# Yambo tutorial: Quasiparticles in the GW approximation

In this tutorial you will learn how to run a GW simulation using Yambo on a HPC machine.

You will compute the quasiparticle corrections to the band structure of a free-standing single layer of MoS$$_2$$ while learning about convergence studies, parallel strategies, and GPU calculations.

In the end, you will obtain a quasiparticle band structure based on the simulations, the first step towards the reproduction of an ARPES spectrum. Beware: we will not use fully converged parameters, so the final result should not be considered very accurate.

MoS$$_2$$ monolayer (top and side views). Gray: Mo atoms, yellow: S atoms.

## Many-body corrections to the DFT band gap

We want to describe the electronic energy levels using a better description of electron-electron interactions than DFT is capable of.

Essentially, we want to solve the non-linear quasiparticle equation at first order in the GW self-energy $$\Sigma$$:

$E^{QP}_{nk}=\epsilon_{nk}+Z_{nk}[\Sigma]\langle\psi_{nk}|\Sigma(\epsilon_{nk})-V_{xc}|\psi_{nk}\rangle$

Here $$\epsilon_{nk}$$ and $$\psi_{nk}$$ are the Kohn-Sham energies and wavefunctions, respectively, while $$V_{xc}$$ is the DFT exchange-correlation potential.

For each electronic state $$nk$$, the self-energy can be separated into two components: a static, gap-opening term called the exchange self-energy ($$\Sigma^x$$), and an energy-dependent, usually gap-closing term called the correlation self-energy ($$\Sigma^c$$). These contributions are tackled separately by the code:

$\Sigma_{nk}(\omega)=\Sigma_{nk}^x+\Sigma^c_{nk}(\omega)$

The energy-dependent dynamical electronic screening $$\varepsilon^{-1}(\omega)$$ is included in $$\Sigma^c$$ and must therefore be calculated as well.

In this way, we can compute the “quasiparticle” corrections $$E^{QP}_{nk}$$ to the single-particle Kohn-Sham eigenvalues $$\epsilon_{nk}$$. The typical workflow for a GW calculation is:

## Set up a Yambo calculation

Go to your user work directory and copy the material from the shared folder

cp $WORK/MoS2_HPC_tutorial_Leonardo.tar.gz .  (as an alternative you can download the materials for the tutorial, 1.2GB, it may take a couple of minutes): wget https://media.yambo-code.eu/educational/tutorials/files/MoS2_HPC_tutorial_Leonardo.tar.gz  After one of the two commands, just extract the data tar -xvzf MoS2_HPC_tutorial_Leonardo.tar.gz  You can now enter the tutorial directory cd MoS2_HPC_tutorial_Leonardo  ### Yambo SAVE folder First of all, we need to convert some of the data produced in a previous non-self-consistent DFT calculation (in this case using Quantum ESPRESSO) into a convenient format for Yambo. The QE save folder for MoS$$_2$$ is already present at 00_QE-DFT. We should move inside it and then run the p2y executable to generate the uninitialised SAVE. But first, we need to access a node interactively: srun --nodes=1 --ntasks-per-node=1 --gres=gpu:1 --cpus-per-task=8 --mem=490000 --account=EUHPC_TD02_030 --partition=boost_usr_prod --qos=boost_qos_dbg --time=0:30:00 --pty /bin/bash  Then we need to load the yambo-specific modules in in the cluster. On Leonardo Booster, we have module load profile/chem-phys module load yambo/5.2.0--openmpi--4.1.4--nvhpc--23.1  Finally: cd 00_QE-DFT/mos2.save mpirun -np 1 p2y  Now, we need to run the initialization step. Every Yambo run must start with this step. This will be automatically performed when you run Yambo on a SAVE directory for the first time. Just type mpirun -np 1 yambo  and then check the yambo log called l_setup. The initialization step determines the $$G$$-vector shells and the $$k$$- and $$q$$-point grids based on the DFT calculation. If you check inside the SAVE you will see two types of databases. The static ones, starting with ns.*, are directly converted in the p2y step, while the dynamical ones, ndb.* are generated during the initialisation. ls SAVE/  ndb.gops # info on G-vector shells, etc ndb.kindx # info and k and q meshes ns.db1 # info on geometry and KS bands ns.kb_pp_pwscf # pseudopotential info ns.wf # wave functions info ...  The databases are written in netCDF format. Yambo has produced also a human readable output, r_setup, reporting relevant information such as lattice parameters, symmetries, atomic positions, k-points, DFT eigenvalues and band gaps. We can have a quick look at the sections. vim r_setup  [02.01] Unit cells # Lattice geometry info ================== ... [02.02] Symmetries # Symmetry ops. written explicitly ================== ... [02.03] Reciprocal space # Reciprocal lattice info ======================== ... [02.04] K-grid lattice # k-point coords. written explicitly ====================== ... [02.05] Energies & Occupations # Info on band gaps and occupations ============================== # DFT eigenvalues ... [03] Transferred momenta grid and indexing # q-points (momentum transfer grid) ==========================================  Finally, let us move the SAVE and the report file to the directory where we will run the first GW calculation. mv SAVE r_setup ../../01_GW_first_run/ cd ../../01_GW_first_run/  ### Yambo input file Now that we have a working SAVE, it is time to generate the input file we will be using for our first GW calculation. This can be done by the yambo executable via command-line instructions. If you type yambo -h  You will get a list of the possibile options:  ___ __ _____ __ __ _____ _____ | Y || _ || Y || _ \ | _ | | | ||. | ||. ||. | / |. | | \ _/ |. _ ||.\ / ||. _ \ |. | | |: | |: | ||: | ||: | \|: | | |::| |:.|:.||:.|:.||::. /|::. | --" -- --"-- --"-----" -----" 'A shiny pot of fun and happiness [C.D.Hogan]' This is : yambo Version : 5.1.0 Revision 21422 Hash fde6e2a07 Configuration: MPI+OpenMP+SLK+HDF5_MPI_IO Help & version: -help (-h) <string> :<string> can be an option (e.g. -h optics) -version :Code version & libraries Input file & Directories: -Input (-F) <string> :Input file -Verbosity (-V) <string> :Input file variables verbosity (more with -h Verbosity) -Job (-J) <string> :Job string -Idir (-I) <string> :Input directory -Odir (-O) <string> :I/O directory -Cdir (-C) <string> :Communication directory Parallel Control: -parenv (-E) <string> :Environment Parallel Variables file -nompi :Switch off MPI support -noopenmp :Switch off OPENMP support Initializations: -setup (-i) :Initialization -coulomb (-r) :Coulomb potential Response Functions: -optics (-o) <string> :Linear Response optical properties (more with -h optics) -X (-d) <string> :Inverse Dielectric Matrix (more with -h X) -dipoles (-q) :Oscillator strenghts (or dipoles) -kernel (-k) <string> :Kernel (more with -h kernel) Self-Energy: -hf (-x) :Hartree-Fock -gw0 (-p) <string> :GW approximation (more with -h gw0) -dyson (-g) <string> :Dyson Equation solver (more with -h dyson) -lifetimes (-l) :GoWo Quasiparticle lifetimes Bethe-Salpeter Equation: -Ksolver (-y) <string> :BSE solver (more with -h Ksolver) Total Energy: -acfdt :ACFDT Total Energy Utilites: -Quiet (-Q) :Quiet input file creation -fatlog :Verbose (fatter) log(s) -DBlist (-D) :Databases properties -walltime <int> :Walltime (more with -h walltime) -memory <int> :Memory (more with -h memory) -slktest :ScaLapacK test YAMBO developers group (http://www.yambo-code.org)  In order to build our input, we need to use the options for a GW calculation. We want to use the plasmon pole approximation for the dynamical screening, solve the quasiparticle equation with the Newton method, and add a truncation of the Coulomb potential which is useful for 2D systems. In addition, we want to set up explicitly the parameters for parallel runs. Therefore we type: yambo -gw0 p -g n -r -V par -F gw.in  Now we can exit the computing node, since all remaining calculations will be run by submitting jobs with the slurm scheduler. exit  You can now inspect the input file gw.in and try to familiarize with some of the parameters. The input will come with default values for many parameters that we might need to change. We discuss them below step by step. ### Parameters for a GW calculation We start with the runlevels: gw0 # [R] GW approximation ppa # [R][Xp] Plasmon Pole Approximation for the Screened Interaction el_el_corr # [R] Electron-Electron Correlation dyson # [R] Dyson Equation solver rim_cut # [R] Coulomb potential HF_and_locXC # [R] Hartree-Fock em1d # [R][X] Dynamically Screened Interaction  Runlevels • The [R] keyword refers to the runlevels: these flag tell Yambo which parts of the code should be executed. Each runlevel enables its own set of input variables. In particular here we have: • rim_cut: Coulomb potential random integration method and cutoff (enables [RIM] and [CUT] variables). • gw0: Yambo learns that it has to run a GW calculation (enables [GW] variables). • HF_and_locXC: calculation of exchange part of the self-energy $$\Sigma^x$$ (i.e., Hartree-Fock approximation). • em1d: enables the calculation of the dynamical screening of the electrons, i.e. the dielectric matrix ([X] variables). In this way Yambo can go beyond Hartree-Fock and compute $$\Sigma^c$$. • ppa: tells Yambo that the dynamical screening should be computed in the plasmon pole approximation ([Xp] variables). • dyson: Yambo will solve the Dyson-like quasiparticle equation. Going through the file we find: EXXRLvcs= 37965 RL # [XX] Exchange RL components VXCRLvcs= 37965 RL # [XC] XCpotential RL components  Recall that we have, for the exchange self-energy: $\Sigma^x_{nk} = - \sum_G\sum_v \int \frac{d^3q}{2\pi^3} v(q+G)|\rho_{nv}(k,q,G)|^2f_{vk-q}$ Exchange self-energy • EXXRLvcs controls the number of Reciprocal Lattice vectors (i.e., G-vectors) used to build the exchange self-energy, while VXCRLvcs does the same for the exchange-correlation potential reconstructed from DFT. Since these two quantities are to be subtracted, it is important to keep the same values here (and possibly not change the default maximum value). Let us now have a look at the parameters for the calculation of the correlation part of the self-energy. Recall that we have: $\Sigma^c_{nk} = i \sum_m \int \frac{d^3q}{2\pi^3} \sum_{GG'} v(q+G) \rho_{nmk}(q,G) \rho^*_{nmk}(q,G') \int d\omega' G^0_{mk-q}(\omega-\omega')\varepsilon^{-1}_{GG'}(q,\omega')$ (Here, the $$\rho$$-terms represent the screening matrix elements which are computed separately by yambo and stored in their own database.) The calculation is divided in two steps. First, the response function in the plasmon pole approximation (em1d ppa), under the keywords [X] and [Xp], i.e., $$\varepsilon^{-1}_{GG'}(q,\omega)$$. Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc % BndsRnXp 1 | 300 | # [Xp] Polarization function bands % NGsBlkXp= 1 RL # [Xp] Response block size % LongDrXp 1.000000 | 0.000000 | 0.000000 | # [Xp] [cc] Electric Field % PPAPntXp= 27.21138 eV # [Xp] PPA imaginary energy XTermKind= "none" # [X] X terminator ("none","BG" Bruneval-Gonze)  Response function • Chimod= "Hartree" indicates that we compute the response function in the Random Phase Approximation (RPA). • BndsRnXp represents the electronic states included in the response function $$\varepsilon$$, and is a convergence parameter. • NGsBlkXp is the number of G-vectors used to calculate the RPA response function $$\varepsilon^{-1}_{GG'}$$. It is a convergence parameter and can be expressed in number of reciprocal lattice vectors (RL) or energy (Ry, suggested). • LongDrXp represents the direction of the long-range auxiliary external electric field used to compute $$\varepsilon^{-1}_{GG'}(q)$$ at $$q,G\rightarrow 0$$. In general you have to be mindful of the system symmetries. In our case, we will put 1 | 1 | 1 to cover all directions. • PPAPntXp= 27.21138 eV is the energy of the plasmon pole. We don’t normally change this. • XTermKind is used to specify a “terminator”: this accelerates numerical convergence with respect to the number of bands BndsRnXp. Next, we have the [GW] group of parameters controlling the next part of the correlation self-energy calculation: % GbndRnge 1 | 300 | # [GW] G[W] bands range % GTermKind= "none" # [GW] GW terminator ("none","BG" Bruneval-Gonze,"BRS" Berger-Reining-Sottile) DysSolver= "n" # [GW] Dyson Equation solver ("n","s","g") %QPkrange # [GW] QP generalized Kpoint/Band indices 1|7|1|300| %  Correlation self-energy • GbndRnge is the number of bands used to build the correlation self-energy. It is a convergence parameter and can be accelerated with the terminator GTermKind. • DysSolver="n" specifies the method used to solve the linearised quasiparticle equation. In most cases, we use the Newton method "n". • QPkrange indicates the range of electronic (nk) states for which the GW correction $$\Sigma_{nk}$$ is computed. The first pair of numbers represents the range of k-point indices, the second pair the range of band indices. We now take a look at the parameters relative to the Coulomb interaction at small momenta and for 2D systems, which we should edit now once and for all. RandQpts=1000000 # [RIM] Number of random q-points in the BZ RandGvec= 100 RL # [RIM] Coulomb interaction RS components CUTGeo= "slab z" # [CUT] Coulomb Cutoff geometry: box/cylinder/sphere/ws/slab X/Y/Z/XY..  Coulomb potential • The [RIM] keyword refers to a Monte Carlo random integration method performed to avoid numerical instabilities close to $$q=0$$ and $$G=0$$ in the $$q$$-integration of the bare Coulomb interaction - i.e. $$4\pi/(q+G)^2$$ - for 2D systems. • The [CUT] keyword refers to the truncation of the Coulomb interaction to avoid spurious interaction between periodically repeated copies of the simulation supercell along the $$z$$-direction (we are working with a plane-wave code). Keep in mind that the vacuum space between two copies of the system should be converged: here we are using 20 bohr but a value of 40 bohr would be more realistic. Finally, we have the parallel parameters. We are going to discuss them at the end of the parallel section, we can skip them for now. X_and_IO_CPU= "" # [PARALLEL] CPUs for each role X_and_IO_ROLEs= "" # [PARALLEL] CPUs roles (q,g,k,c,v) X_and_IO_nCPU_LinAlg_INV=-1 # [PARALLEL] CPUs for Linear Algebra (if -1 it is automatically set) X_Threads=0 # [OPENMP/X] Number of threads for response functions DIP_CPU= "" # [PARALLEL] CPUs for each role DIP_ROLEs= "" # [PARALLEL] CPUs roles (k,c,v) DIP_Threads=0 # [OPENMP/X] Number of threads for dipoles SE_CPU= "" # [PARALLEL] CPUs for each role SE_ROLEs= "" # [PARALLEL] CPUs roles (q,qp,b) SE_Threads=0 # [OPENMP/GW] Number of threads for self-energy  In a GW calculation, the most important parameters to be numerically converged are: • kpoint mesh (requires multiple nscf DFT runs) • BndsRnXp • NGsBlkXp • GbndRnge • [2D system]: vacuum separation with Coulomb cutoff (requires multiple scf+nscf DFT runs) From the above discussion you can easily guess that many-body perturbation theory calculations are much more numerically expensive than DFT calculations. ## The first run We will start by running a single GW calculation. Here we will focus on the magnitude of the quasiparticle gap. This means that we only need to calculate two quasi-particle corrections, i.e., valence and conduction bands at the k-point where the minimum gap occurs. This information can be found by inspecting the report file r_setup produced when the SAVE folder was initialised. Just search for the string ‘Direct Gap’ and you’ll see that the latter occurs at k-point 7 between bands 13 and 14:  [X] Filled Bands : 13 [X] Empty Bands : 14 300 [X] Direct Gap : 1.858370 [eV] [X] Direct Gap localized at k : 7  In addition, we will set the number of bands in BndsRnXp and GbndRnge to a small value, just to have it run fast. Hence, we modify the input file accordingly (check BndsRnXp, GbndRnge, LongDrXp, QPkrange): rim_cut # [R] Coulomb potential gw0 # [R] GW approximation ppa # [R][Xp] Plasmon Pole Approximation for the Screened Interaction dyson # [R] Dyson Equation solver HF_and_locXC # [R] Hartree-Fock em1d # [R][X] Dynamically Screened Interaction X_Threads=0 # [OPENMP/X] Number of threads for response functions DIP_Threads=0 # [OPENMP/X] Number of threads for dipoles SE_Threads=0 # [OPENMP/GW] Number of threads for self-energy RandQpts=1000000 # [RIM] Number of random q-points in the BZ RandGvec= 100 RL # [RIM] Coulomb interaction RS components CUTGeo= "slab z" # [CUT] Coulomb Cutoff geometry: box/cylinder/sphere/ws/slab X/Y/Z/XY.. % CUTBox 0.000000 | 0.000000 | 0.000000 | # [CUT] [au] Box sides % CUTRadius= 0.000000 # [CUT] [au] Sphere/Cylinder radius CUTCylLen= 0.000000 # [CUT] [au] Cylinder length CUTwsGvec= 0.700000 # [CUT] WS cutoff: number of G to be modified EXXRLvcs= 37965 RL # [XX] Exchange RL components VXCRLvcs= 37965 RL # [XC] XCpotential RL components Chimod= "HARTREE" # [X] IP/Hartree/ALDA/LRC/PF/BSfxc % BndsRnXp 1 | 20 | # [Xp] Polarization function bands % NGsBlkXp= 1 RL # [Xp] Response block size % LongDrXp 1.000000 | 1.000000 | 1.000000 | # [Xp] [cc] Electric Field % PPAPntXp= 27.21138 eV # [Xp] PPA imaginary energy XTermKind= "none" # [X] X terminator ("none","BG" Bruneval-Gonze) % GbndRnge 1 | 20 | # [GW] G[W] bands range % GTermKind= "none" # [GW] GW terminator ("none","BG" Bruneval-Gonze,"BRS" Berger-Reining-Sottile) DysSolver= "n" # [GW] Dyson Equation solver ("n","s","g") %QPkrange # [GW] QP generalized Kpoint/Band indices 7|7|13|14| %  We are now ready to run this calculation. Since you should never run a Yambo calculation on the login node, we will need a submission script to add our job to the queue. A submission script optimized for Leonardo Booster (running on GPUs) is provided as an example. Modify it to suit your specific machine. vim run_first_job.sh  #!/bin/bash #SBATCH --nodes=1 #SBATCH --ntasks-per-node=2 #SBATCH --cpus-per-task=8 #SBATCH --partition=boost_usr_prod #SBATCH --time=0:05:00 #SBATCH --gres=gpu:2 #SBATCH --account=EUHPC_TD02_030 #SBATCH --job-name=first_job export OMP_NUM_THREADS=8 export OMP_PLACES=cores export OMP_PROC_BIND=close # load yambo module load profile/chem-phys module load yambo/5.2.0--openmpi--4.1.4--nvhpc--23.1 # run yambo mpirun -np${SLURM_NTASKS} --map-by socket:PE=8 --rank-by core \
yambo -F gw.in -J job_00_first_run -C out_00_first_run


We will ignore all details regarding parallelization, as it will be covered in the next section. Since there are no lowercase flags after yambo, it is not going to generate an input file, but rather, run the one specified by -F. Now, go ahead an submit this job

sbatch run_first_job.sh


The status of the jobs can be monitored via:

squeue -u $USER # to inspect the status of jobs # (hint: make a unix alias, if you like) scancel <jobid> # to delete jobs in the queue  The newly generated databases will be stored in the job directory, as specified by -J, while the report, log and output files will be stored in the communications directory (-C). As this is your first yambo run, take a moment to inspect the report and log files, which you can find inside the -C directory. In these report and log files, you can see the steps performed by yambo. For instance, the code calculates the screening at every k-point and stores it in the PPA database called ndb.pp. By opening the report vim out_00_first_run/r-job_00_first_run_HF_and_locXC_gw0_dyson_rim_cut_em1d_ppa  you will see [07] Dynamic Dielectric Matrix (PPA) ==================================== [WR./job_00_first_run//ndb.pp]-------------------------------  Then, the actual GW section will use this calculated dielectric screening to construct the correlation part of the self-energy: [09.01] G0W0 (W PPA) ==================== [ GW ] Bands range : 1 20 [GW/PPA] G damping : 0.100000 [eV] QP @ state[ 1 ] K range: 7 7 QP @ state[ 1 ] b range: 13 14 [RD./job_00_first_run//ndb.pp]-------------------------------  Now, inspect the output file vim out_00_first_run/o-job_00_first_run.qp  # Vxc =Slater exchange(X)+Perdew & Zunger(C) # Vnlxc=Hartree-Fock # # K-point Band Eo [eV] E-Eo [eV] Sc|Eo [eV] # 7 13 0.000000 -0.025774 0.543987 7 14 1.858370 3.496193 -0.417555 #  In this file, Eo is our starting point (DFT) while E-Eo shows the GW correction one should apply to obtain the quasi-particle energies. In order to calculate the gap (automatically from the command line), we’ll use some simple commands. First, we get everything that is not a # symbol grep -v '#' and we pass that to another command with a “pipe” |. Then, tail -n 1/head -n 1 will retain the first/last line, and awk '{print$3+$4}' will get us the sum of the third and fourth columns. Altogether, this would be as follows grep -v '#' out_00_first_run/o-job_00_first_run.qp|head -n 1| awk '{print$3+$4}' -0.025774 grep -v '#' out_00_first_run/o-job_00_first_run.qp|tail -n 1| awk '{print$3+$4}' 5.35456  These two commands give us the quasiparticle energies we’ve calculated - their difference is the GW-corrected optical gap. ## GW convergence In this part of the tutorial, we will study convergence with respect to some of the parameters mentioned above. In order to complete this tutorial within a single hands-on session, we will restrict ourselves to a very coarse $$k$$-point grid. Hence, we will perform our convergence studies on top of a DFT calculation done with a $$6 \times 6 \times 1$$ k-point grid and without spin-orbit coupling: the SAVE we generated earlier. While this will speed up calculations and could be run even on a single GPU card, you should be aware that such coarse sampling of the BZ is significantly underconverged and should only be used for educational purposes. In addition, spin-orbit interaction is extremely relevant for the valley physics of MoS2 and should not be neglected in realistic calculations. Let’s move into the appropriate directory cd ../02_GW_convergence  ### Response function $$\varepsilon^{-1}_{GG'}$$ - Bands and G-vectors We are now ready to start our convergence tests. We’ll begin with the variables controlling the polarization function, i.e., NGsBlkXp for the number of G-vectors and BndsRnXp for the number of bands. For this, we will keep GbndRnge constant at a reasonably high value - you can inspect the input i01-GW vim i01-GW  and check that you have: % GbndRnge 1 | 80 | # [GW] G[W] bands range %  Since we need to run yambo for several values of NGsBlkXp and BndsRnXp, it makes sense to use two nested loops. That is exactly what we did in the submission script run01_converge_pol.sh. Since this will take a few minutes, save time by submitting it straight away and we will have a look at it while it runs: sbatch run01_converge_pol.sh  Monitoring You can monitor that the job is running by the squeue command sbatch squeue -u$USER


and also by checking the files created in your folder

ls -ltr

i01-GW_Xp_20_bands_6_Ry
job_Xp_20_bands_6_Ry
out_Xp_20_bands_6_Ry
i01-GW_Xp_20_bands_8_Ry
out_Xp_20_bands_8_Ry
job_Xp_20_bands_8_Ry
i01-GW_Xp_20_bands_10_Ry
out_Xp_20_bands_10_Ry
job_Xp_20_bands_10_Ry
i01-GW_Xp_20_bands_12_Ry
out_Xp_20_bands_12_Ry
job_Xp_20_bands_12_Ry
summary_01_20bands.txt
i01-GW_Xp_40_bands_6_Ry
...


Finally you can monitor how runs are proceeding by looking into the log files

tail -f out_Xp_*_bands_*/LOG/*_1

==> out_Xp_20_bands_6_Ry/LOG/l-job_Xp_20_bands_6_Ry_HF_and_locXC_gw0_dyson_rim_cut_em1d_ppa_CPU_1 <==
<15s> P1: [TIMING]                                          io_WF :      1.1353s CPU (34 calls,   0.033 sec avg)
<15s> P1: [TIMING]                                    WF_load_FFT :      1.1538s CPU ( 7 calls,   0.165 sec avg)
<15s> P1: [TIMING]                                    io_KB_pwscf :      1.1918s CPU ( 6 calls,   0.199 sec avg)
<15s> P1: [TIMING]                              DIPOLE_transverse :      2.4771s CPU
<15s> P1: [TIMING]                                    io_fragment :      2.6220s CPU (46 calls,   0.057 sec avg)
<15s> P1: [TIMING]                                           io_X :      3.1953s CPU (19 calls,   0.168 sec avg)
<15s> P1: [TIMING]                                        Dipoles :      4.4012s CPU
<15s> P1: [11] Memory Overview
<15s> P1: [12] Game Over & Game summary
<15s> P1: [TIMING]            [Time-Profile]: 15s

==> out_Xp_20_bands_8_Ry/LOG/l-job_Xp_20_bands_8_Ry_HF_and_locXC_gw0_dyson_rim_cut_em1d_ppa_CPU_1 <==
<11s> P1: X@q[6] |                                        | [000%] --(E) --(X)
<11s> P1: X@q[6] |########################################| [100%] --(E) --(X)
<11s> P1: [PARALLEL distribution for RL vectors(X) on 2 CPU] Loaded/Total (Percentual):31878/64009(50%)
<11s> P1: [PARALLEL distribution for RL vectors(X) on 2 CPU] Loaded/Total (Percentual):18975/64009(30%)
<11s> P1: [X-CG] R(p) Tot o/o(of R):   153   504   100
<11s> P1: Xo@q[7] |                                        | [000%] --(E) --(X)
<11s> P1: Xo@q[7] |########################################| [100%] --(E) --(X)
<11s> P1: [PARALLEL distribution for X Frequencies on 1 CPU] Loaded/Total (Percentual):2/2(100%)
<11s> P1: X@q[7] |                                        | [000%] --(E) --(X)
<11s> P1: X@q[7] |########################################| [100%] --(E) --(X)
<11s> P1: [PARALLEL distribution for RL vectors(X) on 2 CPU] Loaded/Total (Percentual):31878/64009(50%)
<11s> P1: [08] Local Exchange-Correlation + Non-Local Fock


Let’s now have look into the job we just submitted.

vim run01_converge_pol.sh


First, we defined the double loop and we intialize a summary file for each iteration of the outer loop by printing a header to it. The input file i01-GW is used as a template for every calculation in the loops, so we assign it to a variable.

file0='i01-GW'

for POL_BANDS in 20 40 60 80; do

echo 'NGsBlkXp [Ry]   E_vale [eV]   E_cond [eV]' > summary_01_${POL_BANDS}bands.txt for NGsBlkXp_Ry in 6 8 10 12; do (...) done done  Inside the loops, we generate some useful labels which will come in handy to distinguish between runs. Then, we pass the variables from the loops to the sed command, in order to generate new files in an automated way (sed replaces any matching string with whatever is provided by the loop variable). Next, we run yambo using the labels to specify different job -J and communication -C directories every time. Finally, we get the quasiparticle energies with grep commands as shown before and append a new line to the summary file. So, inside each loop, we have label=Xp_${POL_BANDS}_bands_${NGsBlkXp_Ry}_Ry jdir=job_${label}
cdir=out_${label} filein=i01-GW_${label}

sed "s/NGsBlkXp=.*/NGsBlkXp=${NGsBlkXp_Ry} Ry/; /% BndsRnXp/{n;s/.*/ 1 |${POL_BANDS} |/}" $file0 >$filein

# run yambo
mpirun -np ${SLURM_NTASKS} --map-by socket:PE=8 --rank-by core yambo -F$filein -J $jdir -C$cdir

E_GW_v=grep -v '#' ${cdir}/o-${jdir}.qp|head -n 1| awk '{print $3+$4}'
E_GW_c=grep -v '#' ${cdir}/o-${jdir}.qp|tail -n 1| awk '{print $3+$4}'

echo ${NGsBlkXp_Ry} ' '${E_GW_v} '        ' ${E_GW_c} >> summary_01_${POL_BANDS}bands.txt


Finally, let us plot this data. First, check that the job has finished with

squeue -u $USER  [$USER@login01 02_GW_convergence]$squeue -u$USER
JOBID PARTITION     NAME     USER ST       TIME  NODES NODELIST(REASON)
[$USER@login01 02_GW_convergence]$


and verify that the energies were extracted correctly by inspecting the summary files. Remember to load the python module if you haven’t done so yet, and then plot:

module load anaconda3/2023.03
python plot-01.py


The plot will produce a fig-01.png file. You can copy and open it in your local machine with something like

[Run this on another terminal in your local machine, fixing $USER,$LOGIN and $TUTORIALPATH] scp$USER@$LOGIN:$TUTORIALPATH/MoS2_HPC_tutorial_Leonardo/02_GW_convergence/fig-01.png ./


You should get:

For the purpose of the tutorial, we will choose 80 bands and 10 Ry as our converged parameters and move on. An error within 10 meV is usually acceptable. To retain the chosen variables, we’ll make a copy of the corresponding input file:

cp i01-GW_Xp_80_bands_10_Ry i02-GW


### Self-energy $$\Sigma^c$$ - Bands

We will now proceed to converge the number of bands for the correlation part of the self-energy, i.e., GbndRnge. This step is actually simpler, since it only involves one loop. This is coded in the provided script run02_converge_Gbnds.noBG.sh. You can look into it

vim run02_converge_Gbnds.noBG.sh


and go ahead and submit it.

sbatch run02_converge_Gbnds.noBG.sh


While this runs, check if the previous job has finished, i.e., you should have a complete summary_02_noBG.txt file by now. For a visual result, proceed to plot them with

python plot-02.py


You should get

We will end the convergence part of the tutorial with an important consideration about k-points convergence. The latter is the most cumbersome and computationally intensive among the various convergence tests, and it involves re-running the DFT step. For this reason (and for this reason only) it was ignored in this tutorial. However, it absolutely cannot be overlooked since it is crucial for the accuracy of the calculated GW corrections. You can read about k-points convergence in GW and, importantly, a very efficient workaround for 2D systems in a recent publication (here). MoS$$_2$$ was one of the materials studied there, and it shows that our result, obtained with a $$6 \times 6 \times 1$$ k-grid, is simply off the chart (blue line).

Guandalini, D’Amico, Ferretti & Varsano. npj Comput Mater 9

## GW parallel strategies

### MPI parallelization

For this section, let us enter the 03_GW_parallel directory. If you were in the 02_GW_convegence folder just do

cd ../03_GW_parallel


and inspect the input gw.in. You will see that we set low values for most of the convergence parameters except bands:

vim gw.in

FFTGvecs= 40       Ry    # [FFT] Plane-waves
EXXRLvcs= 2        Ry    # [XX] Exchange RL components
VXCRLvcs= 2        Ry    # [XC] XCpotential RL components
% BndsRnXp
1 |  300 |               # [Xp] Polarization function bands
%
NGsBlkXp= 1            Ry    # [Xp] Response block size
% GbndRnge
1 |  300 |               # [GW] G[W] bands range
%
%QPkrange                    # [GW] QP generalized Kpoint/Band indices
1| 19| 23| 30|
%


Note that we also added FFTGvecs to reduce the size of the Fourier transforms (the default corresponds to Quantum ESPRESSO ecutwfc, i.e. 60 Ry in this case).

In addition, we have deleted all the parallel parameters since we will be setting them via the submission script.

Actually we are now dealing with a heavier system than before: as you can see from the QPkrange values, we have switched to a 12x12x1 k-point grid - having 19 points in the irreducible Brillouin zone - and turned spin-orbit coupling on in the DFT calculation (now the top valence band is number 26 instead of 13 because the bands are spin-polarized).

For this part of the tutorial, we will be using the slurm submission script job_parallel.sh, which is available in the calculation directory. If you inspect it, you will see that the script adds additional variables to the yambo input file. These variables control the parallel execution of the code:

DIP_CPU= "1 $ngpu 1" # [PARALLEL] CPUs for each role DIP_ROLEs= "k c v" # [PARALLEL] CPUs roles (k,c,v) DIP_Threads= 0 # [OPENMP/X] Number of threads for dipoles X_and_IO_CPU= "1 1 1$ngpu 1" # [PARALLEL] CPUs for each role
X_and_IO_ROLEs= "q g k c v"   # [PARALLEL] CPUs roles (q,g,k,c,v)
X_and_IO_nCPU_LinAlg_INV=1    # [PARALLEL] CPUs for Linear Algebra (if -1 it is automatically set)
SE_CPU= "1 \$ngpu 1"           # [PARALLEL] CPUs for each role
SE_ROLEs= "q qp b"            # [PARALLEL] CPUs roles (q,qp,b)


The keyword DIP refers to the calculations of the screening matrix elements (also called “dipoles”) needed for the screening function, X is the screening function itself (it stands for $$\chi$$ since it is a response function), SE the self-energy. These three sections of the code can be parallelised independently.

We are running on GPUs. In particular, each node hosts four GPU cards. Yambo is coded in such a way that each MPI task is run on a single card, therefore ntasks=ngpu.

Note

• In this subsection we are mainly concerned with the [PARALLEL] variables which refer to MPI tasks (distributed memory).

• What about [OPENMP] parallelisation (i.e., adding threads with shared memory)? When Yambo is run on GPUs, the explicit threading that you can set in the input and submission script only applies to the very few sections of the code that are not run on GPU cards, but stay on the CPUs. Therefore, in a GPU calculation, CPU-only threads are not going to be a relevant factor in the performance of the code. We keep them fixed to 8 since on Leonardo Booster (32 CPUs and 4 GPUs per node) the best hybrid parallel setup for CPUs is 4 tasks times 8 threads. We will see an example of the impact of threads in a CPU-only calculation later.

We start by calculating the QP corrections using 4 MPI tasks / GPUs. We leave the number of openMP threads at 8, the optimized value for Yambo on Leonardo. Therefore, edit the submission script as:

#!/bin/bash
#SBATCH --nodes=1
...
#SBATCH --gres=gpu:4


and submit the job

sbatch job_parallel.sh


This will create a new input file and run it. The calculation databases and the human-readable files will be put in separate directories. Check the location of the report r-* file and the log l-* files, and inspect them while the calculation runs. For simplicity you can just type

tail -f run_MPI4_OMP8.out/LOG/l-*_CPU_1


to monitor the progress in the master thread (Ctrl+c to exit). As you can see, the run takes some time, even though we are using minimal parameters.

Meanwhile, we can run other jobs increasing the parallelisation. Let’s employ 16 MPI tasks / GPUs (i.e., 4 nodes on Leonardo). To this end modify the job_parallel.sh script changing

#!/bin/bash
#SBATCH --nodes=4
...
#SBATCH --gres=gpu:4


This time the code should be much faster. Once the run is over, try to run the simulation also on 8 MPI tasks by changing nodes appropriately. Finally, you can try to produce a scaling plot.

The timings are contained in the r-* report files. You can already have a look at them typing

grep Time-Profile run_MPI*/r-*


The python script parse_ytiming.py is useful for the post-processing of report files. You can already find it in the directory, together with the reports for the longer calculations with 1 and 2 MPI tasks which have been provided.

module load anaconda3/2023.03


Then, after your jobs have finished, run the script as

python parse_ytiming.py run_MPI


to look for a report file in each run_MPI*.out folder. Make sure you have only one report file per folder. You can also play with the script to make it print detailed timing information, however you should already see that it produced a png plot showing times-to-completion on y axis against number of MPI tasks (i.e., GPUs in this case) on the x axis.

What can we learn from this plot? In particular, try to answer the following questions:

• Up to which number of MPI tasks our system scales efficiently?

• How can we decide at which point adding more nodes to the calculation becomes a waste of resources?

Note

Keep in mind that the MPI scaling we are seeing here is not the true Yambo scaling, but depends on the small size of our tutorial system. In a realistic calculation for a large-sized system, Yambo has been shown to scale well up to tens of thousands of MPI tasks! (See the next optional box for an example)

.

## Full GW band structure

This is the final section of the tutorial, in which we want to compute the full correction to the band structure of single-layer MoS$$_2$$.

This is a massive calculation, so run it right now and we’ll discuss it in the meantime:

cd ../04_GW_bands
sbatch gpu_job.sh


In order to get somewhat realistic results, we will use the larger values for the convergence parameters we have identified in the convergence section. In addition, we also increased the vacuum separation (to 30 au) and the k-point mesh (to 18x18x1) in the DFT calculation, and of course we consider spin-orbit coupling.

Now we have a heavier calculation, and we have to do it not just for the band gap, but the entire band structure which includes 37 kpoints in the irreducible Brillouin zone, two spin-orbit-split valence bands, and two spin-orbit-split conduction bands. Let us check the new input:

vim gw.in

%QPkrange                        # [GW] QP generalized Kpoint/Band indices
1|37|25|28|
%


After about 3 minutes the calculation should be over and the results collected in folder GW_bnds. The quasiparticle corrections are stored in human-readable form in the file o-GW_bnds.QP, and in netCDF format in the quasiparticle database ndb.QP.

In order to visualize the results in the form of a GW band structure, we will first interpolate the calculated points - recall that we have just 37 points, few of which lie on high-symmetry lines - with ypp, the yambo pre- and post-processing executable.

Note

We also have a python-based interface for advanced treatment of all the Yambo databases, called Yambopy. You can check it out here on the Yambo wiki.

Let us enter a computing node interactively

srun --nodes=1 --ntasks-per-node=1 --gres=gpu:1 --cpus-per-task=8 --mem=490000 --account=EUHPC_TD02_030 --partition=boost_usr_prod --qos=boost_qos_dbg --time=0:30:00 --pty /bin/bash


module load profile/chem-phys


We can review the options with ypp -h and generate an input file for band structure interpolation with

ypp -s b -F ypp_bands.in


Let us modify the resulting input file by selecting the ‘boltztrap’ approach to interpolation, the last two valence and first two conduction bands, and a path in the Brillouin zone along the the points $$\Gamma-M-K-\Gamma$$. We also set 100 points for each high-symmetry line.

electrons                        # [R] Electronic properties
bnds                             # [R] Bands
PROJECT_mode= "none"             # Instruct ypp how to project the DOS. ATOM, LINE, PLANE.
INTERP_mode= "BOLTZ"                # Interpolation mode (NN=nearest point, BOLTZ=boltztrap aproach)
INTERP_Shell_Fac= 20.00000       # The bigger it is a higher number of shells is used
INTERP_NofNN= 1                  # Number of Nearest sites in the NN method
OutputAlat= 5.90008             # [a.u.] Lattice constant used for "alat" ouput format
cooIn= "rlu"                     # Points coordinates (in) cc/rlu/iku/alat
cooOut= "rlu"                    # Points coordinates (out) cc/rlu/iku/alat
% BANDS_bands
25 | 28 |                         # Number of bands
%
CIRCUIT_E_DB_path= "none"        # SAVE obtained from the QE bands run (alternative to %BANDS_kpts)
BANDS_path= ""                   # High-Symmetry points labels (G,M,K,L...) also using composed positions (0.5xY+0.5xL).
BANDS_steps= 100                  # Number of divisions
#BANDS_built_in                # Print the bands of the generating points of the circuit using the nearest internal point
%BANDS_kpts                      # K points of the bands circuit
0.00000 |0.00000 |0.00000 |
0.00000 |0.50000 |0.00000 |
0.33333 |0.33333 |0.00000 |
0.00000 |0.00000 |0.00000 |
%


Now, let’s run ypp:

mpirun -np 1 ypp -F ypp_bands.in


This run will produce the file o.bands_interpolated. You can inspect it and see that it contains a plottable band structure, but beware: these are the DFT eigevalues! We didn’t tell ypp where to look for the quasiparticle corrections, so it went into the SAVE folder and interpolated the DFT data. Let’s rename the output:

mv o.bands_interpolated o.DFT_bands
mkdir DFT_bands
mv o.spin* o.magn* DFT_bands/


In order to interpolate the quasiparticle database, we append its location to the ypp input:

vim ypp_bands.in


add this line at the end

...
GfnQPdb= "E < ./GW_bnds/ndb.QP"


and run ypp again.

mpirun -np 1 ypp -F ypp_bands.in


When it’s done, let’s rename the new output as

mv o.bands_interpolated o.GW_bands
mkdir GW_bands
mv o.spin* o.magn* GW_bands/


Now we are ready to visualize the band structures. In order to do so, you can use the script plt_bands.py that should be already available in the directory.

module load anaconda3/2023.03


and run the script as

python plt_bands.py o.DFT_bands o.GW_bands 4


Now we can also exit the computing node

exit


The python script should have produced a GW_bands.png file containing the following visualization, which you can copy and open it in your local machine using scp`:

You may compare this plot with a converged result from this paper (also done with Yambo):

Dashed lines: DFT, thick lines: GW.

As you can see, the general result is not too bad, but there are some differences both at the DFT and GW levels. The magnitude of the band gap is too large, and the relative energy of the two conduction band minima is not correct. One obvious issue is the lack of convergence of our tutorial calculations. As we know, we should include more vacuum space and many, many more k-points. Additionally, this is a transition metal dichalcogenide: for this class of systems, the details of the band structure can strongly depend on small variations in the lattice parameters and on the type of pseudopotential used. A great deal of care must be taken when performing these calculations!

In order to learn more about Yambo, we suggest visiting the Yambo website. For technical information and tutorials, you can check out the Yambo wiki. If you have issues and questions about installing and running the code, you can write about them on the Yambo forum.