrons and the external field is described by means of the [http://www.theochem.unito.it/didattica/tec-comp_sdm/note1.pdf Modern Theory of Polarization]:

[[File:Tdse new.png|center|400px|Time-Dependent Schrödinger Equation]]

Where <math> | v_{i \mathbf{k}} \rangle </math> are the time-dependent Bloch states (corresponding to the filled Bloch-states at zero-field), <math>\mathcal E(t)</math> is the external field and the term <math> i e \partial k</math> is the dipole operator consistent with periodic boundary condition. For more details on this last term, see Ref. <ref>I. Souza, J. Íñiguez, and D. Vanderbilt, [https://cfm.ehu.es/ivo/publications/souza_prb04.pdf PRB 69, 085106 (2004)]</ref> and <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref>.

<math>h_{rt}</math> is the Hamiltonian,
[[File:H mb.png|400px|center|Many-Body Hamiltonian]].
This contains the equilibrium Hamiltonian, that is the zero-field eigenvalues evaluated through DFT; the quasiparticle corrections, evaluated from GW or estimated from experiment, and the variation of the self-energy. The latter is a functional of the density matrix <math>\rho</math>, which is obtained from the Bloch states.

From the solution of the equation of motion for the Bloch states, the polarization is calculated by means of Berry's phase

[[File:Berry polarization.png|center|350px|Berry's polarization]].

One can decide the level of theory by selecting the terms to include in the Hamiltonian:

* by including only the first term, one selects the ''independent-particle approximation''
* by including the first and second terms, one selects the ''independent-quasiparticle approximation''
* by including the first, second terms and the Hartree part of the self-energy, one selects the ''time-dependent Hartree or random-phase approximation''
* by including all terms, one selects the ''time-dependent Screened-exchange or Bethe-Salpeter equation'' level of theory.

The type of "experiment" to perform, or of the response one would like to extract, depend on the time-dependent field in input and the post-processing of the Berry's phase polarization. The induced polarization is the sum of the responses of the system to all orders in the external field:
<math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{P}_n (t) </math>

In the case we are interested in linear response only, we use a delta-like electric field that excites all frequencies of the system. We choose the intensity of the field <math>E</math> to be weak enough so that the response is dominated by the first order term. The response at all frequencies is obtained by taking the Fourier-transform of the polarization as

<math> \chi(\omega) = \frac{P(\omega)}{E} </math>.

The response <math> \chi</math> is related to the dielectric function through the relation <math>\epsilon(\omega) = 1 + 4 \pi \chi(\omega) </math>.
See also Ref. <ref name="Attaccalite2013"></ref>.

== Notes: ==
* Length vs Velocity gauge: ''the equations of motion of Yambo are in the length gauge. Other codes choose instead to work in the velocity gauge. If you want to know more about the advantages and disadvantages of the two gauges read section 2.7 of Ref. <ref>C. Attaccalite, [https://arxiv.org/abs/1609.09639 arXiv 1609.09639 (2017)]</ref>.''

* Advantages and disadvantages with respect to the density-matrix based formalism: ''Yambo implements two distinct formalisms for studying the real-time response, the one based on density-matrix and the one based on Boch states. Which formalism is best to use depends on what one wants to simulate. The dynamics formulation in terms of density matrix or non-equilibrium Green's functions allows a systematic treatment of correlation effects and electron-phonon coupling<ref name="DavideEPL">[https://arxiv.org/abs/1409.1706 D. Sangalli, and A. Marini EPL '''110''', 47004 (2015)]</ref>, but the price to pay is that the coupling with the external field is correct only to the linear order. This formalism is important in all those phenomena where electronic relaxation plays a major role. On the other hand, the formulation in terms of the Bloch states treats coupling with the external field in an exact way even beyond the linear regime, and for this reason, it allows the description of phenomena coherent with the external fields and to access, for example, non-linear responses.''

= Step 1: Prerequisites =
In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations in yambo you need to:
* download input files and Yambo databases for this tutorial here: [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz hBN-2D-RT.tar.gz].
* perform the steps described in [[Prerequisites for Real Time propagation with Yambo]].

= Step 2: Independent-particle approximation =

Similar to what you have seen in the tutorial for the density matrix formalism ([[Real time approach to linear response]]) we start from the lowest level of theory. Here it is assumed you ran the tutorial on the density-matrix formalism and so you are familiar with the optical spectrum of 2D h-BN at this level of theory. You will recognise that the input and output for the Bloch-state formalism are very similar to those for the density-matrix formalism.

== Run the simulations ==
First of all, create a directory for the inputs to be run by <code>yambo_nl</code>:
mkdir Inputs_nl

Then, use the command:
yambo_nl -nl n -F Inputs_nl/01_td_ip.in

to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime= 55.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
-1.000000 |-1.000000 | eV # [NL] Energy range
%
NLEnSteps= 1 # [NL] Energy steps
NLDamping= 0.000000 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= 18475 RL # [HA] Hartree RL components
EXXRLvcs= 18475 RL # [XX] Exchange RL components
% 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= "DELTA" # [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
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

Note that the standard input of <code>yambo_nl</code> has default settings for nonlinear response (e.g. the <code>SOFTSIN</code> for the <code>Field1_kind</code> (that set the time-dependence 'kind' of the field).
Set the field direction <code>Field1_Dir</code> along y, the field kind to <code>DELTA</code>, the length of the simulation <code>NLtime</code> to 55 fs, select the bands from 3 to 6 (<code>NLBands</code>) and set the <code>NLDamping</code> to zero. The fields you need to change with respect to the default settings are shown above in red.

Now run
yambo_nl -F Inputs_nl/01_td_ip.in -J TD-IP_nl -C TD-IP_nl

The code produces different files (in the <code>TD-IP_nl</code> directory, all with the suffix <code>TD-IP_nl</code>: <code>o.polarization_F1</code> that contains the polarization, <code>o.external_potential_F1</code> the external field we used, and finally <code>r_optics_nloptics</code> a report with all information about the simulation.
The simulation takes longer than the density-matrix counterpart. This is the price to pay for computing the polarization as a Berry Phase. While there is no gain for calculating the linear response, this allows computing nonlinear coefficients, which cannot be obtained within the density based approach. Have a look at the time profile of the run towards the end of the <code>r_optics_nloptics</code> (absolute time may differ):
NL Berry Pol NEQ : 2.2409s CPU ( 5502 calls, 0.000 sec avg)
io_WF : 3.8788s CPU ( 757 calls, 0.005 sec avg)
DIPOLE_overlaps : 4.1834s CPU
DIPOLE_buil_covariants : 5.5611s CPU
Dipoles : 7.3264s CPU
Most of the time is spent computing the covariant dipoles and the non-equilibrium Berry Polarization, which involves the inversion of small matrices via the Lapack inversion routine.
Some time is also required by the integrator <code>NL_Integrator</code> of the equation of motion inversion, which is coded via the solution of a linear system of equations as implemented in the Lapack library. This integrator is more accurate than the Runge-Kutta 2nd order employed for the corresponding calculation within the density matrix formalism.

Plot the third column of <code>o-TD-IP_nl.polarization_F1</code> versus the first one (time-variable) to get the time-dependent polarization along the y-direction:
gnuplot> p 'TD-IP_nl/o-TD-IP_nl.polarization_F1' u 1:3 w l

[[File:Bn polarization.png|thumb|center|900px|Real-time polarization in the y-direction]]

For comparison, in the figure above, the polarization is obtained either through the Bloch-states or the density-matrix.
There are few differences which could be due to the following reasons:
* Coupling with the external field: the <code>yambo_rt</code> evaluates the coupling with the external field through the dipoles (by default computing the commutator [x,Hnl]; the <code>yambo_nl</code> evaluates the coupling using an expression based on the Berry Phase. For the latter, a numerical differentiation in reciprocal space is performed. The 10x10 2D k-mesh may not be fine enough. For denser k-mesh, the Berry phase approach converges to the one based on the dipoles. To avoid this one can use <code>DipApproach="Covariant"</code> in the scheme used by yambo_rt.
* Integrator: the integrator used for the <code>yambo_nl</code> is more accurate. The fact that the differences are more pronounced for long times may hint to inaccuracies in the integration of the equation of motions in <code>yambo_rt</code>. One can use a similar integrator with yambo_rt, by setting in input <code>Integrator= "INV RK2"</code>

== Output postprocessing: the dielectric function ==

Once we obtained the polarization, we Fourier transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F Inputs_nl/ypp_abs.in

to generate the input file:

nonlinear # [R] NonLinear Optics Post-Processing
Xorder= 1 # Max order of the response functions
% TimeRange
-1.000000 |-1.000000 | fs # Time-window where processing is done
%
ETStpsRt= 1001 # Total Energy steps
% EnRngeRt
0.00000 | 10.00000 | eV # Energy range
%
DampMode= "LORENTZIAN" # Damping type ( NONE | LORENTZIAN | GAUSSIAN )
DampFactor= 0.10000 eV # Damping parameter

where we set a Lorentzian smearing corresponding to 0.1 eV. Note that due to the finite time of our simulation, we need to introduce a finite smearing to Fourier transform the result.

To run the post-processing, use the command:
ypp_nl -F Inputs_nl/ypp_abs.in -J TD-IP_nl -C TD-IP_nl
and obtain the files for the dielectric constant along with the field direction, the EELS along with the same direction, and the damped polarization.
o-TD-IP_nl.YPP-eps_along_E
o-TD-IP_nl.YPP-eels_along_E
o-TD-IP_nl.YPP-damped_polarization
inside the folder <code>TD-IP_nl</code>

Plot the dielectric constant and compare it with the linear response and the density-matrix formalism (available from the [[Real time approach to linear response]] tutorial):
gnuplot
plot "CHI_IP/o-CHI_IP.eps_q1_ip" u 1:2 w l
rep "TD-IP_rt/o-TD-IP_rt.YPP-eps_along_E" u 1:2 w l
rep "TD-IP_nl/o-TD-IP_nl.YPP-eps_along_E" u 1:2 w l

[[File:Bn optics.png|thumb|900px|center|Imaginary part of the dielectric constant]]

The differences observed with the spectra obtained from either the density-matrix and linear-response formalisms are due to how the dipoles are evaluated. Within the density-matrix and linear-response formalisms, dipoles are computed in the same way. The Bloch-states formalism uses instead covariant dipoles evaluated numerically on the k-mesh. The differences are due to numerical errors in evaluating the covariant dipoles. These differences disappear with a denser k-mesh.

'''It is a good practice to converge the dielectric function obtained from the Bloch-state dynamics with respect to the k-points using the linear-response spectrum as a reference. This gives the minimum number of k-points to be used to obtain accurate results.
'''

= Step 3: Hartree+Screened exchange approximation =

== Evaluating the collisions==

The evaluation of the collisions. (see [[Introduction to Real Time propagation in Yambo]] to remember what are the collisions)
This part is common in between the <code>yambo_nl</code> and the <code>yambo_rt</code>. If you have computed the collisions for the corresponding <code>yambo_rt</code>, you can re-used those results. If not, follow the related steps in [[Real time Bethe-Salpeter Equation (density matrix only)#The_Real_Time_Collisions| the real-time BSE tutorial]].

== Run the simulations ==
To generate the input for the real-time simulation you can run
yambo_nl -nl n -V qp -F Inputs_nl/04_td-sex.in

The input file is

nloptics # [R NL] Non-linear optics
% NLBands
 4 | 5 | # [NL] Bands
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime= 20.00000 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
-1.000000 | -1.000000 | eV # [NL] Energy range
%
NLEnSteps= 1 # [NL] Energy steps
NLDamping= "0.10000 eV # [NL] Damping
#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

The next section of the input is about external quasiparticle corrections (<code>EXTQP G</code>). There change only

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

The latter line introduces a scissor operator (a rigid shift of the conduction bands) of 3.0 eV. In principle, it is possible to perform a G0W0 calculation with Yambo and use the Quasi-particle band structure instead of the rigid shift.

The last section regards the applied field (<code>RT Field1</code>). There change these two lines:

% Field1_Dir
0.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_kind= "DELTA" # [RT Field1] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN)

Run the calculation:

nohup yambo_nl -F Inputs_nl/04_td_sex.in -J "TD-SEX_nl,COLL_HSEX" -C TD-SEX_nl

== Output postprocessing: dielectric function ==

The post-processing of the output does not depend on the level of approximation. The same steps are taken as in the independent-particle approximation.

Use the command:
ypp_nl -F Inputs_nl/ypp_abs.in -J TD-SEX_nl -C TD-SEX_nl

Plot the results (compare with results obtained from the density-matrix formalism):
gnuplot> set xrange [3:10]
gnuplot> set yrange [0:12]
gnuplot> plot "TD-SEX_rt/o-TD-SEX_rt.YPP-eps_along_E" u 1:2 w l
gnuplot> rep "TD-SEX_nl/o-TD-SEX_nl.YPP-eps_along_E" u 1:2 w l
gnuplot> rep "TD-IP_rt/o-TD-IP_rt.YPP-eps_along_E" u ($1+3):2 w l

[[File:HBN HSEX absorption.png|thumb|900px|center|In plane absorption of hBN at the TD-HSEX level compared with IP absorption]]

Similarly to what we have seen in the independent-particle case, the differences between the two approaches are due to the accuracy in evaluating numerically the covariant dipoles in the Bloch-state dynamics approach.

''For comparison, the linear response results at the BSE level can be obtained following the [[How to obtain an optical spectrum|BSE tutorial]]. ''

= References =
<references/>

Second-harmonic generation of 2D-hBN

2023-05-25T08:26:25Z

Fulvio: /* Step 1: Prerequisites */

= Step 0: Theoretical framework =

In this tutorial, we compute the Second-harmonic generation (SHG) from the dynamics of the Bloch-states. The equation-of-motion of the Bloch-states is explained in the tutorial on [[Linear response from Bloch-states dynamics]]. Independently of the 'experiment' we simulate, the part of the integration of motion stays the same. This means that any experiment, including non-linear optics, can be performed at the level of the theory listed for the computation of the dielectric function:
* independent (quasi)particle
* time-dependent Hartree (RPA level)
* time-dependent DFT (and DPFT)
* time-dependent Hartree+Screened exchange (BSE level)

What changes when we want to calculate the SHG (or higher harmonics) is:
* the time-dependence of the input electric field and
* the post-processing of the signal

'''Time dependence of the electric Field''': we choose a sinusoidal electric field, thus a monochromatic laser field. This allows one to expand the polarization in the form <math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{p}_n e^{-i\omega_n t} </math> where the coefficient <math>\bf{p}_1,...,\bf{p}_n </math> are related to <math>\chi^{(1)},...,\chi^{(n)} </math> through a coefficient depending on a power of the strength of the field (with the power depending on the order of the response).

At difference with a delta-like perturbation, a real-time simulation gives the response at the laser-field only. Then, to obtain the spectrum for the desired range of frequency, we have to perform so many simulations as the frequencies in the desired range.

'''Post-processing of the signal''': The switch-on of the electric field corresponds to a weak delta-like kick. Thus, even if it may not be noticeable, the polarization results from both the sinusoidal field and the delta-like kick. The latter excites all eigenfrequencies of the system. Though weak, the resulting signal is comparable with the second-harmonic signal. To eliminate the signal from the eigenfrequencies, we apply a dephasing (<math>\gamma_{deph}</math>). To have a signal useful to sample the second-harmonic signal, we need to wait a time <math>\bar t</math> much larger than the dephasing-time <math>1/\gamma_{deph} </math>. Note that we cannot choose a too short dephasing time (to shorten the simulation time), as this would result in a too large broadening of the spectrum. After <math>\bar t</math>, we sample $2N+1$ times in a period and use discrete Fourier transform to extract <math>{p}_n</math> for <math> n = 0,...,N</math>, where n=2 gives the SHG. This is schematically illustrated in figure:

[[File:Pt analysis.png|600px|center|Non-linear response analysis]]

The scheme from Ref. <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref> summarised the workflow for computing the SHG (or higher-harmonics) in a energy range <math>[\Omega_1,\Omega_2]</math>.

[[File:Scheme nl.png|350px|center|Schematic representation of real-time calculations]]

= Step 1: Prerequisites =

In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations to compute the SHG in <code>yambo</code> you need to download input files and Yambo databases for this tutorial here: [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz hBN-2D-RT.tar.gz].
Unzip the tarball and change directory to <code>YAMBO</code> and run the setup (<code>yambo_nl -nompi</code>).
Execute the command
ypp_nl -fixsym -F 00_removesym.in
to create the input, to remove the symmetries which are not compatible with the field:
fixsyms # [R] Remove symmetries not consistent with an external perturbation
% Efield1
1.000000 | 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]
#RmAllSymm # Remove all symmetries
RmTimeRev # Remove Time Reversal
#RmSpaceInv # Remove Spatial Inversion
#KeepKGrid # Do not expand the k-grid
where we chose the field along the xy direction and we remove time-reversal symmetry.

Then run the pre-processing job:
ypp_nl -F 00_removesym.in J 00_removesym
This creates the directory <code>FixSymm</code>. Change directory to <code>FixSymm</code>. This contains the <code>SAVE</code> directory with reduced symmetries, compatible with the direction of the field.

= Step 2: Independent-particle approximation =

Use the command:
yambo_nl -nl n -F 01_SHG_ip.in
to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime=-1.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
0.5000000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [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= 21817 RL # [HA] Hartree RL components
EXXRLvcs= 21817 RL # [XX] Exchange RL components
% 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= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)
Field1_pol= "linear" # [RT Field1] Pol(linear|circular)
% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

To describe the Bloch-dynamics we will use as a basis the KS-states 3-6 for the k-grid used in the non-scf DFT calculations, that is a 6x6x1 (<code>NLBands</code>). We choose 12 equally spaced frequencies (<code>NLEnSteps</code>) for the applied field in the range (<code>NLEnRange</code>) between 0.5 and 8 eV. We choose a sinusoidal time-dependence (<code>Field1_kind</code>) and set the field direction (<code>Field1_kind</code>) along xy (in plane). The negative simulation time (<code>NLtime=-1.000000</code>) means that the code will choose it based on the value of the <code>NLDamping</code>. While this is a good lower bound for the simulation time, one should check it is sufficient to get accurate results.

Run the simulation, possibly in parallel e. g. in a interactive session on 4 cores, execute the command:
mpirun -n 4 yambo_nl -F 01_SHG_ip.in -J 01_SHG_ip -C 01_SHG_ip_files

Once the run is finished, check the report <code>01_SHG_ip_files/r-01_SHG_ip_nloptics</code>. In particular, in the appended input files, you can see the default parallelization applied by the code for calculating the dipoles and run the time-propagation. For the latter, the main parallelization is on the frequencies and then on the k-points (<code>NL_ROLEs= "w.k"</code>).

| NL_CPU= "2.2" # [PARALLEL] CPUs for each role
| NL_ROLEs= "w.k" # [PARALLEL] CPUs roles (w,k)
| DIP_CPU= "2.2.1" # [PARALLEL] CPUs for each role
| DIP_ROLEs= "v.c.k" # [PARALLEL] CPUs roles (k,c,v)

Also, look at the simulation time. This is approximately 42 fs.

Plot the polarization for a couple of frequencies, e.g. the first and last of the chosen range (the corresponding frequency can be read in the file, search for "Frequency value"):
set ylabel "x-component polarization" offset -0.7
set xlabel "time (fs)"
plot '01_SHG_ip_files/o-01_SHG_ip.polarization_F1' u 1:2 w l lw 2 title "omega = 0.5 eV"
replot '01_SHG_ip_files/o-01_SHG_ip.polarization_F12' u 1:2 w l lw 2 title "omega = 7.375 eV"

[[File:Berry_phase_polarization_of_2D_h-BN_obtained_from_the_previous_run_for_two_laser_frequencies.png|Berry_phase_polarization_of_2D_h-BN_obtained from the run for two laser frequencies]]

After the applied field is 'switched on', the polarization is oscillating at the frequency of the applied field and at the eigenfrequencies of the system as described above in the cartoon. This is visible particularly for the higher frequency. Due to the dephasing, after about 20 fs the polarization is oscillating (at least to the eye) at the frequency of the applied field. Nonlinear components are not visible since they are several orders of magnitude smaller than the first order.

== Output post-processing: the dielectric function ==

Once we obtained the polarization, we Fourier-transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F 02_SHG_ip_pp.in -C 01_SHG_ip_files

to generate the input file:

nonlinear # [R] Non-linear response analysis
 # Max order of the response/exc functions
% TimeRange
-1.000000 |-1.000000 | 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

Where we changed the <code>Xorder</code> to 4. This variable controls the terms in the Fourier expansion of the polarization. We choose up to expand up to the fourth order. This should ensure the second order to be accurate enough. As an exercise, you can change this value (e.g. choose 2,3,5,6) and see how the result below is changing. The time-window where processing is done is chosen automatically, since values are negative. The code chooses by default the last period to carry out the analysis.

To run the post-processing, use the command:
ypp_nl -F Inputs_shg/ypp_shg.in -J TD-IP_nl -C TD-IP_nl.

In <code>01_SHG_ip_files</code> the following files were generated
o-01_SHG_ip.YPP-X_probe_order_1
o-01_SHG_ip.YPP-X_probe_order_2
o-01_SHG_ip.YPP-X_probe_order_3
o-01_SHG_ip.YPP-X_probe_order_4
r-01_SHG_ip_nonlinear

The output files contain respectively the first, second, third and fourth order response(corresponding to <code>Xorder=4</code>) at the 12 frequencies in the chosen range.

Inspect <code>o-01_SHG_ip.YPP-X_probe_order_2</code>
# E [eV] X/Im[cm/stV](x) X/Re[cm/stV](x) X/Im[cm/stV](y) X/Re[cm/stV](y) X/Im[cm/stV](z) X/Re[cm/stV](z)
#
0.50000000 -0.35378838E-09 -0.19774796E-07 0.23591828E-09 -0.89405637E-09 0.0000000 0.0000000
1.1250000 -0.19356524E-08 -0.23160605E-07 -0.16052184E-09 -0.67662771E-09 0.0000000 0.0000000
1.7500000 -0.69446459E-08 -0.35586956E-07 -0.76025706E-09 -0.10205678E-08 0.0000000 0.0000000
2.37500000 -0.181606152E-7 -0.261823658E-8 0.438358596E-8 0.407915226E-8 0.00000000 0.00000000

The first column are the frequencies of the applied field, the second and third is the polarization along x (imaginary and real part), the fourth and fifth is the polarization along y (imaginary and real part)
and the sixth and seventh is the polarization along z (imaginary and real part).

Plot the absolute value of the SHG for the "xxy" component (that is, the applied field is in the x and y directions and one look at the polarization along x):

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '01_SHG_ip_files/o-01_SHG_ip.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File: SHG_intensity_of_2D_h-BN_obtained_from_the_previous_run_(and_compared_with_a_run_on_112_frequencies).png | SHG_intensity_of_2D_h-BN obtained from the current run (12 frequencies) compared with a run with 112 frequencies]]

In the figure above, the results of the current run are compared with those for a run with 112 frequencies (one can obtain these results by repeating the calculations, changing the number of frequencies in <code>NLEnSteps</code> though this take about 10 times more computational time than the run with 12). One can notice some differences between the two runs. In fact, the two runs evaluate different frequnecies and the difference indicates spurious oscillations of the value of the polarizability due to the simulation time being too short and thus to the presence of 'noise' due to the eigenfrecies.
As an exercise, you can repeat the simulation for e,g, a time of <code>60 fs</code>, that is setting <code>NLtime=60</code>. The plot below compare the SHG extracted at 42 fs and 60 fs, where the latter gives clearly more accurate results.

[[File:Comparison of the SHG intensity of 2D h-BN obtained with different simulation time.png|Comparison of the SHG intensity of 2D h-BN obtained with different simulation time]]

As a final note, the SHG depends on the size of the vacuum added in the supercell. Instead, one should consider the surface SHG which obtained by considering an effective thickness for the layer. This is usually chosen to be similar to the layer separation in the bulk counterpart.

= Step 3: Hartree+Screened exchange approximation (BSE level) =

== Hartree+Screened exchange collisions ==
Similarly to the linear part, one needs to generate the Screened exchange collisions. The procedure is exactly the same.

Create the input:
yambo_nl -X s -e -v hsex -F 03_coll_hsex.in
modify the input ad follows
em1s # [R][Xs] Statically Screened Interaction
collisions # [R] Collisions
dipoles # [R] Oscillator strenghts (or dipoles)
Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc
% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 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

Calculate the collisions (you may want to run in parallel):
yambo_nl -F 03_coll_hsex.in -J 03_coll -C 04_hsex_files

== Run the simulation ==

Create the input:
yambo_nl -nl n -v hsex -V qp -F 04_SHG_hsex.in

modify the following parts of the input (by this time all the innput variables should be known from the previous exercises):

% BndsRnXs
1 | 20 | # [Xs] Polarization function bands
%
NGsBlkXs= 1000 mHa # [Xs] Response block size
% LongDrXs
1.000000 | 1.000000 | 0.000000 | # [Xs] [cc] Electric Field
%
% NLBands
4 | 5 | # [NL] Bands range
%

NLintegrator= "CRANKNIC" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "SEX" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")

% NLEnRange
0.500000 |8.000000 | eV # [NL] Energy range
%
NLEnSteps= 12 # [NL] Energy steps

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

Field1_kind= "SIN" # [RT Field1] Kind(SIN|COS|RES|ANTIRES|GAUSS|DELTA|QSSIN)

% Field1_Dir
1.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%

Then run the simulation using e.g. mpi and 4 cores:
mpirun -n 4 yambo_nl -F 04_SHG_hsex.in -J 04_SHG_hsex,03_coll -C 04_SHG_hsex_files

== Postprocessing: obtain the nonlinear response ==

Similarly to what we did for the independent particle case, we create the input:
ypp_nl -nl -F 05_SHG_hsex_pp.in

we change <code>Xorder = 4 </code> and then run:
ypp_nl -F 05_SHG_hsex_pp.in -J 04_SHG_hsex -C 04_hsex_files

Then we plot (as before we compare with a simulation ran for 112 frequnecies).

Plot the results:

set ylabel "{/Symbol=26 \174}{/Symbol=26 c}_{xyx}^{(2)}{/Symbol=26 \174} (pm/V)" offset -0.7
set xlabel "Energy (eV)"
pmVm1toCGS=2.38721e-09
plot '04_hsex_files/o-04_SHG_hsex.YPP-X_probe_order_2' u 1:(sqrt($2**2+$3**2)/pmVm1toCGS) w p title "calc"

[[File:SHG hsex.png|SHG at the Hartree + Screened exchange from current run with 12 frequencies (crosses) and a run with 112 frequencies (keeping the other parameters the same)]]

also in this case we note differences between the two simulations. This is due to the too short simulation time. Repeating the calculations with a longer time gives more accurate results.
In addition, note that in the interest of time, all parameters are underconverged. In principle, to calculate the collisions one should use the same parameters of a converged BSE calculation. As well, one should use the Coulomb cutoff as it was used for this system in the BSE case.

= References =
<references/>

Dielectric function from Bloch-states dynamics

2023-05-25T08:25:54Z

Fulvio: /* Step 1: Prerequisites */

= Step 0: Theoretical background =

The dynamics of Bloch-states is found by integrating a set of Time Dependent Effective Schrödinger Equations (TD-ESEs). Effective means that an effective Hamiltonian is considered. We consider the many-body effective Hamiltonians seen in the [[Linear response from real time simulations]]. In addition, one can also consider effective Hamiltonians based on density-functional theory. The latter choice gives time-dependent density-functional theory.

At difference with what seen in [[Linear response from real time simulations]], the coupling between electrons and the external field is described by means of the [http://www.theochem.unito.it/didattica/tec-comp_sdm/note1.pdf Modern Theory of Polarization]:

[[File:Tdse new.png|center|400px|Time-Dependent Schrödinger Equation]]

Where <math> | v_{i \mathbf{k}} \rangle </math> are the time-dependent Bloch states (corresponding to the filled Bloch-states at zero-field), <math>\mathcal E(t)</math> is the external field and the term <math> i e \partial k</math> is the dipole operator consistent with periodic boundary condition. For more details on this last term, see Ref. <ref>I. Souza, J. Íñiguez, and D. Vanderbilt, [https://cfm.ehu.es/ivo/publications/souza_prb04.pdf PRB 69, 085106 (2004)]</ref> and <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref>.

<math>h_{rt}</math> is the Hamiltonian,
[[File:H mb.png|400px|center|Many-Body Hamiltonian]].
This contains the equilibrium Hamiltonian, that is the zero-field eigenvalues evaluated through DFT; the quasiparticle corrections, evaluated from GW or estimated from experiment, and the variation of the self-energy. The latter is a functional of the density matrix <math>\rho</math>, which is obtained from the Bloch states.

From the solution of the equation of motion for the Bloch states, the polarization is calculated by means of Berry's phase

[[File:Berry polarization.png|center|350px|Berry's polarization]].

One can decide the level of theory by selecting the terms to include in the Hamiltonian:

* by including only the first term, one selects the ''independent-particle approximation''
* by including the first and second terms, one selects the ''independent-quasiparticle approximation''
* by including the first, second terms and the Hartree part of the self-energy, one selects the ''time-dependent Hartree or random-phase approximation''
* by including all terms, one selects the ''time-dependent Screened-exchange or Bethe-Salpeter equation'' level of theory.

The type of "experiment" to perform, or of the response one would like to extract, depend on the time-dependent field in input and the post-processing of the Berry's phase polarization. The induced polarization is the sum of the responses of the system to all orders in the external field:
<math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{P}_n (t) </math>

In the case we are interested in linear response only, we use a delta-like electric field that excites all frequencies of the system. We choose the intensity of the field <math>E</math> to be weak enough so that the response is dominated by the first order term. The response at all frequencies is obtained by taking the Fourier-transform of the polarization as

<math> \chi(\omega) = \frac{P(\omega)}{E} </math>.

The response <math> \chi</math> is related to the dielectric function through the relation <math>\epsilon(\omega) = 1 + 4 \pi \chi(\omega) </math>.
See also Ref. <ref name="Attaccalite2013"></ref>.

== Notes: ==
* Length vs Velocity gauge: ''the equations of motion of Yambo are in the length gauge. Other codes choose instead to work in the velocity gauge. If you want to know more about the advantages and disadvantages of the two gauges read section 2.7 of Ref. <ref>C. Attaccalite, [https://arxiv.org/abs/1609.09639 arXiv 1609.09639 (2017)]</ref>.''

* Advantages and disadvantages with respect to the density-matrix based formalism: ''Yambo implements two distinct formalisms for studying the real-time response, the one based on density-matrix and the one based on Boch states. Which formalism is best to use depends on what one wants to simulate. The dynamics formulation in terms of density matrix or non-equilibrium Green's functions allows a systematic treatment of correlation effects and electron-phonon coupling<ref name="DavideEPL">[https://arxiv.org/abs/1409.1706 D. Sangalli, and A. Marini EPL '''110''', 47004 (2015)]</ref>, but the price to pay is that the coupling with the external field is correct only to the linear order. This formalism is important in all those phenomena where electronic relaxation plays a major role. On the other hand, the formulation in terms of the Bloch states treats coupling with the external field in an exact way even beyond the linear regime, and for this reason, it allows the description of phenomena coherent with the external fields and to access, for example, non-linear responses.''

= Step 1: Prerequisites =
In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations in yambo you need to:
* download input files and Yambo databases for this tutorial here: [https://media.yambo-code.eu/educational/tutorials/files/hBN-2D-RT.tar.gz].
* perform the steps described in [[Prerequisites for Real Time propagation with Yambo]].

= Step 2: Independent-particle approximation =

Similar to what you have seen in the tutorial for the density matrix formalism ([[Real time approach to linear response]]) we start from the lowest level of theory. Here it is assumed you ran the tutorial on the density-matrix formalism and so you are familiar with the optical spectrum of 2D h-BN at this level of theory. You will recognise that the input and output for the Bloch-state formalism are very similar to those for the density-matrix formalism.

== Run the simulations ==
First of all, create a directory for the inputs to be run by <code>yambo_nl</code>:
mkdir Inputs_nl

Then, use the command:
yambo_nl -nl n -F Inputs_nl/01_td_ip.in

to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime= 55.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
-1.000000 |-1.000000 | eV # [NL] Energy range
%
NLEnSteps= 1 # [NL] Energy steps
NLDamping= 0.000000 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= 18475 RL # [HA] Hartree RL components
EXXRLvcs= 18475 RL # [XX] Exchange RL components
% 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= "DELTA" # [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
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

Note that the standard input of <code>yambo_nl</code> has default settings for nonlinear response (e.g. the <code>SOFTSIN</code> for the <code>Field1_kind</code> (that set the time-dependence 'kind' of the field).
Set the field direction <code>Field1_Dir</code> along y, the field kind to <code>DELTA</code>, the length of the simulation <code>NLtime</code> to 55 fs, select the bands from 3 to 6 (<code>NLBands</code>) and set the <code>NLDamping</code> to zero. The fields you need to change with respect to the default settings are shown above in red.

Now run
yambo_nl -F Inputs_nl/01_td_ip.in -J TD-IP_nl -C TD-IP_nl

The code produces different files (in the <code>TD-IP_nl</code> directory, all with the suffix <code>TD-IP_nl</code>: <code>o.polarization_F1</code> that contains the polarization, <code>o.external_potential_F1</code> the external field we used, and finally <code>r_optics_nloptics</code> a report with all information about the simulation.
The simulation takes longer than the density-matrix counterpart. This is the price to pay for computing the polarization as a Berry Phase. While there is no gain for calculating the linear response, this allows computing nonlinear coefficients, which cannot be obtained within the density based approach. Have a look at the time profile of the run towards the end of the <code>r_optics_nloptics</code> (absolute time may differ):
NL Berry Pol NEQ : 2.2409s CPU ( 5502 calls, 0.000 sec avg)
io_WF : 3.8788s CPU ( 757 calls, 0.005 sec avg)
DIPOLE_overlaps : 4.1834s CPU
DIPOLE_buil_covariants : 5.5611s CPU
Dipoles : 7.3264s CPU
Most of the time is spent computing the covariant dipoles and the non-equilibrium Berry Polarization, which involves the inversion of small matrices via the Lapack inversion routine.
Some time is also required by the integrator <code>NL_Integrator</code> of the equation of motion inversion, which is coded via the solution of a linear system of equations as implemented in the Lapack library. This integrator is more accurate than the Runge-Kutta 2nd order employed for the corresponding calculation within the density matrix formalism.

Plot the third column of <code>o-TD-IP_nl.polarization_F1</code> versus the first one (time-variable) to get the time-dependent polarization along the y-direction:
gnuplot> p 'TD-IP_nl/o-TD-IP_nl.polarization_F1' u 1:3 w l

[[File:Bn polarization.png|thumb|center|900px|Real-time polarization in the y-direction]]

For comparison, in the figure above, the polarization is obtained either through the Bloch-states or the density-matrix.
There are few differences which could be due to the following reasons:
* Coupling with the external field: the <code>yambo_rt</code> evaluates the coupling with the external field through the dipoles (by default computing the commutator [x,Hnl]; the <code>yambo_nl</code> evaluates the coupling using an expression based on the Berry Phase. For the latter, a numerical differentiation in reciprocal space is performed. The 10x10 2D k-mesh may not be fine enough. For denser k-mesh, the Berry phase approach converges to the one based on the dipoles. To avoid this one can use <code>DipApproach="Covariant"</code> in the scheme used by yambo_rt.
* Integrator: the integrator used for the <code>yambo_nl</code> is more accurate. The fact that the differences are more pronounced for long times may hint to inaccuracies in the integration of the equation of motions in <code>yambo_rt</code>. One can use a similar integrator with yambo_rt, by setting in input <code>Integrator= "INV RK2"</code>

== Output postprocessing: the dielectric function ==

Once we obtained the polarization, we Fourier transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F Inputs_nl/ypp_abs.in

to generate the input file:

nonlinear # [R] NonLinear Optics Post-Processing
Xorder= 1 # Max order of the response functions
% TimeRange
-1.000000 |-1.000000 | fs # Time-window where processing is done
%
ETStpsRt= 1001 # Total Energy steps
% EnRngeRt
0.00000 | 10.00000 | eV # Energy range
%
DampMode= "LORENTZIAN" # Damping type ( NONE | LORENTZIAN | GAUSSIAN )
DampFactor= 0.10000 eV # Damping parameter

where we set a Lorentzian smearing corresponding to 0.1 eV. Note that due to the finite time of our simulation, we need to introduce a finite smearing to Fourier transform the result.

To run the post-processing, use the command:
ypp_nl -F Inputs_nl/ypp_abs.in -J TD-IP_nl -C TD-IP_nl
and obtain the files for the dielectric constant along with the field direction, the EELS along with the same direction, and the damped polarization.
o-TD-IP_nl.YPP-eps_along_E
o-TD-IP_nl.YPP-eels_along_E
o-TD-IP_nl.YPP-damped_polarization
inside the folder <code>TD-IP_nl</code>

Plot the dielectric constant and compare it with the linear response and the density-matrix formalism (available from the [[Real time approach to linear response]] tutorial):
gnuplot
plot "CHI_IP/o-CHI_IP.eps_q1_ip" u 1:2 w l
rep "TD-IP_rt/o-TD-IP_rt.YPP-eps_along_E" u 1:2 w l
rep "TD-IP_nl/o-TD-IP_nl.YPP-eps_along_E" u 1:2 w l

[[File:Bn optics.png|thumb|900px|center|Imaginary part of the dielectric constant]]

The differences observed with the spectra obtained from either the density-matrix and linear-response formalisms are due to how the dipoles are evaluated. Within the density-matrix and linear-response formalisms, dipoles are computed in the same way. The Bloch-states formalism uses instead covariant dipoles evaluated numerically on the k-mesh. The differences are due to numerical errors in evaluating the covariant dipoles. These differences disappear with a denser k-mesh.

'''It is a good practice to converge the dielectric function obtained from the Bloch-state dynamics with respect to the k-points using the linear-response spectrum as a reference. This gives the minimum number of k-points to be used to obtain accurate results.
'''

= Step 3: Hartree+Screened exchange approximation =

== Evaluating the collisions==

The evaluation of the collisions. (see [[Introduction to Real Time propagation in Yambo]] to remember what are the collisions)
This part is common in between the <code>yambo_nl</code> and the <code>yambo_rt</code>. If you have computed the collisions for the corresponding <code>yambo_rt</code>, you can re-used those results. If not, follow the related steps in [[Real time Bethe-Salpeter Equation (density matrix only)#The_Real_Time_Collisions| the real-time BSE tutorial]].

== Run the simulations ==
To generate the input for the real-time simulation you can run
yambo_nl -nl n -V qp -F Inputs_nl/04_td-sex.in

The input file is

nloptics # [R NL] Non-linear optics
% NLBands
 4 | 5 | # [NL] Bands
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime= 20.00000 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
-1.000000 | -1.000000 | eV # [NL] Energy range
%
NLEnSteps= 1 # [NL] Energy steps
NLDamping= "0.10000 eV # [NL] Damping
#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

The next section of the input is about external quasiparticle corrections (<code>EXTQP G</code>). There change only

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

The latter line introduces a scissor operator (a rigid shift of the conduction bands) of 3.0 eV. In principle, it is possible to perform a G0W0 calculation with Yambo and use the Quasi-particle band structure instead of the rigid shift.

The last section regards the applied field (<code>RT Field1</code>). There change these two lines:

% Field1_Dir
0.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_kind= "DELTA" # [RT Field1] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN)

Run the calculation:

nohup yambo_nl -F Inputs_nl/04_td_sex.in -J "TD-SEX_nl,COLL_HSEX" -C TD-SEX_nl

== Output postprocessing: dielectric function ==

The post-processing of the output does not depend on the level of approximation. The same steps are taken as in the independent-particle approximation.

Use the command:
ypp_nl -F Inputs_nl/ypp_abs.in -J TD-SEX_nl -C TD-SEX_nl

Plot the results (compare with results obtained from the density-matrix formalism):
gnuplot> set xrange [3:10]
gnuplot> set yrange [0:12]
gnuplot> plot "TD-SEX_rt/o-TD-SEX_rt.YPP-eps_along_E" u 1:2 w l
gnuplot> rep "TD-SEX_nl/o-TD-SEX_nl.YPP-eps_along_E" u 1:2 w l
gnuplot> rep "TD-IP_rt/o-TD-IP_rt.YPP-eps_along_E" u ($1+3):2 w l

[[File:HBN HSEX absorption.png|thumb|900px|center|In plane absorption of hBN at the TD-HSEX level compared with IP absorption]]

Similarly to what we have seen in the independent-particle case, the differences between the two approaches are due to the accuracy in evaluating numerically the covariant dipoles in the Bloch-state dynamics approach.

''For comparison, the linear response results at the BSE level can be obtained following the [[How to obtain an optical spectrum|BSE tutorial]]. ''

= References =
<references/>

Dielectric function from Bloch-states dynamics

2023-05-25T08:19:36Z

Fulvio: /* Output postprocessing: dielectric function */

= Step 0: Theoretical background =

The dynamics of Bloch-states is found by integrating a set of Time Dependent Effective Schrödinger Equations (TD-ESEs). Effective means that an effective Hamiltonian is considered. We consider the many-body effective Hamiltonians seen in the [[Linear response from real time simulations]]. In addition, one can also consider effective Hamiltonians based on density-functional theory. The latter choice gives time-dependent density-functional theory.

At difference with what seen in [[Linear response from real time simulations]], the coupling between electrons and the external field is described by means of the [http://www.theochem.unito.it/didattica/tec-comp_sdm/note1.pdf Modern Theory of Polarization]:

[[File:Tdse new.png|center|400px|Time-Dependent Schrödinger Equation]]

Where <math> | v_{i \mathbf{k}} \rangle </math> are the time-dependent Bloch states (corresponding to the filled Bloch-states at zero-field), <math>\mathcal E(t)</math> is the external field and the term <math> i e \partial k</math> is the dipole operator consistent with periodic boundary condition. For more details on this last term, see Ref. <ref>I. Souza, J. Íñiguez, and D. Vanderbilt, [https://cfm.ehu.es/ivo/publications/souza_prb04.pdf PRB 69, 085106 (2004)]</ref> and <ref name="Attaccalite2013">C. Attaccalite and M. Gruning [https://arxiv.org/abs/1309.4012v2 Rev. B, '''88''', 235113 (2013)]</ref>.

<math>h_{rt}</math> is the Hamiltonian,
[[File:H mb.png|400px|center|Many-Body Hamiltonian]].
This contains the equilibrium Hamiltonian, that is the zero-field eigenvalues evaluated through DFT; the quasiparticle corrections, evaluated from GW or estimated from experiment, and the variation of the self-energy. The latter is a functional of the density matrix <math>\rho</math>, which is obtained from the Bloch states.

From the solution of the equation of motion for the Bloch states, the polarization is calculated by means of Berry's phase

[[File:Berry polarization.png|center|350px|Berry's polarization]].

One can decide the level of theory by selecting the terms to include in the Hamiltonian:

* by including only the first term, one selects the ''independent-particle approximation''
* by including the first and second terms, one selects the ''independent-quasiparticle approximation''
* by including the first, second terms and the Hartree part of the self-energy, one selects the ''time-dependent Hartree or random-phase approximation''
* by including all terms, one selects the ''time-dependent Screened-exchange or Bethe-Salpeter equation'' level of theory.

The type of "experiment" to perform, or of the response one would like to extract, depend on the time-dependent field in input and the post-processing of the Berry's phase polarization. The induced polarization is the sum of the responses of the system to all orders in the external field:
<math>\bf{P}(t) = \sum_{n=-\infty}^{+\infty} \bf{P}_n (t) </math>

In the case we are interested in linear response only, we use a delta-like electric field that excites all frequencies of the system. We choose the intensity of the field <math>E</math> to be weak enough so that the response is dominated by the first order term. The response at all frequencies is obtained by taking the Fourier-transform of the polarization as

<math> \chi(\omega) = \frac{P(\omega)}{E} </math>.

The response <math> \chi</math> is related to the dielectric function through the relation <math>\epsilon(\omega) = 1 + 4 \pi \chi(\omega) </math>.
See also Ref. <ref name="Attaccalite2013"></ref>.

== Notes: ==
* Length vs Velocity gauge: ''the equations of motion of Yambo are in the length gauge. Other codes choose instead to work in the velocity gauge. If you want to know more about the advantages and disadvantages of the two gauges read section 2.7 of Ref. <ref>C. Attaccalite, [https://arxiv.org/abs/1609.09639 arXiv 1609.09639 (2017)]</ref>.''

* Advantages and disadvantages with respect to the density-matrix based formalism: ''Yambo implements two distinct formalisms for studying the real-time response, the one based on density-matrix and the one based on Boch states. Which formalism is best to use depends on what one wants to simulate. The dynamics formulation in terms of density matrix or non-equilibrium Green's functions allows a systematic treatment of correlation effects and electron-phonon coupling<ref name="DavideEPL">[https://arxiv.org/abs/1409.1706 D. Sangalli, and A. Marini EPL '''110''', 47004 (2015)]</ref>, but the price to pay is that the coupling with the external field is correct only to the linear order. This formalism is important in all those phenomena where electronic relaxation plays a major role. On the other hand, the formulation in terms of the Bloch states treats coupling with the external field in an exact way even beyond the linear regime, and for this reason, it allows the description of phenomena coherent with the external fields and to access, for example, non-linear responses.''

= Step 1: Prerequisites =
In this example, we will consider a single layer of hexagonal boron nitride (hBN). Before running real time simulations in yambo you need to:
* download input files and Yambo databases for this tutorial here: [http://www.yambo-code.org/educational/tutorials/files/hBN-2D-RT.tar.gz hBN-2D-RT.tar.gz].
* perform the steps described in [[Prerequisites for Real Time propagation with Yambo]].

= Step 2: Independent-particle approximation =

Similar to what you have seen in the tutorial for the density matrix formalism ([[Real time approach to linear response]]) we start from the lowest level of theory. Here it is assumed you ran the tutorial on the density-matrix formalism and so you are familiar with the optical spectrum of 2D h-BN at this level of theory. You will recognise that the input and output for the Bloch-state formalism are very similar to those for the density-matrix formalism.

== Run the simulations ==
First of all, create a directory for the inputs to be run by <code>yambo_nl</code>:
mkdir Inputs_nl

Then, use the command:
yambo_nl -nl n -F Inputs_nl/01_td_ip.in

to generate the input:

nloptics # [R] Non-linear spectroscopy
% NLBands
3 | 6 | # [NL] Bands range
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime= 55.000000 fs # [NL] Simulation Time
NLintegrator= "INVINT" # [NL] Integrator ("EULEREXP/RK2/RK4/RK2EXP/HEUN/INVINT/CRANKNIC")
NLCorrelation= "IPA" # [NL] Correlation ("IPA/HARTREE/TDDFT/LRC/LRW/JGM/SEX")
NLLrcAlpha= 0.000000 # [NL] Long Range Correction
% NLEnRange
-1.000000 |-1.000000 | eV # [NL] Energy range
%
NLEnSteps= 1 # [NL] Energy steps
NLDamping= 0.000000 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= 18475 RL # [HA] Hartree RL components
EXXRLvcs= 18475 RL # [XX] Exchange RL components
% 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= "DELTA" # [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
%
Field1_Tstart= 0.010000 fs # [RT Field1] Initial Time

Note that the standard input of <code>yambo_nl</code> has default settings for nonlinear response (e.g. the <code>SOFTSIN</code> for the <code>Field1_kind</code> (that set the time-dependence 'kind' of the field).
Set the field direction <code>Field1_Dir</code> along y, the field kind to <code>DELTA</code>, the length of the simulation <code>NLtime</code> to 55 fs, select the bands from 3 to 6 (<code>NLBands</code>) and set the <code>NLDamping</code> to zero. The fields you need to change with respect to the default settings are shown above in red.

Now run
yambo_nl -F Inputs_nl/01_td_ip.in -J TD-IP_nl -C TD-IP_nl

The code produces different files (in the <code>TD-IP_nl</code> directory, all with the suffix <code>TD-IP_nl</code>: <code>o.polarization_F1</code> that contains the polarization, <code>o.external_potential_F1</code> the external field we used, and finally <code>r_optics_nloptics</code> a report with all information about the simulation.
The simulation takes longer than the density-matrix counterpart. This is the price to pay for computing the polarization as a Berry Phase. While there is no gain for calculating the linear response, this allows computing nonlinear coefficients, which cannot be obtained within the density based approach. Have a look at the time profile of the run towards the end of the <code>r_optics_nloptics</code> (absolute time may differ):
NL Berry Pol NEQ : 2.2409s CPU ( 5502 calls, 0.000 sec avg)
io_WF : 3.8788s CPU ( 757 calls, 0.005 sec avg)
DIPOLE_overlaps : 4.1834s CPU
DIPOLE_buil_covariants : 5.5611s CPU
Dipoles : 7.3264s CPU
Most of the time is spent computing the covariant dipoles and the non-equilibrium Berry Polarization, which involves the inversion of small matrices via the Lapack inversion routine.
Some time is also required by the integrator <code>NL_Integrator</code> of the equation of motion inversion, which is coded via the solution of a linear system of equations as implemented in the Lapack library. This integrator is more accurate than the Runge-Kutta 2nd order employed for the corresponding calculation within the density matrix formalism.

Plot the third column of <code>o-TD-IP_nl.polarization_F1</code> versus the first one (time-variable) to get the time-dependent polarization along the y-direction:
gnuplot> p 'TD-IP_nl/o-TD-IP_nl.polarization_F1' u 1:3 w l

[[File:Bn polarization.png|thumb|center|900px|Real-time polarization in the y-direction]]

For comparison, in the figure above, the polarization is obtained either through the Bloch-states or the density-matrix.
There are few differences which could be due to the following reasons:
* Coupling with the external field: the <code>yambo_rt</code> evaluates the coupling with the external field through the dipoles (by default computing the commutator [x,Hnl]; the <code>yambo_nl</code> evaluates the coupling using an expression based on the Berry Phase. For the latter, a numerical differentiation in reciprocal space is performed. The 10x10 2D k-mesh may not be fine enough. For denser k-mesh, the Berry phase approach converges to the one based on the dipoles. To avoid this one can use <code>DipApproach="Covariant"</code> in the scheme used by yambo_rt.
* Integrator: the integrator used for the <code>yambo_nl</code> is more accurate. The fact that the differences are more pronounced for long times may hint to inaccuracies in the integration of the equation of motions in <code>yambo_rt</code>. One can use a similar integrator with yambo_rt, by setting in input <code>Integrator= "INV RK2"</code>

== Output postprocessing: the dielectric function ==

Once we obtained the polarization, we Fourier transform the polarization to obtain the dielectric function.

Use the command:

ypp_nl -nl -F Inputs_nl/ypp_abs.in

to generate the input file:

nonlinear # [R] NonLinear Optics Post-Processing
Xorder= 1 # Max order of the response functions
% TimeRange
-1.000000 |-1.000000 | fs # Time-window where processing is done
%
ETStpsRt= 1001 # Total Energy steps
% EnRngeRt
0.00000 | 10.00000 | eV # Energy range
%
DampMode= "LORENTZIAN" # Damping type ( NONE | LORENTZIAN | GAUSSIAN )
DampFactor= 0.10000 eV # Damping parameter

where we set a Lorentzian smearing corresponding to 0.1 eV. Note that due to the finite time of our simulation, we need to introduce a finite smearing to Fourier transform the result.

To run the post-processing, use the command:
ypp_nl -F Inputs_nl/ypp_abs.in -J TD-IP_nl -C TD-IP_nl
and obtain the files for the dielectric constant along with the field direction, the EELS along with the same direction, and the damped polarization.
o-TD-IP_nl.YPP-eps_along_E
o-TD-IP_nl.YPP-eels_along_E
o-TD-IP_nl.YPP-damped_polarization
inside the folder <code>TD-IP_nl</code>

Plot the dielectric constant and compare it with the linear response and the density-matrix formalism (available from the [[Real time approach to linear response]] tutorial):
gnuplot
plot "CHI_IP/o-CHI_IP.eps_q1_ip" u 1:2 w l
rep "TD-IP_rt/o-TD-IP_rt.YPP-eps_along_E" u 1:2 w l
rep "TD-IP_nl/o-TD-IP_nl.YPP-eps_along_E" u 1:2 w l

[[File:Bn optics.png|thumb|900px|center|Imaginary part of the dielectric constant]]

The differences observed with the spectra obtained from either the density-matrix and linear-response formalisms are due to how the dipoles are evaluated. Within the density-matrix and linear-response formalisms, dipoles are computed in the same way. The Bloch-states formalism uses instead covariant dipoles evaluated numerically on the k-mesh. The differences are due to numerical errors in evaluating the covariant dipoles. These differences disappear with a denser k-mesh.

'''It is a good practice to converge the dielectric function obtained from the Bloch-state dynamics with respect to the k-points using the linear-response spectrum as a reference. This gives the minimum number of k-points to be used to obtain accurate results.
'''

= Step 3: Hartree+Screened exchange approximation =

== Evaluating the collisions==

The evaluation of the collisions. (see [[Introduction to Real Time propagation in Yambo]] to remember what are the collisions)
This part is common in between the <code>yambo_nl</code> and the <code>yambo_rt</code>. If you have computed the collisions for the corresponding <code>yambo_rt</code>, you can re-used those results. If not, follow the related steps in [[Real time Bethe-Salpeter Equation (density matrix only)#The_Real_Time_Collisions| the real-time BSE tutorial]].

== Run the simulations ==
To generate the input for the real-time simulation you can run
yambo_nl -nl n -V qp -F Inputs_nl/04_td-sex.in

The input file is

nloptics # [R NL] Non-linear optics
% NLBands
 4 | 5 | # [NL] Bands
%
NLverbosity= "high" # [NL] Verbosity level (low | high)
NLtime= 20.00000 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
-1.000000 | -1.000000 | eV # [NL] Energy range
%
NLEnSteps= 1 # [NL] Energy steps
NLDamping= "0.10000 eV # [NL] Damping
#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

The next section of the input is about external quasiparticle corrections (<code>EXTQP G</code>). There change only

% GfnQP_E
3.000000 | 1.000000 | 1.000000 | # [EXTQP G] E parameters (c/v) eV|adim|adim
%

The latter line introduces a scissor operator (a rigid shift of the conduction bands) of 3.0 eV. In principle, it is possible to perform a G0W0 calculation with Yambo and use the Quasi-particle band structure instead of the rigid shift.

The last section regards the applied field (<code>RT Field1</code>). There change these two lines:

% Field1_Dir
0.000000 | 1.000000 | 0.000000 | # [RT Field1] Versor
%
Field1_kind= "DELTA" # [RT Field1] Kind(SIN|SOFTSIN|RES|ANTIRES|GAUSS|DELTA|QSSIN)

Run the calculation:

nohup yambo_nl -F Inputs_nl/04_td_sex.in -J "TD-SEX_nl,COLL_HSEX" -C TD-SEX_nl

== Output postprocessing: dielectric function ==

The post-processing of the output does not depend on the level of approximation. The same steps are taken as in the independent-particle approximation.

Use the command:
ypp_nl -F Inputs_nl/ypp_abs.in -J TD-SEX_nl -C TD-SEX_nl

Plot the results (compare with results obtained from the density-matrix formalism):
gnuplot> set xrange [3:10]
gnuplot> set yrange [0:12]
gnuplot> plot "TD-SEX_rt/o-TD-SEX_rt.YPP-eps_along_E" u 1:2 w l
gnuplot> rep "TD-SEX_nl/o-TD-SEX_nl.YPP-eps_along_E" u 1:2 w l
gnuplot> rep "TD-IP_rt/o-TD-IP_rt.YPP-eps_along_E" u ($1+3):2 w l

[[File:HBN HSEX absorption.png|thumb|900px|center|In plane absorption of hBN at the TD-HSEX level compared with IP absorption]]

Similarly to what we have seen in the independent-particle case, the differences between the two approaches are due to the accuracy in evaluating numerically the covariant dipoles in the Bloch-state dynamics approach.

''For comparison, the linear response results at the BSE level can be obtained following the [[How to obtain an optical spectrum|BSE tutorial]]. ''

= References =
<references/>