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tutorial is result of my personal notes while trying (which I still do) to learn Density Functional Theory calculations myself. I am no expert in this subject. I am sharing this notes here, just in case it helps you getting started. I will cite numerous other resources that I am following. Hope you will find this tutorial helpful. The quantum espresso input files, jupyter notebooks (containing python code for visualizations), and other source files related to this tutorial can be found on GitHub: pranabdas/espresso. You may clone the repository to your local machine: git clone https://github.com/pranabdas/espresso.git","s":"Density Functional Theory using Quantum Espresso","u":"/espresso/","h":"","p":1},{"i":4,"t":"Lately, I decided to follow specific pattern for the filenames, but you can choose whatever works best for you. Note that all example files in this tutorial does not follow this convention yet. {program}.{calculation}.{system}.{in, out} {program}.{calculation}.{system_description}.{in, out} {pw, pp, ...}.{scf, bands, ...}.{silicon, al_slab}.{in, out} Example: pw.bands.silicon.in → input file for the bands calculation using PWscf program for silicon. For PWTK scripts, we will use .pwtk extension (e.g., silicon_vc_relax.pwtk).","s":"Filename conventions","u":"/espresso/","h":"#filename-conventions","p":1},{"i":6,"t":"1 Bohr = 0.529177249 Å 1 Rydberg (Ry) = 13.6056981 eV. Angstrom to Bohr converter: lattice constants are often provided in angstrom, you can use following utility to convert to Bohr. Å = 1.8897259886 Bohr. Copy","s":"Unit conversions","u":"/espresso/","h":"#unit-conversions","p":1},{"i":8,"t":"Topological insulators are a special class of material that is insulating in the bulk, however exhibit conducting states in the surface. Bi2Se3 is such a material. Spin orbit coupling and breaking of the inversion symmetry at the surface of the crystal is crucial to the existence of the Dirac surface state. Here we will calculate the bandstructure step by step: first for the bulk, next including SOC, and finally for the slab. Please check the respective input files. I followed the following steps: # SCF calculation for bulk mpirun -np 24 pw.x -i pw.scf.bi2se3_01.in > pw.scf.bi2se3_01.out # bands calculation for bulk mpirun -np 24 pw.x -i pw.bands.bi2se3_01.in > pw.bands.bi2se3_01.out # post processing for bulk bands mpirun -np 24 bands.x -i pp.bands.bi2se3_01.in > pp.bands.bi2se3_01.out # for bulk with SOC mpirun -np 24 pw.x -i pw.scf.bi2se3_02.in > pw.scf.bi2se3_02.out mpirun -np 24 pw.x -i pw.bands.bi2se3_02.in > pw.bands.bi2se3_02.out mpirun -np 24 bands.x -i pp.bands.bi2se3_02.in > pp.bands.bi2se3_02.out # slab calculation mpirun -np 24 pw.x -i pw.scf.bi2se3_03.in > pw.scf.bi2se3_03.out mpirun -np 24 pw.x -i pw.bands.bi2se3_03.in > pw.bands.bi2se3_03.out mpirun -np 24 bands.x -i pp.bands.bi2se3_03.in > pp.bands.bi2se3_03.out # DOS mpirun -np 24 pw.x -i pw.nscf.bi2se3_04.in > pw.nscf.bi2se3_04.out mpirun -np 24 dos.x -i pp.dos.bi2se3_04.in > pp.dos.bi2se3_04.out For the slab calculation the periodicity of the lattice was broken along the c-axis to artificially add 10 Å vacuum. In above calculation electronic spin was not considered (meaning the states are degenerate with spin up and down). If starting_magnetization is set to zero (or not given) the code makes a spin-orbit calculation without spin magnetization. It assumes that time reversal symmetry holds and it does not calculate the magnetization. The states are still two-component spinors but the total magnetization is zero. Notice that for the Dirac surface states the gap did not completely close at the Fermi energy. This is possibly due to finite size effect. We could repeat the calculation with larger vacuum, and see what happens. Also the Fermi energy estimation seems incorrect. In order to sample the Γ\\GammaΓ point for our DOS calculation, an odd k-grid mesh (25✕25✕5) was used. The signature of Dirac cone is evident from the DOS figure.","s":"Bandstructure of topological insulating Bi2Se3","u":"/espresso/hands-on/Bi2Se3","h":"","p":7},{"i":10,"t":"https://docs.quantumatk.com/tutorials/topological_insulator_bi2se3/","s":"Resources","u":"/espresso/hands-on/Bi2Se3","h":"#resources","p":7},{"i":13,"t":"First we are going to relax the cell and choose appropriate lattice constant for our chosen pseudo potential. In case of metals, it is important to provide smearing parameters in the input file. src/al/al_vc_relax.in &CONTROL calculation= 'vc-relax', prefix= 'al', outdir= '/tmp/' pseudo_dir = '../pseudos/' etot_conv_thr= 1e-6, forc_conv_thr= 1e-5 / &SYSTEM ibrav= 2, celldm(1)= 7.652, nat= 1, ntyp= 1, ecutwfc = 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / &IONS / &CELL cell_dofree= 'ibrav' / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS (automatic) 10 10 10 0 0 0 We run pw.x to perform variable cell relaxation calculation: pw.x < al_vc_relax.in > al_vc_relax.out Now you may open the output file in vi editor and invoke search by pressing / and type Final enthalpy You will find the final lattice parameters below it.","s":"Variable cell relaxation","u":"/espresso/hands-on/aluminum","h":"#variable-cell-relaxation","p":11},{"i":15,"t":"We obtain relaxed lattice constant = 7.652 * 0.498611683 / 0.5 = 7.63075 Bohr. We will use this value for our next step, self consistent calculation. src/al/al_scf.in &CONTROL calculation= 'scf', restart_mode= 'from_scratch', prefix= 'al', outdir= '/tmp/', pseudo_dir= '../pseudos/' / &SYSTEM ibrav= 2, celldm(1) = 7.63075, nat= 1, ntyp= 1, ecutwfc= 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS (automatic) 10 10 10 0 0 0 We run our self consistent calculation: pw.x < al_scf.in > al_scf.out","s":"Self consistent field (SCF) calculation","u":"/espresso/hands-on/aluminum","h":"#self-consistent-field-scf-calculation","p":11},{"i":17,"t":"Inspect the output file, and let's proceed to next step non-self consistent calculation: src/al/al_nscf.in &CONTROL calculation= 'nscf', restart_mode= 'from_scratch', prefix= 'al', outdir= '/tmp/', pseudo_dir= '../pseudos/' / &SYSTEM ibrav= 2, celldm(1) = 7.63075, nat= 1, ntyp= 1, ecutwfc= 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS (automatic) 40 40 40 0 0 0 Note the changes in input file. The calculation changed to nscf and we are now using a higher number of k-points grid. pw.x < al_nscf.in > al_nscf.out","s":"Non-self consistent field calculation","u":"/espresso/hands-on/aluminum","h":"#non-self-consistent-field-calculation","p":11},{"i":19,"t":"Next we go ahead with our density of states calculation: src/al/al_dos.in &DOS prefix= 'al', outdir= '/tmp/', fildos= 'al_dos.dat', emin= -10, emax= 35 / We run dos.x with DOS inputs: dos.x < al_dos.in > al_dos.out Note from our al_nscf.out that our Fermi energy is at 7.9421 eV. We plot our density of states:","s":"Density of states","u":"/espresso/hands-on/aluminum","h":"#density-of-states","p":11},{"i":21,"t":"We prepare the input file the same as the case of our previous example silicon: src/al/al_bands.in &CONTROL calculation= 'bands', restart_mode= 'from_scratch', prefix= 'al', outdir= '/tmp/', pseudo_dir= '../pseudos/' / &SYSTEM ibrav= 2, celldm(1) = 7.63075, nat= 1, ntyp= 1, ecutwfc= 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS {crystal_b} 5 00.000 0.500 00.000 20 !L 00.000 0.000 00.000 30 !G -0.500 0.000 -0.500 10 !X -0.375 0.250 -0.375 30 !U 00.000 0.000 00.000 20 !G Followed by run pw.x: pw.x < al_bands.in > al_bands.out Now we proceed with post-processing: src/al/al_bands_pp.in &BANDS prefix = 'al' outdir = '/tmp/' filband = 'al_bands.dat' / And run bands.x: bands.x < al_bands_pp.in > al_bands_pp.out We obtain the following bandstructure:","s":"Bandstructure calculation","u":"/espresso/hands-on/aluminum","h":"#bandstructure-calculation","p":11},{"i":23,"t":"Smearing is a technique used for suppressing unstable electron density in the calculation of metals. Such a problem occurs in metals (and semimetals) because the valence bands that cross Fermi level are partially occupied. Due to numerical accuracy, the electrons may occupy the unoccupied states during some iterations, making the algorithm unstable. In order to stablize the algorithm without using excessive number of k-points, smearing technique is used, which replaces the occupation number (either 0 or 1) is replaced by a smoothly varying function of energy. Such a smearing function could be Fermi Dirac distribution, instead of a step function (T = 0 K), we can use the finite temperature form. Below we will test the convergence using PWTK against the number of k-points, three different smearing functions (Gauss, Methfessel-Paxton, and Marzari-Vanderbilt), and for various smearing values. pwtk al.degauss.pwtk We see that the m-v and m-p broadening allow for faster and smother convergence while depending less on degauss value than Gaussian broadening. The number suffix next to the legend labels are number of uniform k-points in Monkhorst-Plank grid.","s":"Importance of smearing in convergence","u":"/espresso/hands-on/aluminum","h":"#importance-of-smearing-in-convergence","p":11},{"i":25,"t":"Before we can run bands calculation, we need to perform single-point self consistent field calculation. We have our input scf file with some new parameters: src/silicon/si_bands_scf.in &CONTROL calculation = 'scf', restart_mode = 'from_scratch', prefix = 'silicon', outdir = './tmp/' pseudo_dir = './pseudos/' verbosity = 'high' / &SYSTEM ibrav = 2, celldm(1) = 10.2076, nat = 2, ntyp = 1, ecutwfc = 50, ecutrho = 400, nbnd = 8, ! occupations = 'smearing', ! smearing = 'gaussian', ! degauss = 0.005 / &ELECTRONS conv_thr = 1e-8, mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 8 8 8 0 0 0 Run the scf calculation: pw.x < si_bands_scf.in > si_bands_scf.out Next step is our band calculation (non-self consistent field) calculation. The bands calculation is non self-consistent and reads/uses the ground state electron density, Hartree, exchange and correlation potentials obtained in the previous step (scf calculation). In case of non self-consistent calculation, the pw.x program determines the Kohn-Sham eigenfunction and eigenvalues without updating Kohn-Sham Hamiltonian at every iteration. We need to specify the k-points for which we want to calculate the eigenvalues. You may use the See-K-path tool by materials cloud to visualize the K-path. We can specify nbnd, by default it calculates half the number of valence electrons, i.e., only the occupied ground state bands. Usually we are interested also in the unoccupied bands above the Fermi energy. Number of occupied bands can be found in the scf output as number of Kohn-Sham states. Below is a sample input file for the band calculation: src/silicon/si_bands.in &control calculation = 'bands', restart_mode = 'from_scratch', prefix = 'silicon', outdir = './tmp/' pseudo_dir = './pseudos/' verbosity = 'high' / &system ibrav = 2, celldm(1) = 10.2076, nat = 2, ntyp = 1, ecutwfc = 50, ecutrho = 400, nbnd = 8 / &electrons conv_thr = 1e-8, mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 Si 0.25 0.25 0.25 K_POINTS {crystal_b} 5 0.0000 0.5000 0.0000 20 !L 0.0000 0.0000 0.0000 30 !G -0.500 0.0000 -0.500 10 !X -0.375 0.2500 -0.375 30 !U 0.0000 0.0000 0.0000 20 !G Run pw.x with bands calculation input file: pw.x < si_bands.in > si_bands.out After the bands calculation is performed, we need some postprocessing using bands.x utility in order to obtain the data in more usable format. Input file for bands.x postprocessing: src/silicon/si_bands_pp.in &BANDS prefix = 'silicon' outdir = './tmp/' filband = 'si_bands.dat' / Run bands.x from post processing (PP) module: bands.x < si_bands_pp.in > si_bands_pp.out Finally, we run plotband.x to visualize bandstructure. We can either run it interactively (as described below) or provide an input file. In order to run interactively, type plotband.x in your terminal. Input file > si_bands.dat Reading 8 bands at 91 k-points Range: -5.8300 16.3420eV Emin, Emax > -6, 16 high-symmetry point: 0.5000 0.5000 0.5000 x coordinate 0.0000 high-symmetry point: 0.0000 0.0000 0.0000 x coordinate 0.8660 high-symmetry point: 1.0000 0.0000 0.0000 x coordinate 1.8660 high-symmetry point: 1.0000 0.2500 0.2500 x coordinate 2.2196 high-symmetry point: 0.0000 0.0000 0.0000 x coordinate 3.2802 output file (gnuplot/xmgr) > si_bands.gnuplot bands in gnuplot/xmgr format written to file si_bands.gnuplot output file (ps) > si_bands.ps Efermi > 6.6416 deltaE, reference E (for tics) 4, 0 bands in PostScript format written to file si_bands.ps You will have si_bands.ps with band diagram. Alternatively, you can use your favorite plotting program to make the plots. Below is an example of using Python matplotlib. notebooks/si-bands.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline plt.rcParams[\"figure.dpi\"]=150 plt.rcParams[\"figure.facecolor\"]=\"white\" plt.rcParams[\"figure.figsize\"]=(8, 6) # load data data = np.loadtxt('../src/silicon/si_bands.dat.gnu') k = np.unique(data[:, 0]) bands = np.reshape(data[:, 1], (-1, len(k))) for band in range(len(bands)): plt.plot(k, bands[band, :], linewidth=1, alpha=0.5, color='k') plt.xlim(min(k), max(k)) # Fermi energy plt.axhline(6.6416, linestyle=(0, (5, 5)), linewidth=0.75, color='k', alpha=0.5) # High symmetry k-points (check bands_pp.out) plt.axvline(0.8660, linewidth=0.75, color='k', alpha=0.5) plt.axvline(1.8660, linewidth=0.75, color='k', alpha=0.5) plt.axvline(2.2196, linewidth=0.75, color='k', alpha=0.5) # text labels plt.xticks(ticks= [0, 0.8660, 1.8660, 2.2196, 3.2802], \\ labels=['L', '$\\Gamma$', 'X', 'U', '$\\Gamma$']) plt.ylabel(\"Energy (eV)\") plt.text(2.3, 5.6, 'Fermi energy', fontsize= small) plt.show() info The k values corresponding to high symmetry points (such as Γ\\GammaΓ, X, U, L) which we need to label in our band diagram, can be found in the post-processing output file (si_bands_pp.out). Bandgap value can be determined from the highest occupied, lowest unoccupied level values printed in scf calculation output.","s":"Bandstructure Calculation","u":"/espresso/hands-on/bands","h":"","p":24},{"i":27,"t":"Usually, band gaps computed using common exchange-correction functionals such as LDA or GGA are severely underestimated compared to actual experimental values. This discrepancy is mainly due to (1) approximations used in the exchange correction functional and (2) a derivative discontinuity term, originating from the density functional being discontinuous with the total number of electrons in the system. The second contribution is larger contributor to the error. It can be partly addressed by a variety of techniques such as the GW approximation. Strategies to improve band gap prediction at moderate to low computational cost now been developed by several groups, including Chan and Ceder (delta-sol)1, Heyd et al. (hybrid functionals)2, and Setyawan et al. (empirical fits)3.","s":"Note on bandgap","u":"/espresso/hands-on/bands","h":"#note-on-bandgap","p":24},{"i":29,"t":"https://docs.materialsproject.org/methodology/materials-methodology/electronic-structure#accuracy-of-band-structures See K-pat online tool M. Chan, G. Ceder, Efficient Band Gap Predictions for Solids, Physical Review Letters 19 (2010) https://doi.org/10.1103/PhysRevLett.105.196403↩ J. Heyd, J.E. Peralta, G.E. Scuseria, R.L. Martin, Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional, Journal of Chemical Physics 123 (2005) https://doi.org/10.1063/1.2085170↩ W. Setyawan, R.M. Gaume, S. Lam, R. Feigelson, S. Curtarolo, High-throughput combinatorial database of electronic band structures for inorganic scintillator materials., ACS Combinatorial Science. (2011) https://doi.org/10.1021/co200012w.↩","s":"Resources","u":"/espresso/hands-on/bands","h":"#resources","p":24},{"i":32,"t":"We can automate the previous self consistent calculation by varying a certain parameter. Say we want to check the total energy of the system for various values of ecutwfc. We can do that by using pwtk script. src/silicon/si_scf_ecutoff.pwtk # load the pw.x input from file load_fromPWI pw.scf.silicon.in # open a file for writing resulting total energies set fid [open etot_vs_ecutwfc.dat w] # loop over different \"ecut\" values foreach ecut { 12 16 20 24 28 32 } { # name of I/O files: $name.in & $name.out set name si_scf_ecutwfc-$ecut # set the pw.x \"ecutwfc\" variable SYSTEM \"ecutwfc = $ecut\" # run the pw.x calculation runPW $name.in # extract the \"total energy\" and write it to file set Etot [::pwtk::pwo::totene $name.out] puts $fid \"$ecut $Etot\" } close $fid To run the above script: pwtk si_scf_ecutoff.pwtk Now we can plot the total energy with respect to ecutwfc. The data is in etot-vs-ecutwfc.dat We will use matplotlib to make the plots. Here is the python code for plotting: notebooks/si-plots.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline plt.rcParams[\"figure.dpi\"]=150 plt.rcParams[\"figure.facecolor\"]=\"white\" x, y = np.loadtxt('../src/silicon/etot-vs-ecutwfc.dat', delimiter=' ', unpack=True) plt.plot(x, y, \"o-\", markersize=5, label='Etot vs ecutwfc') plt.xlabel('ecutwfc (Ry)') plt.ylabel('Etot (Ry)') plt.legend(frameon=False) plt.show()","s":"Convergence with cutoff energy using PWTK","u":"/espresso/hands-on/convergence","h":"#convergence-with-cutoff-energy-using-pwtk","p":30},{"i":34,"t":"We can do the convergence test with various parameters. We can calculate the total energy of the system by varying various parameters. We will use the shell script to automate the process with different cutoff energy values. src/silicon/si_script.sh #!/bin/sh NAME=\"ecut\" for CUTOFF in 10 15 20 25 30 35 40 do cat > ${NAME}_${CUTOFF}.in << EOF &control calculation = 'scf', prefix = 'silicon' outdir = './tmp/' pseudo_dir = './pseudos/' / &system ibrav = 2, celldm(1) = 10.0, nat = 2, ntyp = 1, ecutwfc = $CUTOFF / &electrons mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 6 6 6 1 1 1 EOF pw.x < ${NAME}_${CUTOFF}.in > ${NAME}_${CUTOFF}.out echo ${NAME}_${CUTOFF} grep ! ${NAME}_${CUTOFF}.out done Make sure the file has executable permission for the user: chmod 700 si_script.sh Run the script file: ./si_script.sh # or sh si_script.sh We can plot the energy vs cutoff energy, and choose a reasonable value. caution Initially, I had problem in running the script in macOS. The problem occurred because the script file format was set to DOS. The file format can be checked in following way: Open the file in vi editor. vi si_script.sh Now in vi editor command mode (ESC key), type :set ff? This would tell you the file format. Now to change file format, use the command :set fileformat=unix","s":"Convergence test using UNIX shell script","u":"/espresso/hands-on/convergence","h":"#convergence-test-using-unix-shell-script","p":30},{"i":36,"t":"We can run similar convergence test against another parameter, and choose the best value of that particular parameter. Here we will try to calculate the number of k-points in the Monkhorst-Pack mesh. src/silicon/si_scf_kpoints.pwtk load_fromPWI pw.scf.silicon.in set fid [open etot-vs-kpoint.dat w] foreach k { 2 4 6 8 } { set name si_scf_kpoints-$k K_POINTS automatic \"$k $k $k 1 1 1\" runPW $name.in set Etot [::pwtk::pwo::totene $name.out] puts $fid \"$k $Etot\" } close $fid Run pwtk program: pwtk si_scf_kpoints.pwtk notebooks/silicon-scf.ipynb x, y = np.loadtxt('../src/silicon/etot-vs-kpoint.dat', delimiter=' ', unpack=True) plt.plot(x, y, \"o-\", markersize=5, label='Etot vs kpoints') plt.xlabel('# kpoints') plt.ylabel('Etot (Ry)') plt.legend(frameon=False) plt.show()","s":"Convergence test against the number of k-points","u":"/espresso/hands-on/convergence","h":"#convergence-test-against-the-number-of-k-points","p":30},{"i":38,"t":"Calculating total energy with respect to varying lattice constant. src/silicon/si_scf_alat.pwtk load_fromPWI pw.scf.silicon.in # please uncomment & insert value as determined in the \"ecutwfc\" exercise SYSTEM { ecutwfc = 30 } # please uncomment & insert values as determined in the \"kpoints\" exercise K_POINTS automatic { 6 6 6 1 1 1 } set fid [open etot-vs-alat.dat w] foreach alat { 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 } { set name si_scf_alat-$alat SYSTEM \"celldm(1) = $alat\" runPW $name.in set Etot [::pwtk::pwo::totene $name.out] puts $fid \"$alat $Etot\" } close $fid Run the above code: pwtk si_scf_alat.pwtk notebooks/silicon-scf.ipynb x, y = np.loadtxt('../src/silicon/etot-vs-alat.dat', delimiter=' ', unpack=True) plt.plot(x, y, \"o-\", markersize=5, label='Etot vs alat') plt.xlabel('alat (Bohr)') plt.ylabel('Etot (Ry)') plt.legend(frameon=False) plt.show()","s":"Convergence against lattice constant","u":"/espresso/hands-on/convergence","h":"#convergence-against-lattice-constant","p":30},{"i":40,"t":"CPU time is proportional to the number of plane waves used for the calculation. Number of plane wave is proportional to the (ecutwfc)3/2 CPU time is proportional to the number if inequivalent k-points CPU time increases as N3, where N is the number of atoms in the system.","s":"Note on CPU time","u":"/espresso/hands-on/convergence","h":"#note-on-cpu-time","p":30},{"i":42,"t":"Electronic structure for transition metals (with localized ddd or fff electrons) is not accurately described by standard DFT, and therefore the need for DFT+U formulation. &SYSTEM ... lda_plus_u = .TRUE. Hubbard_u(i) = 2.0 ... / Here i refers to the atomic index in the &ATOMIC_SPECIES card corresponding to each ntyp. We can specify Hubbard_u(i) corresponding to more than one atom in separate lines. There is also Ueff=U−JU_{eff} = U - JUeff​=U−J implementation in QE. JJJ represents on-site exchange interaction. Number of JJJ terms depends on the manifold of localized electrons. For ppp, we have 1; for ddd, we have 2; and for fff, we have 3 terms. ... lda_plus_u = .TRUE. lda_plus_u_kind = 1 Hubbard_u(i) = U Hubbard_J(k, i) = J_{ki} ... COMMON ERRORS If you add Hubbard_u for elements that is not implemented to have UUU term in QE, you might see a \"pseudopotential not yet inserted\" error.","s":"DFT+U calculation","u":"/espresso/hands-on/dft-u","h":"","p":41},{"i":44,"t":"Starting from Quantum Espresso version 7.1, there are changes to input syntax for DFT+U calculations. In the new version, instead of defining the necessary DFT+U parameters, now there is a new Hubbard card. &system ... - lda_plus_u = .true., - lda_plus_u_kind = 0, - U_projection_type = 'atomic', - Hubbard_U(1) = 4.6 - Hubbard_U(2) = 4.6 ... / + HUBBARD (ortho-atomic) + U Fe1-3d 4.6 + U Fe2-3d 4.6 Please refer to the qe-x.x/Doc/Hubbard_input.pdf for details.","s":"Changes to input syntax in v7.1","u":"/espresso/hands-on/dft-u","h":"#changes-to-input-syntax-in-v71","p":41},{"i":46,"t":"We will first perform the standard DFT calculation. Perform the SCF calculation: pw.x -in feo_scf.in > feo_scf.out Perform NSCF calculation with denser k-grid: pw.x -in feo_nscf.in > feo_nscf.out Perform P-DOS calculation: projwfc.x -in feo_projwfc.in > feo_projwfc.out This gives us metallic density of states. In practice we get insulating FeO.","s":"DFT calculation for FeO","u":"/espresso/hands-on/dft-u","h":"#dft-calculation-for-feo","p":41},{"i":48,"t":"src/FeO/feo_hp.in &inputhp prefix = 'FeO' outdir = './tmp/' nq1 = 1, nq2 = 1, nq3 = 1 / Perform a linear-response calculation using hp.x program: hp.x -in feo_hp.in > feo_hp.out Check the file FeO.Hubbard_parameters.dat. info We need to check the convergence against q-mesh (as well as k-mesh in SCF calculation). Here 1×1×11\\times 1\\times 11×1×1 mesh is used. Important: lda_plus_u must be set to .true. during the SCF calculation, UUU may be set to zero. We can update the obtained UUU value in our SCF calculation, and repeat linear response calculation until we have reached self consistency in UUU value. To go even further one can check the convergence of geometry during UUU updates. There is also inter-site Hubbard correction DFT+U+V calculation. The results could be more closer to hybrid functionals like GW. The VVV can also be calculated using Quantum Espresso hp.x code. Obtained value of UUU depends on pseudopotential, Hubbard manifold (whether atomic, ortho-atomic etc.). danger The above hp.x code is not suitable for closed cell systems (e.g., fully occupied d-shell element), in such cases this linear response method gives unrealistically large UUU value.","s":"Calculating Hubbard U","u":"/espresso/hands-on/dft-u","h":"#calculating-hubbard-u","p":41},{"i":49,"t":"We repeat the calculation after setting in the &SYSTEM card: Hubbard_U(1) = 4.6 Hubbard_U(2) = 4.6 We repeat the above calculation and plot the results. Now we find insulating ground state. info U_projection_type = 'ortho-atomic' might give more realistic result than the default 'atomic'. When performing DFT+UDFT+UDFT+U calculation, the ground state might get stuck in a local minimum, in such cases we need to provide starting_ns_eigenvalue to help calculation reach desired/actual ground state. Please see these slides by Dr. Iurii Timrov for a relevant example. tip Here we have plotted the lpdos (local density of states). If we want to know the contribution of dz2,dyz,dx2−z2d_{z^2}, d_{yz}, d_{x^2-z^2}dz2​,dyz​,dx2−z2​ ect., we can find them from the pdos columns. Also there arise important Lowdin charges information in the feo_projwfc.out file.","s":"DFT+U calculation","u":"/espresso/hands-on/dft-u","h":"#dftu-calculation","p":41},{"i":51,"t":"Hands-on DFT+U by Iurii Timrov and Matteo Cococcioni Hubbard parameter calculation","s":"Resources","u":"/espresso/hands-on/dft-u","h":"#resources","p":41},{"i":53,"t":"Electronic density of states is an important property of a material. ρ(E)dE\\rho(E)dEρ(E)dE = number of electronic states in the energy interval (E,E+dE)(E, E + dE)(E,E+dE) Before we can run the Density of States (DOS) calculation, we need Perform fixed-ion self consistent filed (scf) calculation. In plane-wave based DFT calculations the electronic density is expressed by functions of the form exp⁡(ik⋅r)\\exp (i \\textbf{k} \\cdot \\textbf{r})exp(ik⋅r) with energy given by E=ℏ2k2/2mE = \\hbar^2k^2/2mE=ℏ2k2/2m. Perform non-self consistent field (nscf) calculation with denser k-point grid. A large number of kkk points are required DOS calculation, as the accuracy of DOS depends on the integration in kkk space. Finally, the DOS can be determined by integrating the electron density in kkk space. I have created a new input file (si_scf_dos.in) which is very much the same as our previous scf input file except some parameters are modified. You can find all the input files in my GitHub repository. We used the lattice constant value that we obtained from the relaxation calculation. We should not directly use the experimental/real lattice constant values. Depending on the method and pseudo-potential, it might result stress in the system. We have increased the ecutwfc to have better precision. We run the scf calculation: pw.x < si_scf_dos.in > si_scf_dos.out Next, we have prepared the input file for the nscf calculation. Where is have added occupations in the &system card as tetrahedra (appropriate for DOS calculation). We have increased the number of k-points to 12 × 12 × 12 with automatic option. Also specify nosym = .TRUE. to avoid generation of additional k-points in low symmetry cases. outdir and prefix must be the same as in the scf step, some of the inputs and output are read from previous step. Here we can specify a larger number of nbnd to calculate unoccupied bands. Number of occupied bands can be found in the scf output as number of Kohn-Sham states. pw.x < si_nscf_dos.in > si_nscf_dos.out Now our final step is to calculate the density of states. The DOS input file as follows: src/silicon/si_dos.in &DOS prefix='silicon', outdir='./tmp/', fildos='si_dos.dat', emin=-9.0, emax=16.0 / We run: dos.x < si_dos.in > si_dos.out The DOS data in the si_dos.dat file that we specified in our input file. We can plot the DOS: notebooks/silicon-dos.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline # load data energy, dos, idos = np.loadtxt('../src/silicon/si_dos.dat', unpack=True) # make plot plt.figure(figsize = (12, 6)) plt.plot(energy, dos, linewidth=0.75, color='red') plt.yticks([]) plt.xlabel('Energy (eV)') plt.ylabel('DOS') plt.axvline(x=6.642, linewidth=0.5, color='k', linestyle=(0, (8, 10))) plt.xlim(-6, 16) plt.ylim(0, ) plt.fill_between(energy, 0, dos, where=(energy < 6.642), facecolor='red', alpha=0.25) plt.text(6, 1.7, 'Fermi energy', fontsize= med, rotation=90) plt.show() Important For a set of calculation, we must keep the prefix same. For example, the nscf or bands calculation uses the wavefunction calculated by the scf calculation. When performing different calculations, for example you change a parameter and want to see the changes, you must use different output folder or unique prefix for different calculations so that the outputs do not get mixed. tip Sometimes it is important to sample the Γ\\GammaΓ point for DOS calculation (e.g., the conducting bands cross the Fermi surface only at Γ\\GammaΓ point). In such cases, we need to use odd k-grid (e.g., 9✕9✕5).","s":"Density of States calculation","u":"/espresso/hands-on/dos","h":"","p":52},{"i":55,"t":"First we perform our self consistent field calculation: mpirun -np 4 pw.x -in pw.scf.silicon.in > pw.scf.silicon.out Next step we perform nscf calculation: src/silicon/pw.nscf.silicon_epsilon.in &CONTROL calculation = 'nscf', prefix = 'silicon', outdir = '/tmp/silicon/' pseudo_dir = '../pseudos/' verbosity = 'high' / &SYSTEM ibrav = 2, celldm(1) = 10.26, nat = 2, ntyp = 1, ecutwfc = 30 nbnd = 16 nosym = .true. noinv = .true. / &ELECTRONS mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 30 30 30 0 0 0 Especially, notice following changes: nbnd = 16 nosym = .true. noinv = .true. We turn off the automatic reduction of k-points that pw.x does by using crystal symmetries (nosym = .true. and noinv = .true.). This is because epsilon.x does not recognize crystal symmetries, therefore the entire list of k-points in the grid is needed. Secondly, we calculate a larger number of bands (16), since we are interested in interband transitions. The final step is to prepare the input file for epsilon.x: src/silicon/epsilon.si.in &inputpp outdir = \"/tmp/silicon/\" prefix = \"silicon\" calculation = \"eps\" / &energy_grid smeartype = \"gauss\" intersmear = 0.2 wmin = 0.0 wmax = 30.0 nw = 500 / The variables smeartype and intersmear define the numerical approximation used to represent the Dirac delta functions in the expression for ϵ2\\epsilon_2ϵ2​ given above. The variables wmin, wmax and nw define the energy grid for the dielectric function. All the energy variables are in eV. mpirun -np 1 epsilon.x < epsilon.si.in > epsilon.si.out We will see the results are saved in separate .dat files. We can plot the real (ϵ1\\epsilon_1ϵ1​) and imaginary (ϵ2\\epsilon_2ϵ2​) parts of dielectric constants: import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline plt.rcParams[\"figure.dpi\"]=150 plt.rcParams[\"figure.facecolor\"]=\"white\" data_r = np.loadtxt('../src/silicon/epsr_silicon.dat') data_i = np.loadtxt('../src/silicon/epsi_silicon.dat') energy_r, epsilon_r = data_r[:, 0], data_r[:, 2] energy_i, epsilon_i = data_i[:, 0], data_i[:, 2] plt.plot(energy_r, epsilon_r, lw=1, label=\"$\\\\epsilon_1$\") plt.plot(energy_i, epsilon_i, lw=1, label=\"$\\\\epsilon_2$\") plt.xlim(0, 15) plt.xlabel(\"Energy (eV)\") plt.ylabel(\"$\\\\epsilon_1~/~\\\\epsilon_2$\") plt.legend(frameon=False) plt.show() danger Ultra-soft pseudopotentials do not work with epsilon.x.","s":"Dielectric constant","u":"/espresso/hands-on/epsilon","h":"","p":54},{"i":57,"t":"I am following this example from the ICTP online school 2021. We will perform the SCF KS orbital calculations on magnetic (nspin=2) iron. Since the d-orbitals of Fe atom are localized/ hard, we will use ultra-soft pseudo potential (USPP). note If we have crystal structure with only one atom per unit cell, or only one type of atoms, the only possible ordering is ferromagnetic. In such cases, we need to form supercell with more number of atoms or label multiple atoms separately, so that their magnetic orientation could be different thus having the possibility of ferro- or antiferromagnetic final states. Run the SCF calculations for both ferro- and antiferromagnetic structures. Notice that for ferromagnetic, we have BCC structure with only one type of atom, while we use simple cubic structure for antiferromagnetic case with two different atomic labels. For antiferromagnetic calculation, we need to start with opposite initial spins. pw.x -i pw.scf.fe_fm.in > pw.scf.fe_fm.out pw.x -i pw.scf.fe_afm.in > pw.scf.fe_afm.out note In case of the AFM calculation, if we have started with FM (say, for both atom types starting_magnetization=0.6 ), the calculation would still converge to AFM state as it is the true ground state for this system, albeit it would take more iteration to converge. If a system has complex potential surface with local minima, it it possible to get different final state magnetization depending on the starting magnetization. In such cases, a stricter convergence criteria might help. info In case of ultrasoft pseudo potentials, the Quantum Espresso default of ecutrho 4 times of ecutoff is not sufficient. We need to set ecutrho 8 or even 12 times that of ecutoff. We must test the convergence for our set values.","s":"Magnetic system: bulk iron","u":"/espresso/hands-on/fe","h":"","p":56},{"i":59,"t":"Below is the PWTK script file: src/fe/fe_ecut.pwtk # load the pw.x input from file load_fromPWI fe_scf_fm.in # dual is the ratio ecutrho/ecutwfc foreach dual { 4 8 12 } { set fid [open etot-vs-ecutwfc.dual$dual.dat w] foreach ecutwfc [seq 25 5 50] { set name pw.Fe.scf.ecutwfc-$ecutwfc.dual-$dual SYSTEM \"ecutwfc = $ecutwfc ecutrho = $ecutwfc*$dual \" runPW $name.in set Etot [pwo_totene $name.out] puts $fid \"$ecutwfc $Etot\" } close $fid } Run the script: pwtk fe_ecut.pwtk","s":"Convergence test for USPP","u":"/espresso/hands-on/fe","h":"#convergence-test-for-uspp","p":56},{"i":61,"t":"PWTK script to calculate DOS and p-DOS: src/fe/fe_dos.pwtk load_fromPWI fe_scf_fm.in SYSTEM \" ecutwfc = 40 ecutrho = 320 \" set name Fe runPW pw.$name.scf.in CONTROL { calculation = 'nscf' } SYSTEM { occupations = 'tetrahedra' , degauss = , ! this is how variable is unset in PWTK } K_POINTS automatic { 12 12 12 1 1 1 } runPW pw.$name.nscf.in DOS \" fildos = '$name.dos.dat' Emin = 5.0 Emax = 20.0, DeltaE = 0.1 \" runDOS dos.$name.in PROJWFC \" filpdos = '$name.pdos.dat' Emin = 5.0 Emax = 20.0, DeltaE = 0.1 \" runPROJWFC projwfc.$name.in Below is the plots of total and projected density of states. Also see bandstructure of Fe with and without SOC.","s":"Density of states calculation","u":"/espresso/hands-on/fe","h":"#density-of-states-calculation","p":56},{"i":63,"t":"Paramagnetic materials have fluctuating magnetic moments that may not be properly described DFT. One approach is to model paramagnetic materials in DFT calculation by building a large supercell and assign randomly oriented magnetic moments. Also note that DFT assumes zero temperature, so it makes sense to perform FM or AFM calculation for magnetic systems.","s":"Paramagnetism","u":"/espresso/hands-on/fe","h":"#paramagnetism","p":56},{"i":65,"t":"We can use XCrySDen to visualize the orientation of magnetic moments. XCrySDen cannot directly read the Quantum Espresso output files for magnetic moment vectors, instead we need to create the input .xsf file with magnetic moments as force vector. You can also change the background color from black from the Palette Menu which is located in the left of File menu. src/fe/fe.xsf # this is a specification for crystal structure CRYSTAL # primitive lattice vectors (in Angstroms) PRIMVEC 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 # conventional lattice vectors (in Angstroms) CONVVEC 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 # First number stands for number of atoms in primitive cell # the second number is always 1 for PRIMCOORD coordinates # followed by atomic coordinates (in Angstroms) and forces: # AtNum X Y Z Fx Fy Fz PRIMCOORD 2 1 26 0.0000000000 0.0000000000 0.0000000000 0.00 0.00 0.01 26 1.4340702350 1.4340702350 1.4340702350 0.00 0.00 -0.01 Open the file from XCrySDen Menu: File → Open Structure → Open XSF. Then go to Display menu and select Forces. If you want to adjust scaling for the force vectors, go to Modify → Force Settings and set suitable Length factor.","s":"Visualizing magnetic moments","u":"/espresso/hands-on/fe","h":"#visualizing-magnetic-moments","p":56},{"i":67,"t":"Here we will calculate Fermi surface of copper. First step is to perform self- consistent field calculation. src/cu/pw.scf.cu.in &control calculation = \"scf\", prefix = \"cu\", pseudo_dir = \"../pseudos/\", outdir = \"/tmp/cu/\" / &system ibrav = 2, celldm(1) = 6.678, nat = 1, ntyp = 1, ecutwfc = 40, ecutrho = 300, occupations = \"smearing\", smearing = \"mp\", degauss = 0.01, nbnd = 16 / &electrons conv_thr = 1e-9, / ATOMIC_SPECIES Cu 63.546 Cu_ONCV_PBE-1.0.oncvpsp.upf ATOMIC_POSITIONS alat Cu 0.00 0.00 0.00 K_POINTS automatic 8 8 8 1 1 1 mpirun -np 4 pw.x -in pw.scf.cu.in > pw.scf.cu.out Next we perform bands calculation over dense uniform k-grid: src/cu/pw.bands.cu.in &control calculation = \"bands\", prefix = \"cu\", pseudo_dir = \"../pseudos/\", outdir = \"/tmp/cu/\" / &system ibrav = 2, celldm(1) = 6.678, nat = 1, ntyp = 1, ecutwfc = 40, ecutrho = 300, occupations = \"smearing\", smearing = \"mp\", degauss = 0.01, nbnd = 16 / &electrons conv_thr = 1e-9, / ATOMIC_SPECIES Cu 63.546 Cu_ONCV_PBE-1.0.oncvpsp.upf ATOMIC_POSITIONS alat Cu 0.00 0.00 0.00 K_POINTS automatic 30 30 30 0 0 0 mpirun -np 4 pw.x -in pw.bands.cu.in > pw.bands.cu.out Finally, we process the data with fs.x post processing tool. Below is the input file: src/cu/fs.cu.in &fermi outdir = \"/tmp/cu/\" prefix = \"cu\" / mpirun -np 4 fs.x -in fs.cu.in > fs.cu.out We can visualize the output file cu_fs.bxsf using xcrysdens program: xcrysden --bxsf cu_fs.bxsf","s":"Fermi Surface","u":"/espresso/hands-on/fermi-surface","h":"","p":66},{"i":69,"t":"Now that we have calculated the bandstructure of silicon (semiconductor) and aluminum (metal), let us proceed with a compound which has two different atoms. We follow the steps like before: First check the lattice constant with cell relaxation according to our chosen pseudo potential. We use that lattice constant in our next steps. Our lattice constant = 10.6867 * 0.508176602 / 0.5 = 10.861462. pw.x < pw.relax.GaAs.in > pw.relax.GaAs.out Perform self consistent field calculation: pw.x < pw.scf.GaAs.in > pw.scf.GaAs.out Give denser k-points and perform non-self consistent calculation. This step is only necessary if you need to obtain density of states. pw.x < pw.nscf.GaAs.in > pw.nscf.GaAs.out Perform bands calculation pw.x < pw.bands.GaAs.in > pw.bands.GaAs.out Post process the data and plot the bandstructure. bands.x < pp.bands.GaAs.in > pp.bands.GaAs.out If everything goes well, you will get the bandstructure as below: Warning Sometimes a calculation with the same inputs converges in one computer, while fails in another due to library configuration or even due to floating point approximations. The final output numbers will always vary slightly for different machines, or even among different runs in the same machine. Also check the Quantum Espresso version for reproducibility. Also see the bandstructure of GaAs with SOC.","s":"Bandstructure of GaAs","u":"/espresso/hands-on/GaAs","h":"","p":68},{"i":71,"t":"I am following this example from the ICTP online school 2021. Graphene is single layer of carbon atoms. First perform the self consistent field calculation to obtain the Kohn-Sham orbitals. Please check the input files in GitHub repository. Run pw.x: pw.x -i graphene_scf.in > graphene_scf.out Next increase the k-grid, and perform the non-self-consistent field calculation. pw.x -i graphene_nscf.in > graphene_nscf.out","s":"DOS and Bandstructure of Graphene","u":"/espresso/hands-on/graphene","h":"","p":70},{"i":73,"t":"dos.x -i graphene_dos.in > graphene_dos.out","s":"DOS calculation","u":"/espresso/hands-on/graphene","h":"#dos-calculation","p":70},{"i":75,"t":"First run the bands calculation for given k-path: pw.x -i graphene_bands.in > graphene_bands.out Followed by the postprocessing to collect the bands: bands.x -i graphene_bands_pp.in > graphene_bands_pp.out Make plots: notebooks/graphene.ipynb import numpy as np import matplotlib.pyplot as plt data = np.loadtxt('../src/graphene/graphene_bands.dat.gnu') k = np.unique(data[:, 0]) bands = np.reshape(data[:, 1], (-1, len(k))) for band in range(len(bands)): plt.plot(k, bands[band, :], linewidth=1, alpha=0.5, color='k') plt.xlim(min(k), max(k)) # Fermi energy plt.axhline(0.921, linestyle=(0, (8, 10)), linewidth=0.75, color='k', alpha=0.5) # High symmetry k-points (check bands_pp.out) plt.axvline(0.6667, linewidth=0.75, color='k', alpha=0.5) plt.axvline(1, linewidth=0.75, color='k', alpha=0.5) # text labels plt.xticks(ticks= [0, 0.6667, 1, 1.5774], labels=['$\\Gamma$', 'K', 'M', '$\\Gamma$']) plt.ylabel(\"Energy (eV)\") plt.show()","s":"Bandstructure calculation","u":"/espresso/hands-on/graphene","h":"#bandstructure-calculation","p":70},{"i":77,"t":"Here we will calculate k-resolved density of states for silicon. First we begin with self consistent field calculation. Here is the input: pw.x -inp si_scf.in > si_scf.out Followed by the bands calculation. Note that for bands calculation I have doubled the number of k-points compared to our previous bands calculation. pw.x -inp si_bands.in > si_bands.out Calculate the orbital projections with k-resolved information: src/silicon/si_projwfc.in &projwfc outdir = './tmp/' prefix = 'silicon' ngauss = 0 degauss = 0.036748 DeltaE = 0.005 kresolveddos = .true. filpdos = 'silicon.k' / projwfc.x -inp si_projwfc.in > si_projwfc.out This will give separate orbital projections, as well as total sum for k-resolved DOS. Make plots: notebooks/silicon-kpdos.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np import zipfile %matplotlib inline # data file was compressed to reduce file size zipobj = zipfile.ZipFile('../src/silicon/silicon.k.pdos_tot.zip', 'r') zipdata = zipobj.open('silicon.k.pdos_tot') data = np.loadtxt(zipdata) k = np.unique(data[:, 0]) # k values e = np.unique(data[:, 1]) # dos energy values dos = np.zeros([len(k), len(e)]) for i in range(len(data)): e_index = int(i % len(e)) k_index = int(data[i][0] - 1) dos[k_index, e_index] = data[i][2] plt.pcolormesh(k, e, dos.T, cmap='magma', shading='auto') # plt.ylim(-2, 10) plt.xticks([]) plt.ylabel('Energy (eV)') plt.xticks([]) plt.gcf().text(0.12, 0.06, 'L', fontsize=16, fontweight='normal') plt.gcf().text(0.29, 0.06, '$\\Gamma$', fontsize=16, fontweight='normal') plt.gcf().text(0.55, 0.06, 'X', fontsize=16, fontweight='normal') plt.gcf().text(0.63, 0.06, 'U', fontsize=16, fontweight='normal') plt.gcf().text(0.892, 0.06, '$\\Gamma$', fontsize=16, fontweight='normal') plt.axvline(21, c='yellow', lw=1, alpha=0.5) plt.axvline(51, c='yellow', lw=1, alpha=0.5) plt.axvline(61, c='yellow', lw=1, alpha=0.5) plt.show() info If you are using ibrav=0, you can calculate projwfc with lsym=.false. option. If we have contribution from multiple orbitals, we can sum desired projections using sumpdos.x program. For example: sumpdos.x *\\(Cl\\)*\\(p\\) > Cl_2p_tot.dat This way we can plot different orbital projections along with energy and k-resolution.","s":"k-resolved DOS","u":"/espresso/hands-on/kpdos","h":"","p":76},{"i":79,"t":"We prepare the input file pw_scf_ni.in and run the calculation: mpirun -np 8 pw.x -i pw_scf_ni.in > pw_scf_ni.out Prepare the input file for bands calculation pw_bands_ni.in with our desired k-path and run: mpirun -np 8 pw.x -i pw_bands_ni.in > pw_bands_ni.out Now we perform the bands.x calculation with spin_component=1 to process only the spin up bands: src/ni/bands_ni_up.in &BANDS outdir='./tmp/', prefix='ni', filband='ni_bands_up.dat', spin_component = 1, / Run the calculation: mpirun -np 8 bands.x -i bands_ni_up.in > bands_ni_up.out Similarly, we process the spin down bands spin_component=2 and plot them.","s":"Spin polarized bandstructure calculation for nickel","u":"/espresso/hands-on/ni","h":"","p":78},{"i":81,"t":"Here we continue with our Aluminum example. Often it is needed to know the contribution from each individual atoms and/or each of their orbital contributions. We can achieve that using projwfc.x code. First, we must perform the self consistent field calculation followed by the non-self consistent field calculation with denser k-points. pw.x < al_scf.in > al_scf.out pw.x < al_nscf.in > al_nscf.out Then we prepare the input file for projwfc.x: src/al/al_projwfc.in &PROJWFC prefix= 'al', outdir= '/tmp/', filpdos= 'al_pdos.dat' / Perform the calculation: projwfc.x < al_projwfc.in > al_projwfc.out Output data format: the DOS values are written in the file {filpdos}.pdos_atm#N(X)_wfc#M(l), where N is atom number, X is atom symbol, M is wfc number, and l=s,p,d,f one file for each atomic wavefunction read from pseudopotential file. The header of file looks like (for spin polarized calculations, we have separate up and down columns): E LDOS(E) PDOS_1(E) ... PDOS_{2l+1}(E) LDOS=∑m=12l+1PDOSm(E)LDOS = \\sum\\limits_{m=1}^{2l+1} PDOS_m (E)LDOS=m=1∑2l+1​PDOSm​(E) PDOSm(E)→PDOS_m (E) \\rightarrowPDOSm​(E)→ projected DOS on atomic wfc with component mmm. Orbital order: for l=1l=1l=1: pz (m=0)p_z~(m=0)pz​ (m=0) pxp_xpx​ (real combination of m=±1m=\\pm 1m=±1 with cosine) pyp_ypy​ (real combination of m=±1m=\\pm 1m=±1 with sine) for l=2l=2l=2: dz2 (m=0)d_{z^2}~(m=0)dz2​ (m=0) dzxd_{zx}dzx​ (real combination of m=±1m=\\pm 1m=±1 with cosine) dzyd_{zy}dzy​ (real combination of m=±1m=\\pm 1m=±1 with sine) dx2−y2d_{x^2-y^2}dx2−y2​ (real combination of m=±2m=\\pm 2m=±2 with cosine) dxyd_{xy}dxy​ (real combination of m=±2m=\\pm 2m=±2 with sine) Let's make our plots: src/notebooks/al-pdos.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline # load data def data_loader(fname): import numpy as np data = np.loadtxt(fname) energy = data[:, 0] pdos = data[:, 2] return energy, pdos energy, pdos_s = data_loader('../src/al/al_pdos.dat.pdos_atm#1(Al)_wfc#1(s)') _, pdos_p = data_loader('../src/al/al_pdos.dat.pdos_atm#1(Al)_wfc#2(p)') _, pdos_tot = data_loader('../src/al/al_pdos.dat.pdos_tot') # make plots plt.figure(figsize = (8, 4)) plt.plot(energy, pdos_s, linewidth=0.75, color='#006699', label='s-orbital') plt.plot(energy, pdos_p, linewidth=0.75, color='r', label='p-orbital') plt.plot(energy, pdos_tot, linewidth=0.75, color='k', label='total') plt.yticks([]) plt.xlabel('Energy (eV)') plt.ylabel('DOS') plt.axvline(x= 7.9421, linewidth=0.5, color='k', linestyle=(0, (8, 10))) plt.xlim(-5, 27) plt.ylim(0, ) plt.fill_between(energy, 0, pdos_s, where=(energy < 7.9421), facecolor='#006699', alpha=0.25) plt.fill_between(energy, 0, pdos_p, where=(energy < 7.9421), facecolor='r', alpha=0.25) plt.fill_between(energy, 0, pdos_tot, where=(energy < 7.9421), facecolor='k', alpha=0.25) # plt.text(6.5, 0.52, 'Fermi energy', fontsize= small, rotation=90) plt.legend(frameon=False) plt.show() Here is how our projected density of states plot looks like: We can perform sums of specific atom or orbital contributions using sumpdos.x code if there are multiple sss or ppp orbitals: sumpdos.x *\\(Al\\)* > atom_Al_tot.dat sumpdos.x *\\(Al\\)*\\(s\\) > atom_Al_s.dat sumpdos.x *\\(Al\\)*\\(p\\) > atom_Al_p.dat","s":"Projected Density of States","u":"/espresso/hands-on/pdos","h":"","p":80},{"i":83,"t":"In Quantum Espresso, phonon dispersion is calculated using ph.x program, which is implementation of density functional perturbation theory (DFPT). Here are the steps for calculating phonon dispersion: (1) perform SCF calculation using pw.x src/GaAs-phonon/pw.scf.GaAs.in &control calculation = 'scf' prefix = 'GaAs' pseudo_dir = '../pseudos/' outdir = './tmp/' verbosity = 'high' wf_collect = .true. / &system ibrav = 2 celldm(1) = 10.861462 nat = 2 ntyp = 2 ecutwfc = 80 ecutrho = 640 / &electrons mixing_mode = 'plain' mixing_beta = 0.7 conv_thr = 1.0e-8 / ATOMIC_SPECIES Ga 69.723 Ga.pbe-dn-kjpaw_psl.1.0.0.UPF As 74.921595 As.nc.z_15.oncvpsp3.dojo.v4-std.upf ATOMIC_POSITIONS Ga 0.00 0.00 0.00 As 0.25 0.25 0.25 K_POINTS {automatic} 8 8 8 0 0 0 We perform the SCF calculation: mpirun -np 4 pw.x -i pw.scf.GaAs.in > pw.scf.GaAs.out info Usually higher energy cutoff values are used for phonon calculation to get better accuracy. In case of two dimensional systems, use assume_isolated = '2D' in the SYSTEM namelist to avoid negative or imaginary acoustic frequencies near Γ\\GammaΓ point. Read more here. (2) calculate the dynamical matrix on a uniform mesh of q-points using ph.x src/GaAs-phonon/ph.GaAs.in &INPUTPH outdir = './tmp/' prefix = 'GaAs' tr2_ph = 1d-14 ldisp = .true. ! recover = .true. nq1 = 6 nq2 = 6 nq3 = 6 fildyn = 'GaAs.dyn' / Run the calculation: mpirun -np 4 ph.x -i ph.GaAs.in > ph.GaAs.out The above calculation is computationally demanding. Our example calculation took about a whole day on a 2.6 GHz quad core processor. info You can restart an interrupted ph.x calculation with recover = .true. in the INPUTPH namelist. You can cleanly exit an ongoing calculation by creating an empty file with name {prefix}.EXIT. (3) perform inverse Fourier transform of the dynamical matrix to obtain inverse Fourier components in real space using q2r.x. Below is our input file: src/GaAs-phonon/q2r.GaAs.in &INPUT fildyn = 'GaAs.dyn' zasr = 'crystal' flfrc = 'GaAs.fc' / mpirun -np 4 q2r.x -i q2r.GaAs.in > q2r.GaAs.out (4) Finally, perform Fourier transformation of the real space components to get the dynamical matrix at any q by using matdyn.x. src/GaAs-phonon/matdyn.GaAs.in &INPUT asr = 'crystal' flfrc = 'GaAs.fc' flfrq = 'GaAs.freq' flvec = 'GaAs.modes' ! loto_2d = .true. q_in_band_form = .true. / 5 0.500 0.500 0.500 20 ! L 0.000 0.000 0.000 20 ! G 0.500 0.000 0.500 20 ! X 0.375 0.375 0.750 20 ! K 0.000 0.000 0.000 1 ! G mpirun -np 4 matdyn.x -i matdyn.GaAs.in > matdyn.GaAs.out We can now plot the phonon dispersion of GaAs: notebooks/GaAs-phonon.ipynb import numpy as np import matplotlib.pyplot as plt data = np.loadtxt(\"../src/GaAs-phonon/GaAs.freq.gp\") nbands = data.shape[1] - 1 for band in range(nbands): plt.plot(data[:, 0], data[:, band], linewidth=1, alpha=0.5, color='k') # High symmetry k-points (check matdyn.GaAs.in) plt.axvline(x=data[0, 0], linewidth=0.5, color='k', alpha=0.5) plt.axvline(x=data[20, 0], linewidth=0.5, color='k', alpha=0.5) plt.axvline(x=data[40, 0], linewidth=0.5, color='k', alpha=0.5) plt.axvline(x=data[60, 0], linewidth=0.5, color='k', alpha=0.5) plt.xticks(ticks= [0, data[20, 0], data[40, 0], data[60, 0], data[-1, 0]], \\ labels=['L', '$\\Gamma$', 'X', 'U,K', '$\\Gamma$']) plt.ylabel(\"Frequency (cm$^{-1}$)\") plt.xlim(data[0, 0], data[-1, 0]) plt.ylim(0, ) plt.show() tip We may need to lower the value of conv_thr in scf calculation for more accurate result.","s":"Phonon dispersion","u":"/espresso/hands-on/phonon","h":"","p":82},{"i":85,"t":"Input file for phonon DOS calculation: src/GaAs-phonon/matdyn.dos.GaAs.in &INPUT asr = 'crystal' flfrc = 'GaAs.fc' flfrq = 'GaAs.dos.freq' flvec = 'GaAs.dos.modes' dos = .true. fldos = 'GaAs.dos' nk1 = 25 nk2 = 25 nk3 = 25 / Plot phonon DOS: notebooks/GaAs-phonon.ipynb freq, dos, pdos_Ga, pdos_As = np.loadtxt(\"../src/GaAs-phonon/GaAs.dos\", unpack=True) plt.plot(freq, dos, c='k', lw=0.5, label='Total') plt.plot(freq, pdos_Ga, c='b', lw=0.5, label='Ga') plt.plot(freq, pdos_As, c='r', lw=0.5, label='As') plt.xlabel('$\\\\Omega~(cm^{-1}$)') plt.ylabel('Phonon DOS (state/cm$^{-1}/u.c.$)') plt.legend(frameon=False, loc='upper left') plt.xlim(freq[0], freq[-1]) plt.show()","s":"Phonon Density of States","u":"/espresso/hands-on/phonon","h":"#phonon-density-of-states","p":82},{"i":87,"t":"School on Electron-Phonon Physics from First Principles (2018) (Video lectures on YouTube) https://github.com/nguyen-group/QE-SSP","s":"Resources","u":"/espresso/hands-on/phonon","h":"#resources","p":82},{"i":89,"t":"We need to provide various important parameters for the self consistent calculation (solves the Kohn-Sham equation self-consistently) via an input file. In QE input files, there are NAMELISTS and INPUT_CARDS. NAMELISTS variables have default values, and new values can be provided as required for a specific calculation. The variables can be declared in any specific order. On the other hand, the variables in the INPUT_CARDS has always to be specified and in specific order. Logically independent INPUT_CARDS may be organized in any order. There are three mandatory NAMELISTS in PWscf: (1) &CONTROL: specifies the flux of computation, (2) &SYSTEM: specifies the system, and (3) &ELECTRONS: specifies the algorithms used to solve the Kohn-Sham equation. There are two other NAMELISTS: &IONS and &CELLS, which need to be specified depending on the calculation. Three INPUT_CARDS: ATOMIC_SPECIES, ATOMIC_POSITIONS, and K_POINTS in PWscf are mandatory. There are few others that must be provided in certain calculations. Below is our input file pw.scf.silicon.in for silicon in standard diamond (FCC) structure. The input files are typically named with .in prefix, while output files are named with .out prefix for their easier identification. The input parameters are organized in &namelists followed by their fields or cards. The &control, &system, and &electrons namelists are required. There are also optional &cell and &ions, you must provide them if your calculation require them. Most parameters in the namelists have default values (which may or may not suit your needs), however some variables you must always provide. Comment lines can be added with lines starting with a ! like in Fortran. Also, parameter names are not case-sensitive, i.e., &control and &CONTROL are the same. src/silicon/pw.scf.silicon.in &CONTROL ! we want to perform self consistent field calculation calculation = 'scf', ! prefix is reference to the output files prefix = 'silicon', ! output directory. Note that it is deprecated. outdir = './tmp/' ! directory for the pseudo potential directory pseudo_dir = '../pseudos/' ! verbosity high will give more details on the output file verbosity = 'high' / &SYSTEM ! Bravais lattice index, which is 2 for FCC structure ibrav = 2, ! Lattice constant in BOHR celldm(1) = 10.26, ! number of atoms in an unit cell nat = 2, ! number of different types of atom in the cell ntyp = 1, ! kinetic energy cutoff for wavefunction in Ry ecutwfc = 30 ! number of bands to calculate nbnd = 8 / &ELECTRONS ! Mixing factor used in the self-consistent method mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 6 6 6 0 0 0 I am using the pseudo potential file (Si.pz-vbc.UPF) downloaded from Quantum Espresso Website. You must read the PWscf user manual for in-depth understanding. Check the qe-x.x/PW/Doc/ folder under your installation directory. There is also another file INPUT_PW.html regarding the details of input parameters. PW stands for plane waves. Run pw.x in self consistent mode for silicon. pw.x < pw.scf.silicon.in > pw.scf.silicon.out # For parallel execution mpirun -np 4 pw.x -inp pw.scf.silicon.in > pw.scf.silicon.out note Note that I have added the executable path to my bash/zsh profile, otherwise you have to provide the full path where the pw.x executable is located. Now let’s look at the output file pw.scf.silicon.out and see how the convergence is reached: grep -e 'total energy' -e estimate pw.scf.silicon.out and you should see something like this: total energy = -15.85014573 Ry Harris-Foulkes estimate = -15.86899637 Ry estimated scf accuracy < 0.06093037 Ry total energy = -15.85194177 Ry Harris-Foulkes estimate = -15.85292281 Ry estimated scf accuracy < 0.00462014 Ry total energy = -15.85218359 Ry Harris-Foulkes estimate = -15.85220235 Ry estimated scf accuracy < 0.00011293 Ry ! total energy = -15.85219789 Ry Harris-Foulkes estimate = -15.85219831 Ry estimated scf accuracy < 0.00000099 Ry The total energy is the sum of the following terms: It is important to note that the absolute value of DFT total energy is not with respect to the vacuum reference, and depends on the chosen pseudopotential. The meaningful measure is the difference in total energy, where various offsets cancel out. note In the above calculation, if you check the output file pw.scf.silicon.out, you will find: highest occupied, lowest unoccupied level (eV): 6.2117 6.8442. Therefore, the bandgap is 0.6325 eV, which is an underestimation of actual bandgap (1.12 eV). Tips on convergence Reduce mixing_beta value, especially if there is an oscillation around the convergence energy. If it is a metallic system, use smearing and degauss. In this case, the SCF accuracy gradually goes down then suddenly increases (due to slight change in Fermi energy highest occupied/ lowest unoccupied levels change). Increase energy and charge density cutoffs (make sure they are sufficient). Certain pseudo potential files have issues, you may try with pseudo potentials from different libraries. Suggested values for the conv_thr: for energy and eigenvalues (scf calculation) 1.0d-7, for forces (relax calculation) 1.0d-8, for stress (vc-relax calculation) 1.0d-9 Ry. For certain calculation convergence might be very slow for the first iteration, one can start the calculation with a higher threshold, after few iterations reduce it and restart the calculation. There are several other important information is printed on the output file. Exchange correlation used in the calculation: Exchange-correlation= SLA PZ NOGX NOGC Where SLA → Slater exchange; PZ → Perdew-Zunger parametrization of the LDA; NOGX and NOGC indicates that density gradients are not taken into account. We can see the total number of plane waves (1067) uses in our calculation: Parallelization info -------------------- sticks: dense smooth PW G-vecs: dense smooth PW Min 108 108 34 1489 1489 266 Max 109 109 35 1492 1492 267 Sum 433 433 139 5961 5961 1067 Number of Kohn-Sham states: number of electrons = 8.00 number of Kohn-Sham states= 8 In our calculation we have specified the number of bands = 8. Otherwise, there would be 4 bands for 8 electrons in case of non spin-polarized systems.","s":"Self consistent field calculation for silicon","u":"/espresso/hands-on/scf","h":"","p":88},{"i":91,"t":"https://www.quantum-espresso.org/Doc/pw_user_guide/ Quantum Espresso Input Generator (can help crating QE input files)","s":"Resources","u":"/espresso/hands-on/scf","h":"#resources","p":88},{"i":93,"t":"In order to consider spin orbit coupling effect in our electronic structure calculation in quantum espresso, we need to use a full relativistic pseudo potential. Following settings are needed in the &SYSTEM card: &SYSTEM ... noncolin = .true. lspinorb = .true. ... /","s":"Spin-Orbit Coupling","u":"/espresso/hands-on/soc","h":"","p":92},{"i":95,"t":"In simple spin polarized calculation (nspin=2), the spin quantum number (up or down) is considered in the calculation. In non-collinear case, the spin has more degrees of freedom, and can be oriented in any direction. Non-collinear magnetism is quite common in nature, where the spins are not parallel (ferromagnetic) or anti-parallel (antiferromagnetic), rather they orient in spirals, helicoids, canted or disordered. Non-collinear magnetism can occur because of geometric frustration of magnetic interaction. It can also occur due to the magnetocrystalline anisotropy which is the result of interaction between the spin and lattice interaction. This relativistic effect comes via spin-orbit coupling. We can constrain the magnetic moment: &SYSTEM ... constrained_magnetization = 'atomic direction' ... / Starting magnetization can be specified by angle1 (angle with zzz axis) and angle2 (angle of projection in xyxyxy-plane and with xxx-axis). Also check the penalty function (λ\\lambdaλ). &SYSTEM ... angle1(i) = 0.0d0 angle2(i) = 0.0d0 lambda = 0.5 ... / i is the index of the atom in ATOMIC_SPECIES card.","s":"Non collinear spin","u":"/espresso/hands-on/soc","h":"#non-collinear-spin","p":92},{"i":97,"t":"Spin-orbit coupling calculations are often hard to converge. Use a smaller mixing_beta for such calculations. First perform a collinear calculation with non-relativistic pseudopotential, and then start from the obtained charge density to perform non-colinear spin orbit calculation. &ELECTRONS ... mixing_beta = 1.0000000000d-01 startingpot = 'file' / When starting with non-collinear calculation from an existing charge density file from a collinear lsda calculation, we need to set lforcet=.true.. It assumes previous density points in z direction, and rotates in the direction specified by angle1 (initial magnetization angle with zzz-axis in degrees), and angle2 (angle in degrees for projections in xyxyxy-plane and with xxx-axis). &SYSTEM ... angle1(i) = 0.0 angle2(i) = 0.0 lforcet = .true. / Also, make sure that energy and charge density cutoffs are sufficient. Certain pseudo potentials might have issues, try with pseudo potentials from a different library. In case of metallic systems, remember to apply smearing. Common Errors S matrix not positive definite: This error might appear due to numerical instability from overlapping atoms. Check atomic positions carefully. In one my calculations, this error was resolved after setting higher ecutrho. Simplified LDA+U not implemented with noncol magnetism, use lda_plus_u_kind=1.","s":"Strategy for convergence","u":"/espresso/hands-on/soc","h":"#strategy-for-convergence","p":92},{"i":99,"t":"src/fe/pw.scf.fe_soc.in &control calculation='scf' pseudo_dir = '../pseudos/', outdir='./tmp/' prefix='fe' / &system ibrav = 3, celldm(1) = 5.39, nat= 1, ntyp= 1, noncolin=.true., lspinorb=.true., starting_magnetization(1)=0.3, ecutwfc = 70, ecutrho = 850.0, occupations='smearing', smearing='marzari-vanderbilt', degauss=0.02 / &electrons diagonalization='david' conv_thr = 1.0e-8 mixing_beta = 0.7 / ATOMIC_SPECIES Fe 55.845 Fe.rel-pbe-spn-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS alat Fe 0.0 0.0 0.0 K_POINTS AUTOMATIC 14 14 14 1 1 1 Run the scf calculation: mpirun -np 8 pw.x -i pw.scf.fe_soc.in > pw.scf.fe_soc.out Prepare the input file for nscf bands calculation: src/fe/pw.bands.fe_soc.in &control calculation='bands' pseudo_dir = '../pseudos/', outdir='./tmp/' prefix='fe' / &system ibrav = 3, celldm(1) = 5.39, nat= 1, ntyp= 1, noncolin=.true., lspinorb=.true., starting_magnetization(1)=0.3, ecutwfc = 70, ecutrho = 850.0, occupations='smearing', smearing='marzari-vanderbilt', degauss=0.02 / &electrons diagonalization='david' conv_thr = 1.0e-8 mixing_beta = 0.7 / ATOMIC_SPECIES Fe 55.845 Fe.rel-pbe-spn-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS alat Fe 0.0 0.0 0.0 K_POINTS tpiba_b 6 0.000 0.000 0.000 40 !gamma 0.000 1.000 0.000 40 !H 0.500 0.500 0.000 30 !N 0.000 0.000 0.000 30 !gamma 0.500 0.500 0.500 30 !P 0.000 1.000 0.000 1 !H Run the bands calculation: mpirun -np 8 pw.x -i pw.bands.fe_soc.in > pw.bands.fe_soc.out Finally post process the bandstructure data: src/fe/pp.bands.fe_soc.in &BANDS outdir='./tmp/', prefix='fe', filband='fe_bands_soc.dat', / In this case spin_component has been removed and we add lsigma(3)=.true. that instructs the program to compute the expectation value for the z component of the spin operator for each eigenfunction and save all values in the file fe.noncolin.data.3. All values in this case are either +1/2 or -1/2. mpirun -np 8 bands.x -i pp.bands.fe_soc.in > pp.bands.fe_soc.out","s":"Bandstructure of Fe with SOC","u":"/espresso/hands-on/soc","h":"#bandstructure-of-fe-with-soc","p":92},{"i":101,"t":"Please check the respective input files. mpirun -np 8 pw.x -i pw.scf.GaAs_soc.in > pw.scf.GaAs_soc.out mpirun -np 8 pw.x -i pw.bands.GaAs_soc.in > pw.bands.GaAs_soc.out mpirun -np 8 bands.x -i pp.bands.GaAs_soc.in > pp.bands.GaAs_soc.out","s":"SOC calculation for GaAs","u":"/espresso/hands-on/soc","h":"#soc-calculation-for-gaas","p":92},{"i":104,"t":"Perform scf calculation using Quantum Espresso pw.x QE_PATH=\"/workspaces/q-e-qe-7.2/bin\" mpirun -np 4 ${QE_PATH}/pw.x -i pw.scf.silicon.in > pw.scf.silicon.out Perform nscf calculation using pw.x. Instead of automatic k-grid, we need to provide explicit list of k-points. Such explicit list of k-points can be generated using perl script included in the Wannier package under utility. WANNIER_PATH=\"/workspaces/wannier90-3.1.0\" # directly append the k-points to the input file ${WANNIER_PATH}/utility/kmesh.pl 4 4 4 >> pw.nscf.silicon.in Run nscf calculation: mpirun -np 4 ${QE_PATH}/pw.x -i pw.nscf.silicon.in > pw.nscf.silicon.out Prepare input file for wannier90 (silicon.win). Here we need the k-points list without the weights: ${WANNIER_PATH}/utility/kmesh.pl 4 4 4 wan Generate nnkp input: # we can just provide the seedname or seedname.win ${WANNIER_PATH}/wannier90.x -pp silicon Create input file for pw2wan, and generate initial projections: mpirun -np 4 ${WANNIER_PATH}/pw2wannier90.x -i pw2wan.silicon.in > pw2 wan.silicon.out Run wannier calculation: mpirun -np 4 ${WANNIER_PATH}/wannier90.x silicon","s":"Obtain bandstructure of Silicon","u":"/espresso/hands-on/wannier","h":"#obtain-bandstructure-of-silicon","p":102},{"i":106,"t":"https://sites.google.com/view/hubbard-koopmans/program","s":"Resources","u":"/espresso/hands-on/wannier","h":"#resources","p":102},{"i":108,"t":"There are two types of structural optimization calculations in Quantum espresso: (1) relax: where only the atomic positions are allowed to vary, and (2) vc-relax: which allows to vary both the atomic positions and lattice constants. src/silicon/si_relax.in &control calculation = 'vc-relax' prefix = 'silicon' outdir = './tmp/' pseudo_dir = './pseudos/' etot_conv_thr = 1e-5 forc_conv_thr = 1e-4 / &system ibrav=2, celldm(1) =14, nat=2, ntyp=1, ecutwfc=30 / &electrons conv_thr=1e-8 / &ions / &cell cell_dofree='ibrav' / ATOMIC_SPECIES Si 28.0855 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 0 0 0 Si 0.25 0.25 0.25 0 0 0 K_POINTS (automatic) 6 6 6 1 1 1 Perform the plane wave calculation: pw.x -inp si_relax.in > si_relax.out This produces following output (see the si_relax.out for more details, look for \"Final enthalpy\"): Final enthalpy = -15.8536258868 Ry Begin final coordinates new unit-cell volume = 265.89380 a.u.^3 ( 39.40140 Ang^3 ) density = 2.36728 g/cm^3 CELL_PARAMETERS (alat= 14.00000000) -0.364556379 0.000000000 0.364556379 0.000000000 0.364556379 0.364556379 -0.364556379 0.364556379 0.000000000 ATOMIC_POSITIONS (alat) Si 0.0000000000 0.0000000000 0.0000000000 0 0 0 Si 0.1822781896 0.1822781896 0.1822781896 0 0 0 End final coordinates Lattice constant = 0.364556379 * 14 / 0.5 = 10.2076 Bohr.","s":"Structure optimization","u":"/espresso/hands-on/structure-optimization","h":"","p":107},{"i":110,"t":"This work is licensed under a Creative Commons Attribution 4.0 International License. Any third party materials in this work are not included in the article’s Creative Commons license, and users will need to obtain permission from the respective license holder to reproduce such materials. You are free to: Share — copy and redistribute the material in any medium or format. Adapt — remix, transform, and build upon the material for any purpose, even commercially. Under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. Notices: No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. To view full copy of this license, visit http://creativecommons.org/licenses/by/4.0/","s":"License","u":"/espresso/license","h":"","p":109},{"i":112,"t":"Quantum Espresso Tutorials ICTP Quantum Espresso workshop 2021 Ljubljana QE summer school 2019 MASTANI Summer School, IISER Pune 2014 (archive) Examples included in Quantum Espresso (qe-x.x/PW/examples/). QE mailing list archive Materials square blog Cornell PARADIM Summer School School on Electron-Phonon Physics from First Principles (2018)","s":"Resources","u":"/espresso/resources","h":"","p":111},{"i":114,"t":"QUANTUM ESPRESSO for quantum simulations of materials Advanced capabilities for materials modelling with QE Hubbard parameters from density-functional perturbation theory Self-consistent Hubbard parameters from DFPT","s":"Papers describing DFT implementations in QE","u":"/espresso/resources","h":"#papers-describing-dft-implementations-in-qe","p":111},{"i":116,"t":"A bird's–eye view of DFT Density Functional Theory: A Practical Introduction by Sholl and Steckel Materials Modelling using Density Functional Theory by Feliciano Giustino Electronic Structure: Basic Theory and Practical Method by Richard M. Martin Electronic Structure Calculations for Solids and Molecules by Jorge Kohanoff PhD Thesis of Dominik Bogdan Jochym","s":"Books","u":"/espresso/resources","h":"#books","p":111},{"i":118,"t":"We can install Quantum Espresso on our personal laptops or desktops to run relatively less computationally intensive calculations. If we intend to perform computationally heavy tasks, we would need access to better computing resources with large number of CPU (or GPU) cores, memory, bandwidth, and disc IO. Throughout this tutorial, I will be using a Ubuntu system for smaller calculations while other computationally intensive calculations will be done in HPC clusters. Perhaps the easiest way to install Quantum Espresso is from the package manager of respective Linux distribution. This should work fine for us and this is recommended option. Following commands are for Ubuntu / Debian. First make sure your system is up-to-date. sudo apt update && sudo apt upgrade Install Quantum Espresso from apt repository: sudo apt install --no-install-recommends \\ libfftw3-dev \\ quantum-espresso If you want to compile from the source yourself, here are the installation steps for the Quantum Espresso version 7.2 in a Ubuntu (LTS 22.04) system. I will be compiling for single processor. First install the recommended libraries and dependencies: sudo apt install --no-install-recommends \\ autoconf \\ build-essential \\ ca-certificates \\ gfortran \\ libblas3 \\ libc6 \\ libfftw3-dev \\ libgcc-s1 \\ liblapack-dev \\ wget If you want to compile for parallel processing, you also need to install: sudo apt install --no-install-recommends \\ libopenmpi-dev \\ libscalapack-openmpi-dev \\ libelpa17 # use libelpa4 on Ubuntu 20.04 Download Quantum Espresso (latest version 7.2 at the time of writing): wget https://gitlab.com/QEF/q-e/-/archive/qe-7.2/q-e-qe-7.2.tar.gz Un-tar the source files: tar -zxvf q-e-qe-7.2.tar.gz Go to the qe directory and issue configure: cd q-e-qe-7.2 ./configure Here we can provide various configuration options. Read the manual in oder to properly understand. But in most cases we will be just fine with the defaults, it should detect the system configuration automatically, in case you don't get what you want, try the various configuration flags with configure. caution Note that certain programs/utilities bundled with Quantum Espresso might not work correctly in parallel compilation, so we may need serial compilation for those by ./configure --disable-parallel option in case parallel option is automatically detected. Finally, compile the source files and create the binary executables: # compile individual packages make pw # or compile everything make all # we can parallelize e.g., below command uses 4 CPUs make -j4 all Now, the binary files or their symbolic links (shortcuts) would be placed in the bin directory. It would be good idea to include the executable path to your .bashrc (or .zshrc or whatever shell you use) file: # use the correct path if it differs from mine echo 'export PATH=\"/root/q-e-qe-7.2/bin:$PATH\"' >> ~/.bashrc Finally, you may need to restart your terminal or source .bashrc. source ~/.bashrc You can compile the documentation by going to particular directory (e.g., PW or PP) and execute (you need to have LaTeX installed in your system): make doc If you want docs in PDF format, you can use latex commands to create them as well: pdflatex filename.tex We are now ready to run Quantum Espresso pw.x (or any other program) using mpirun by following command: pw.x -inp inputfile > outputfile # For parallel version mpirun -np 12 pw.x -inp inputfile > outputfile Where -np 12 specifies the number of processors. -inp stands for input file. Alternatively, we can use -i, or -in, or -input, or even standard input redirect <. But beware some systems may not interpret all the different options, I think safe option is to use -i. Once installation is completed, optionally we can run tests if everything went OK. Go to the test-suite directory and run make run-tests If all is well, we will see Passed messages and we are good to go. caution Note that the above installation steps may not be the most optimal way to run Quantum Espresso in your computer. There are multiple implementations of same library. For example, you can replace openmpi libraries with Intel MKL or MPICH implementations. Please do research yourself or ask help from someone who has knowledge about high performance computing.","s":"Quantum Espresso installation","u":"/espresso/setup/install","h":"","p":117},{"i":120,"t":"We will install a very hand scripting package PWscf Toolkit (PWTK). First we need to install following dependencies: sudo apt install tcl tcllib Download the file from - http://pwtk.ijs.si/download/pwtk-2.0.tar.gz wget \"http://pwtk.ijs.si/download/pwtk-2.0.tar.gz\" Above command will download and save the file to your current directory. Next we need to just un-tar (no need to compile): tar -zxvf pwtk-2.0.tar.gz Add the path (modify below as appropriate) to .bashrc: echo 'export PATH=\"/root/pwtk-2.0:$PATH\"' >> ~/.bashrc source ~/.bashrc","s":"Installing PWTK","u":"/espresso/setup/install","h":"#installing-pwtk","p":117},{"i":122,"t":"In order to perform computationally heavy calculations, we would require access to high performance computing facilities.","s":"High Performance Computing","u":"/espresso/setup/hpc","h":"","p":121},{"i":124,"t":"Connect to a login node via ssh: ssh {username}@atlas9.nus.edu.sg Secure copy files between local and remote machines: scp {username}@10.10.0.2:/remote/file.txt /local/directory scp local/file.txt {username}@10.10.0.2:/remote/directory Check disk usage: du -hs . du -hs /path/ Rsync to synchronize two folders: rsync -azhv --delete /source/my_project/ /destination/my_project","s":"Useful UNIX commands","u":"/espresso/setup/hpc","h":"#useful-unix-commands","p":121},{"i":126,"t":"Check your storage quota: hpc s PBS commands: hpc pbs summary Example scrips for job submissions: hpc pbs script parallel20 hpc pbs vasp List available modules: module avail Load a module: module load {module-name} Purge loaded modules: module purge Quantum Espresso is already installed in NUS HPC clusters. Here is a sample job script for NUS HPC clusters: scripts/pbs_job.sh #!/bin/bash #PBS -q parallel24 #PBS -l select=2:ncpus=24:mpiprocs=24:mem=96GB #PBS -j eo #PBS -N qe-project-xx source /etc/profile.d/rec_modules.sh module load espresso6.5-intel_18 ## module load espresso6.5-Centos6_Intel cd $PBS_O_WORKDIR; np=$( cat ${PBS_NODEFILE} |wc -l ); mpirun -np $np -f ${PBS_NODEFILE} pw.x -inp qe-scf.in > qe-scf.out info Notice that the lines beginning with #PBS are actually PBS commands, not comments. For comments, I am using ##. Query about a queue system: qstat -q Check status of a particular queue system: qstat -Qx parallel24 Submitting a job: qsub pbs_job.sh Check running jobs: qstat Details about a job: qstat -f {job-id} Stopping a job: qdel {job-id}","s":"Running jobs at NUS HPC","u":"/espresso/setup/hpc","h":"#running-jobs-at-nus-hpc","p":121},{"i":128,"t":"If you need to modify certain parameters while the program is running, e.g., you want to change the mixing_beta value because SCF accuracy is oscillation without any sign of convergence. Create an empty file named {prefix}.EXIT in the directory where you have the input file or in the outdir as set in the &CONTROL card of input file. touch {prefix}.EXIT That will stop the program on the next iteration, and save the state. In order to restart, set the restart_mode in &CONTROL card to 'restart' and re-run after necessary changes. You must re-submit the job with the same number of processors. &CONTROL ... restart_mode = 'restart' ... /","s":"Abort and restart a calculation","u":"/espresso/setup/hpc","h":"#abort-and-restart-a-calculation","p":121},{"i":130,"t":"If you need a newer or specific version of Quantum Espresso that is not installed in the NUS clusters or you have modified the source codes yourself, here are the steps that I followed to successfully compile. info Quantum Espresso project is primarily hosted on GitLab, and its mirror is maintained at GitHub. You may check their repository at GitLab for more up to date information. The releases via GitLab can be found under: https://gitlab.com/QEF/q-e/-/releases Download and decompress the source files. wget https://gitlab.com/QEF/q-e/-/archive/qe-7.2/q-e-qe-7.2.tar.gz tar -zxvf q-e-qe-7.2.tar.gz Load the necessary modules (applicable for NUS clusters, last checked in Jun 2022): module load xe_2018 module load fftw/3.3.7 Go to QE directory and run configure: cd q-e-qe-7.2 ./configure You will see output something like: ... BLAS_LIBS= -lmkl_intel_lp64 -lmkl_sequential -lmkl_core LAPACK_LIBS= FFT_LIBS= ... For me, the LAPACK_LIBS and FFT_LIBS libs were not automatically detected. We need to specify them manually. First, get the link libraries line specific to your version of MKL and other configurations from the Intel link advisor. For my case, the link line was: -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl We need to insert the link for BLAS_LIBS, LAPACK_LIBS, and SCALAPACK_LIBS. We also need to find out where is the FFTW lib located. In NUS HPC, we can use module avail command to see where a particular module is located, usually under /app1/modules/. Open make.inc and make the following changes: make.inc # ... CFLAGS = -O2 $(DFLAGS) $(IFLAGS) CFLAGS = -O3 $(DFLAGS) $(IFLAGS) F90FLAGS = $(FFLAGS) -nomodule -fpp $(FDFLAGS) $(CUDA_F90FLAGS) $(IFLAGS) $(MODFLAGS) # compiler flags with and without optimization for fortran-77 # the latter is NEEDED to properly compile dlamch.f, used by lapack - FFLAGS = -O2 -assume byterecl -g -traceback + FFLAGS = -O3 -assume byterecl -g -traceback FFLAGS_NOOPT = -O0 -assume byterecl -g -traceback # ... # If you have nothing better, use the local copy # BLAS_LIBS = $(TOPDIR)/LAPACK/libblas.a - BLAS_LIBS = -lmkl_intel_lp64 -lmkl_sequential -lmkl_core + BLAS_LIBS = -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl # If you have nothing better, use the local copy # LAPACK = liblapack # LAPACK_LIBS = $(TOPDIR)/external/lapack/liblapack.a - LAPACK = + LAPACK = liblapack - LAPACK_LIBS = + LAPACK_LIBS = -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl - SCALAPACK_LIBS = + SCALAPACK_LIBS = -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl # nothing is needed here if the internal copy of FFTW is compiled # (needs -D__FFTW in DFLAGS) - FFT_LIBS = + FFT_LIBS = -L/app1/centos6.3/gnu/fftw/3.3.7/lib/ -lmpi # ... Now we are ready to compile: make -j8 all I am parallelizing with 8 processors to speed things up. You may add the q-e-qe-7.2/bin path to your .bashrc: echo 'export PATH=\"/home/svu/{username}/q-e-qe-7.2/bin:$PATH\"' >> ~/.bashrc And don't forget to load dependencies before calling QE executables. module load xe_2018 module load fftw/3.3.7 note If you are submitting job via PBS queue, you need to provide full path of the QE executables, e.g., /home/svu/{username}/q-e-qe-7.2/bin/pw.x. PBS system won't read your bash settings, neither the relative paths of your login node would apply.","s":"Compiling Quantum Espresso using Intel® Math Kernel Library (MKL)","u":"/espresso/setup/hpc","h":"#compiling-quantum-espresso-using-intel-math-kernel-library-mkl","p":121},{"i":132,"t":"If you need to install Intel oneAPI libraries yourself, following instructions might be useful. Please refer to Intel website for up to date information. Intel oneAPI Base Toolkit:​ wget https://registrationcenter-download.intel.com/akdlm/IRC_NAS/992857b9-624c-45de-9701-f6445d845359/l_BaseKit_p_2023.2.0.49397_offline.sh # requires gnu-awk sudo apt update && sudo apt install -y --no-install-recommends gawk gcc g++ # interactive cli installation sudo apt install -y --no-install-recommends ncurses-term sudo sh ./l_BaseKit_p_2023.2.0.49397_offline.sh -a --cli # list components included in oneAPI Base Toolkit sh ./l_BaseKit_p_2023.2.0.49397_offline.sh -a --list-components # install a subset of components with silent/unattended option sudo sh ./l_BaseKit_p_2023.2.0.49397_offline.sh -a --silent --eula accept --components intel.oneapi.lin.dpcpp-cpp-compiler:intel.oneapi.lin.mkl.devel note If you install oneAPI without sudo privilege, it will be installed under the user directory: /home/{username}/intel/oneapi/. After installation is completed, the setup script will print the installation location. HPC Toolkit​ wget https://registrationcenter-download.intel.com/akdlm/IRC_NAS/0722521a-34b5-4c41-af3f-d5d14e88248d/l_HPCKit_p_2023.2.0.49440_offline.sh sudo sh ./l_HPCKit_p_2023.2.0.49440_offline.sh -a --silent --eula accept Intel MKL library​ Installing individual components: wget https://registrationcenter-download.intel.com/akdlm/IRC_NAS/adb8a02c-4ee7-4882-97d6-a524150da358/l_onemkl_p_2023.2.0.49497_offline.sh sudo sh ./l_onemkl_p_2023.2.0.49497_offline.sh -a --silent --eula accept After installation, do not forget to source the environment variables before using: source /opt/intel/oneapi/setvars.sh Compile quantum espresso: wget https://gitlab.com/QEF/q-e/-/archive/qe-7.2/q-e-qe-7.2.tar.gz tar -zxvf q-e-qe-7.2.tar.gz rm q-e-qe-7.2.tar.gz cd q-e-qe-7.2 ./configure \\ F90=mpiifort \\ MPIF90=mpiifort \\ CC=mpicc CXX=icc \\ F77=mpiifort \\ FFLAGS=\"-O3 -assume byterecl -g -traceback\" \\ LAPACK_LIBS=\"-L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl\" \\ BLAS_LIBS=\"-L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl\" \\ SCALAPACK_LIBS=\"-L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl\" make -j4 all","s":"Installing Intel oneAPI libraries","u":"/espresso/setup/hpc","h":"#installing-intel-oneapi-libraries","p":121},{"i":134,"t":"Please check out the official documentation for more details. It requires cmake version 3.14 or later. apt update && apt install autoconf cmake gawk gcc g++ make I used following steps to successfully compile Quantum Espresso using 2023 versions of Intel libraries in Ubuntu 22.04 system: cd q-e-qe-7.2 mkdir build && cd build cmake -DCMAKE_C_COMPILER=mpiicc -DCMAKE_Fortran_COMPILER=mpiifort -DQE_ENABLE_SCALAPACK=ON .. make -j4 mv bin .. cd .. rm -rf build","s":"Compiling Quantum Espresso with CMake","u":"/espresso/setup/hpc","h":"#compiling-quantum-espresso-with-cmake","p":121},{"i":136,"t":"https://nusit.nus.edu.sg/services/getting-started/introductory-guide-for-new-hpc-users/ https://help.nscc.sg/pbspro-quickstartguide/ https://www.youtube.com/watch?v=doudMLEaq3w","s":"Resources","u":"/espresso/setup/hpc","h":"#resources","p":121},{"i":139,"t":"Density functional theory (DFT) calculations are ab-initio meaning the calculation is done from the scratch based on given input parameters. We need to provide the crystal structure in order to calculate DFT. Crystal structures are widely available in Crystallographic Information File (.CIF) format. There are several databases where you can look for crystal structures. http://crystallography.net/cod/ https://materialsproject.org https://mpds.io/ https://icsd.fiz-karlsruhe.de/index.xhtml http://aflowlib.org/CrystalDatabase/ http://crystdb.nims.go.jp/crystdb/search-materials In Quantum Espresso, the structure information is provided by ibrav number, and corresponding celldm values or lattice constants and cosines of angle between the axes. It is also possible to set ibrav=0 and provide lattice vectors in CELL_PARAMETERS. danger When set ibrav=0, the lattice vectors must be provided with sufficiently large number of decimal accuracy, otherwise symmetry detection may fail and strange problems may arrise. ibrav numbers for different lattice types: ibrav Lattice type 1 Simple cubic 2 Face centered cubic 3,-3 Body centered cubic 4 Hexagonal 5 Trigonal with c as 3-fold axis -5 Trigonal with <111> as 3-fold axis 6 Simple tetragonal 7 Centered tetragonal 8 Simple orthorhombic 9,-9,91 One-face centered orthorhombic 10 Face centered orthorhombic 11 Body centered orthorhombic 12 Simple monoclinic, c unique -12 Simple monoclinic, b unique 13 One base centered monoclinic, c unique -13 One base centered monoclinic, b unique 14 Triclinic","s":"Structure databases","u":"/espresso/setup/crystal-structure","h":"#structure-databases","p":137},{"i":141,"t":"Vesta - https://jp-minerals.org/vesta/en/. It helps you visualize crystal structure, create and modify supercells, crystal structures, and many other useful functionalities. We can prepare our Quantum Espresso input file using cif2cell utility. If you do not have cif2cell installed, you can use pip to install: sudo pip3 install cif2cell You may need to add it to the path in your .bashrc manually: export PATH=\"/home/pranab/.local/lib/python3.8/site-packages/:$PATH\" Running cif2cell command: cif2cell file.cif -p quantum-espresso -o inputfile.in","s":"Useful tools:","u":"/espresso/setup/crystal-structure","h":"#useful-tools","p":137},{"i":143,"t":"You can explore the crystal structure, find out k-path and many more using Xcrysdens application - http://www.xcrysden.org For certain functionality, Xcrysdens requires basic calculator program. On Ubuntu/ Debian: sudo apt update sudo apt install bc xcrysden Manual installation: # install dependencies sudo apt install --no-install-recommends bc tk libglu1-mesa libtogl2 \\ libfftw3-3 libxmu6 imagemagick openbabel libgfortran5 # download the latest version of xcrysden and extract wget http://www.xcrysden.org/download/xcrysden-1.6.2-linux_x86_64-shared.tar.gz tar -zxvf xcrysden-1.6.2-linux_x86_64-shared.tar.gz # launch (provided you extracted under your home directory) ~/xcrysden-1.6.2-bin-shared/xcrysden If you are on WSL, you need to install X-server (X-ming for Windows) on the host and set export DISPLAY=:0 in your WSL instance.","s":"Xcrysdens","u":"/espresso/setup/crystal-structure","h":"#xcrysdens","p":137},{"i":145,"t":"You can generate PWscf input files using tools in this website as well https://www.materialscloud.org/work/tools/qeinputgenerator The same website also has a tool for k-path visualization and generation https://www.materialscloud.org/work/tools/seekpath","s":"QE Input generator","u":"/espresso/setup/crystal-structure","h":"#qe-input-generator","p":137},{"i":147,"t":"Supercell construction using Vesta","s":"Resources","u":"/espresso/setup/crystal-structure","h":"#resources","p":137},{"i":149,"t":"There are several ways you can run Jupyterlab in your computer.","s":"Jupyter notebooks","u":"/espresso/setup/jupyter","h":"","p":148},{"i":151,"t":"Install Python 3 in your computer # on ubuntu / debian apt install python3 python3-pip Install the required python packages on your computer pip3 install --upgrade -r requirements.txt # or pip3 install --upgrade numpy scipy matplotlib jupyterlab Run Jupyterlab jupyter-lab # or the classic jupyter notebook jupyter-notebook","s":"1. Install on your computer","u":"/espresso/setup/jupyter","h":"#1-install-on-your-computer","p":148},{"i":153,"t":"Install Python 3 and virtualenv on your computer pip3 install --upgrade virtualenv create virtual environment in the project directory cd qe-dft virtualenv venv activate virtual env source venv/bin/activate Install required python packages under virtualenv Launch Jupyterlab Once done, deactivate virtualenv deactivate","s":"2. Install python packages via virtualenv","u":"/espresso/setup/jupyter","h":"#2-install-python-packages-via-virtualenv","p":148},{"i":155,"t":"Install Docker Create an image with Python and the required packages installed # build using the Dockerfile included in my github repo: # https://github.com/pranabdas/espresso # (adjust the Dockerfile according to your needs) docker build -t espresso . Run a container with port forwarding docker run -it --rm -p 8888:8888 -v ${PWD}:/home espresso bash Launch Jupyterlab jupyter-lab","s":"3. Run on a container","u":"/espresso/setup/jupyter","h":"#3-run-on-a-container","p":148},{"i":157,"t":"In Quantum Espresso, pseudopotential replaces the actual electron-ion interaction. The pseudopotential describes the atomic nucleus and all the electrons except the outermost valence shell. The rapidly changing potential field near the atomic core is replaced by a smoother function that simulates the potential field far from the core very well. By doing so, it requires less number plane wave basis for wavefunction expansion. We can choose form various pseudopotential libraries. Choice of pseudopotential depends on the problem we are investigating, e.g., if there is a heavy element present in our system and we are interested in the spin-orbit coupling effects, we should choose a full relativistic pseudopotential. We need to be careful whether our chosen pseudopotential correctly reproduces physical properties. Various pseudopotential libraries: https://www.quantum-espresso.org/pseudopotentials https://www.materialscloud.org/discover/sssp/table/efficiency http://www.pseudo-dojo.org https://www.physics.rutgers.edu/gbrv/ https://nninc.cnf.cornell.edu http://www.quantum-simulation.org/potentials/ BLYP pseudopotentials SCAN pseudopotentials Pseudopotential naming conventions in PSLibrary: an example pseudopotential filename is O.rel-pbe-n-rrkjus_psl.1.0.0.UPF. O → denotes the atomic species rel → full relativistic (optional) pbe → exchange correlation functional n → non-linear core correction (optional) rrkus → pseudopotential type Exchange correlation functionals: Identifier Functional pz Perdew-Zunger (LDA) pbe Perdew-Burke-Ernzerhof (GGA) pw91 Perdew-Wang 91 (GGA) blyp Becke-Lee-Yang-Parr (GGA) Pseudopotential types: Identifier PP types ae all-electron rrkj Rappe-Rabe-Kaxiras-Joannopoulos (Norm conserving) rrkjus Rappe-Rabe-Kaxiras-Joannopoulos (Ultrasoft) kjpaw Kresse-Joubert (PAW) Ultra soft pseudopotentials are computationally efficient than the norm conserving pseudopotentials. You will find the recommended ecutwfc in the header of each pseudopotential file. If you choose an ultra-soft pseudopotential, you will need ecutrho about 8 times the value of ecutwfc. The default ecutrho is 4 times ecutwfc in Quantum Espresso code, which is a good choice for norm conserving pseudopotentials. You should check energy convergence against ecutwfc for your system. By using pseudopotential, we want to get rid of the core electrons that do not participate in the chemical properties of material. This is known also as rigid core approximation. Instead of accounting the nucleus and core electrons separately, we want to have a pseudopotential that interacts in a similar way with the valence electrons. info We can mix different types of pseudo potentials (e.g., norm conserving, ultra-soft, or PAW), but we cannot mix different exchange correlation functional (e.g., PBE and LDA). Exchange correlation functional can be read from the pseudopotential file or be provided via input_dft parameter in Quantum Espresso. \"sol\" in PBE-sol stands for solid. For bulk systems PBE-sol should be used, while PBE is appropriate for molecules. In case of 2D materials generally PBE is chosen, but one can check PBE-sol. Common error If you mix PBE with PBE-sol type, it results in Error: conflicting values for igcx. However, it is allowed to mix those two types of pseudo. We can set desired exchange correlation functional via input_dft instead of reading from the pseudopotential file.","s":"Pseudo potentials","u":"/espresso/setup/pseudo-potential","h":"","p":156},{"i":159,"t":"Naming convention for PP files","s":"Resources","u":"/espresso/setup/pseudo-potential","h":"#resources","p":156},{"i":161,"t":"Density functional theory (DFT) approaches the many-body problem by focusing on the electronic density which is a function of three spatial coordinates instead of finding the wave functions. DFT tries to minimize the energy of a system (ground state) in a self consistent way, and it is very successful in calculating the electronic structure of solid state systems. info A functional is a function whose argument is itself a function. f(x)f(x)f(x) is a function of the variable xxx while F[f]F[f]F[f] is a functional of the function fff. y=f(x)y = f(x)y=f(x) fff is a function, it takes a number xxx as input and output yyy is also a number. y=F[f(x)]y = F[f(x)]y=F[f(x)] FFF is a functional it takes function f(x)f(x)f(x) as input and output yyy is a number.","s":"Introduction to Density Functional Theory","u":"/espresso/theory/dft","h":"","p":160},{"i":163,"t":"The ground state density n(r)n(\\textbf{r})n(r) determines the external potential energy v(r)v(\\textbf{r})v(r) to within a trivial additive constant. So what Hohenberg-Kohn theorem says, may not sound very trivial. Schrödinger equation says how we can get the wavefunction from a given potential. Once solved the wavefunction (which could be difficult), we can determine the density or any other properties. Now Hohenberg and Kohn theorem says the opposite is also true. For a given density, the potential can be uniquely determined. For non-degenerate ground states, two different Hamiltonian cannot have the same ground-state electron density. It is possible to define the ground-state energy as a function of electronic density.","s":"Hohenberg-Kohn Theorem 1","u":"/espresso/theory/dft","h":"#hohenberg-kohn-theorem-1","p":160},{"i":165,"t":"Total energy of the system E(n)E(n)E(n) is minimal when n(r)n(\\textbf{r})n(r) is the actual ground-state density, among all possible electron densities. The ground state energy can therefore be found by minimizing E(n)E(n)E(n) instead of solving for the many-electron wavefunction. However, note that HK theorems do not tell us how the energy depends on the electron density. In reality, apart from some special cases, the exact E(n)E(n)E(n) is unknown and only approximate functionals are used. The essence of the HK theorem is that the non-degenerate ground-state wave function is a unique functional of the ground-state density: Ψ0(r1,r2,…,rN)=Ψ[n0(r)]\\Psi_0(\\textbf{r}_1, \\textbf{r}_2, \\dots, \\textbf{r}_N) = \\Psi[n_0(\\textbf{r})]Ψ0​(r1​,r2​,…,rN​)=Ψ[n0​(r)]","s":"Hohenberg-Kohn Theorem 2","u":"/espresso/theory/dft","h":"#hohenberg-kohn-theorem-2","p":160},{"i":167,"t":"For any system of NNN interacting electrons in a given external potential vext(r)v_{ext} (\\textbf{r})vext​(r), there is a virtual system of NNN non-interacting electrons with exactly the same density as the interacting one. The non-interacting electrons subjected to a different external (single particle) potential. [−ℏ2∇22m+vs(r)]ψi(r)=ϵiψi(r)\\left[-\\frac{\\hbar^2 \\nabla^2}{2m} + v_s(\\textbf{r}) \\right] \\psi_i(\\textbf{r}) = \\epsilon_i \\psi_i(\\textbf{r})[−2mℏ2∇2​+vs​(r)]ψi​(r)=ϵi​ψi​(r) vs(r)=vext(r)+e2∫d3r′n(r)∣r−r′∣+vxc(r;[n])v_s(\\textbf{r}) = v_{ext}(\\textbf{r}) + e^2 \\int d^3r' \\frac{n(\\textbf{r})}{|\\textbf{r} - \\textbf{r}'|} + v_{xc}(\\textbf{r}; [n])vs​(r)=vext​(r)+e2∫d3r′∣r−r′∣n(r)​+vxc​(r;[n]) n(r)=∑ifi∣ψi(r)∣2n(\\textbf{r}) = \\sum_i f_i |\\psi_i (\\textbf{r})|^2n(r)=i∑​fi​∣ψi​(r)∣2 where fif_ifi​ is the occupation factor of electrons (0≤fi≤20 \\le f_i \\le 20≤fi​≤2). The KS equation looks like single particle Schrödinger equation, however e2∫d3r′n(r)∣r−r′∣e^2 \\int d^3r' \\frac{n(\\textbf{r})}{|\\textbf{r} - \\textbf{r}'|}e2∫d3r′∣r−r′∣n(r)​ (the Hartree energy due to electrostatic interaction of electronic cloud) and vxc(r;[n])v_{xc} (\\textbf{r}; [n])vxc​(r;[n]) (exchange-correlation potential, reminiscence from Hartree-Fock theory, it includes all the remaining/unknown energy corrections) terms depend on n(r)n(\\textbf{r})n(r) i.e., on ψi\\psi_iψi​ which in turn depends on vextv_{ext}vext​. Therefore the problem is non-linear. It is usually solved computationally by starting from a trial potential and iterate to self-consistency. Also note that we have not included the kinetic energy term for the nucleus. This is because the nuclear mass is about three orders of magnitude heavier than the electronic mass (M≫mM \\gg mM≫m), so essentially electronic dynamics is much faster than the nuclear dynamics (see Born-Oppenheimer approximation). Now we are left with the task of solving a non-interacting Hamiltonian. info vext(r)v_{ext}(\\textbf{r})vext​(r) includes the potential energy due to nuclear field, and external electric and magnetic fields if present.","s":"Kohn-Sham hypothesis","u":"/espresso/theory/dft","h":"#kohn-sham-hypothesis","p":160},{"i":170,"t":"Energy functional is a function of the local charge density: Exc=∫n(r)ϵxc(n(r))drE_{xc} = \\int n(\\textbf{r}) \\epsilon_{xc}(n(\\textbf{r})) d\\textbf{r}Exc​=∫n(r)ϵxc​(n(r))dr vxc(r)=ϵxc(n(r))+n(r)dϵxc(n)dn∣n=n(r)v_{xc}(\\textbf{r}) = \\epsilon_{xc}(n(\\textbf{r})) + n(\\textbf{r})\\frac{d\\epsilon_{xc}(n)}{dn}\\bigg\\rvert_{n=n(\\textbf{r})}vxc​(r)=ϵxc​(n(r))+n(r)dndϵxc​(n)​​n=n(r)​ where ϵxc(n)\\epsilon_{xc}(n)ϵxc​(n) is obtained for the homogeneous electron gas of density nnn (using Quantum Monte Carlo techniques) and fitted to some analytic form.","s":"Local Density Approximation (LDA)","u":"/espresso/theory/dft","h":"#local-density-approximation-lda","p":160},{"i":172,"t":"These are a family of functionals that depends on the local density and the local gradient of the density: Exc=∫n(r)ϵGGA(n(r),∣∇n(r)∣)drE_{xc} = \\int n(\\textbf{r}) \\epsilon_{GGA}(n(\\textbf{r}), |\\nabla n(\\textbf{r})|) d\\textbf{r}Exc​=∫n(r)ϵGGA​(n(r),∣∇n(r)∣)dr There are many flavor of this functional. There are also more advanced functionals: Meta-GGA (e.g., SCAN), hybrids (e.g., B3LYP), nonlocal functionals for van der Waals forces, Grimme's DFT+D (a semi-empirical correction to GGA). They usually produces more accurate result, but computationally more expensive and sometimes numerically unstable.","s":"Generalized Gradient Approximation (GGA)","u":"/espresso/theory/dft","h":"#generalized-gradient-approximation-gga","p":160},{"i":174,"t":"We can write our Schrödinger in Dirac Bra-Ket notation: H^∣ψ⟩=E∣ψ⟩\\hat{H} \\ket{\\psi} = E\\ket{\\psi}H^∣ψ⟩=E∣ψ⟩ we are going to solve non-interacting single particle Hamiltonian in terms of known basis functions (plane waves) with unknown coefficients. We start with an initial guess for the electron density n(r)n(\\textbf{r})n(r), and construct a pseudo potential for the nuclear potential. In turn, we have the Hamiltonian. Solve for ψi(r)\\psi_i(\\textbf{r})ψi​(r), subsequently n(r)n(\\textbf{r})n(r), and iterate until self consistency is achieved. Self consistency loop in DFT calculation. The above screenshot was taken from lecture slide of Professor Ralph Gevauer from ICTP MAX School 2021. The potential due to the ions is replaced by the pseudo potentials which removes the oscillations near the atomic core (reducing number of required plane wave basis vectors) and simulates the exact behavior elsewhere. The pseudo potential is also different for different exchange correlation functional, and it is specified in the pseudo potential file. If a system had more than one type of atom, always choose the pseudo potentials with same exchange correlation (e.g., PBE). It is important to note that DFT is calculations are not exact solution to the real systems because exact functional (vxcv_{xc}vxc​) we need to solve the Kohn-Sham equation is not known. Therefore, we have to compare the results with experimental observations. The Kohn-Sham wavefunction of orbitals is not an approximation to the exact wavefunction. Rather it is precisely defined property of any electronic system, which is uniquely determined by the density. The in-exactness of DFT results come from the fact that we do not know the exact correlation functional that truly describes real systems.","s":"Algorithmic implementation","u":"/espresso/theory/dft","h":"#algorithmic-implementation","p":160},{"i":176,"t":"The wavefunctions are expanded in terms of a basis set. In quantum espresso, the the basis function is plane waves. There exists other DFT codes that use localized basis function as well. Plane waves are simpler but generally requires much large number of them compared to other localized basis sets. ψi(r)=∑α=1Nbciαfα(r)\\psi_i(\\textbf{r}) = \\sum_{\\alpha = 1} ^{N_b} c_{i\\alpha} f_{\\alpha}(\\textbf{r})ψi​(r)=α=1∑Nb​​ciα​fα​(r) Where NbN_bNb​ is the size basis set. Then the eigenvalue equation becomes: ∑βHαβciβ=ϵiciα\\sum_{\\beta} \\rm{H}_{\\alpha\\beta} c_{i\\beta} = \\epsilon_i c_{i\\alpha}β∑​Hαβ​ciβ​=ϵi​ciα​ ⇒(H11...H1b.........Hb1...Hbb)(c1...cb)=ϵi(c1...cb)\\Rightarrow \\begin{pmatrix} H_{11} & ... & H_{1b} \\\\ ... & ... & ... \\\\ H_{b1} & ... & H_{bb} \\end{pmatrix} \\begin{pmatrix} c_1 \\\\ ... \\\\ c_b \\end{pmatrix} = \\epsilon_i \\begin{pmatrix} c_1 \\\\ ... \\\\ c_b \\end{pmatrix}⇒​H11​...Hb1​​.........​H1b​...Hbb​​​​c1​...cb​​​=ϵi​​c1​...cb​​​ This is a linear algebra problem, solving the above involves diagonalization of (Nb×NbN_b \\times N_bNb​×Nb​) matrix which gives us corresponding eigenvalue and eigenfunction. Apart from plane waves, various localized basis set could be used, e.g., Linear Combination of Atomic Orbitals (LCAO), Gaussian-type Orbitals (GTO), Linearized Muffin-Tin Orbitals (LMTO). Once could also consider mixed basis sets, such as the Linearized Augmented Plane Waves (LAPW). Localized sets are smaller in size, they can be used for both finite and periodic systems, however they are difficult to use/calculate. In case of plane waves, we need larger basis set, and requires periodicity. Need to construct supercell for finite systems. Use of pseudopotential reduces the number of required plane waves.","s":"Plane-wave expansion","u":"/espresso/theory/dft","h":"#plane-wave-expansion","p":160},{"i":178,"t":"Finding the ground state: E[Φ]=⟨Φ∣H^∣Φ⟩⟨Φ∣Φ⟩E[\\Phi] = \\frac{\\braket{\\Phi | \\hat H | \\Phi}}{\\braket{\\Phi|\\Phi}}E[Φ]=⟨Φ∣Φ⟩⟨Φ∣H^∣Φ⟩​ E[Φ]≥E0E[\\Phi] \\ge E_0E[Φ]≥E0​","s":"Variational Principle","u":"/espresso/theory/dft","h":"#variational-principle","p":160},{"i":180,"t":"ψk(r)=eik⋅ruk(r)\\psi_k(r) = e^{i \\textbf{k} \\cdot \\textbf{r}} u_k(\\textbf{r})ψk​(r)=eik⋅ruk​(r) uk(r)=uk(r+R)u_k(\\textbf{r}) = u_k(\\textbf{r} + \\textbf{R})uk​(r)=uk​(r+R) R\\textbf{R}R is lattice vector. Fourier expansion: uk(r)=1Ω∑Gck,GeiG⋅ru_k(\\textbf{r}) = \\frac{1}{\\Omega} \\sum_G c_{\\textbf{k,G}} e^{i \\textbf{G} \\cdot \\textbf{r}}uk​(r)=Ω1​G∑​ck,G​eiG⋅r G\\textbf{G}G is reciprocal lattice vector. ψk(r)=1Ω∑Gck,Gei(k + G)⋅r\\psi_k(\\textbf{r}) = \\frac{1}{\\Omega} \\sum_G c_{\\textbf{k,G}} e^{i (\\textbf{k + G}) \\cdot \\textbf{r}}ψk​(r)=Ω1​G∑​ck,G​ei(k + G)⋅r Contribution from higher Fourier components are small, we can limit the sum at finite ∣k + G∣|\\textbf{k + G}|∣k + G∣ ℏ2∣k + G∣2m≤Ecutoff\\frac{\\hbar^2 |\\textbf{k + G}|}{2m} \\le E_{\\text{cutoff}}2mℏ2∣k + G∣​≤Ecutoff​ The charge density can be obtained from: n(r)=∑kψk∗(r)ψk(r)n(\\textbf{r}) = \\sum_k \\psi_k^*(\\textbf{r}) \\psi_k(\\textbf{r})n(r)=k∑​ψk∗​(r)ψk​(r) We need two sets of basis vectors: one to store the wavefunctions, and another for the charge density. info We need about 4 times the cutoff for the charge density compared to the cutoff for the wavefunction. In case of ultrasoft pseudo potentials, we require a lower cutoff for energy, therefore ecutrho might require 8 or 12 times higher than the ecutwfc.","s":"Bloch theorem","u":"/espresso/theory/dft","h":"#bloch-theorem","p":160},{"i":182,"t":"MIT Course Quantum Espresso Tutorials Introduction to DFT by Paolo Giannozzi http://compmatphys.epotentia.com","s":"Resources","u":"/espresso/theory/dft","h":"#resources","p":160},{"i":184,"t":"Hatree-Fock theory is foundational to many subsequent electronic structure theories. It is an independent particle model or mean filed theory. Consider we have two non-interacting electrons. In that case, the Hamiltonian would be separable, and the total wavefunction Ψ(r1,r2)\\Psi(\\textbf{r}_1, \\textbf{r}_2)Ψ(r1​,r2​) would be product of the individual wave function. Now if we consider two electrons are forming a single system, then there are two issues. (1) We can no longer ignore the electron-electron interaction. (2) The wavefunction describing fermions must be antisymmetric with respect to the interchange of any set of space-spin coordinates. A simple Hartree product fails to satisfy that condition: ΨHP(r1,r2,⋯ ,rN)=ϕ1(r1)ϕ2(r2)⋯ϕN(rN)\\Psi_{HP}(\\textbf{r}_1, \\textbf{r}_2, \\cdots, \\textbf{r}_N) = \\phi_1(\\textbf{r}_1) \\phi_2(\\textbf{r}_2) \\cdots \\phi_N(\\textbf{r}_N)ΨHP​(r1​,r2​,⋯,rN​)=ϕ1​(r1​)ϕ2​(r2​)⋯ϕN​(rN​) In order to satisfy the antisymmetry condition, for our two electron system we can formulate a total wavefunction of the form: Ψ(r1,r2)=12[χ1(r1)χ2(r2)−χ1(r2)χ2(r1)]\\Psi(\\textbf{r}_1, \\textbf{r}_2) = \\frac{1}{\\sqrt{2}} [\\chi_1(\\textbf{r}_1) \\chi_2(\\textbf{r}_2) - \\chi_1(\\textbf{r}_2)\\chi_2(\\textbf{r}_1)]Ψ(r1​,r2​)=2​1​[χ1​(r1​)χ2​(r2​)−χ1​(r2​)χ2​(r1​)]","s":"Hartree-Fock Theory","u":"/espresso/theory/hartree-fock","h":"","p":183},{"i":186,"t":"The above equation can be written as: Ψ(r1,r2)=12∣χ1(r1)χ2(r1)χ1(r2)χ2(r2)∣\\Psi(\\textbf{r}_1, \\textbf{r}_2) = \\frac{1}{\\sqrt{2}} \\begin{vmatrix} \\chi_1(\\textbf{r}_1) & \\chi_2(\\textbf{r}_1) \\\\ \\chi_1(\\textbf{r}_2) & \\chi_2(\\textbf{r}_2) \\end{vmatrix}Ψ(r1​,r2​)=2​1​​χ1​(r1​)χ1​(r2​)​χ2​(r1​)χ2​(r2​)​​ Now what happens if we have more than two electrons? We can generalize the above determinant form to NNN electrons: Ψ=1N!∣χ1(r1)χ2(r1)⋯χN(r1)χ1(r2)χ2(r2)⋯χN(r2)⋮⋮⋱⋮χ1(rN)χ2(rN)⋯χN(rN)∣\\Psi = \\frac{1}{\\sqrt{N!}} \\begin{vmatrix} \\chi_1(\\textbf{r}_1) & \\chi_2(\\textbf{r}_1) & \\cdots & \\chi_N(\\textbf{r}_1) \\\\ \\chi_1(\\textbf{r}_2) & \\chi_2(\\textbf{r}_2) & \\cdots & \\chi_N(\\textbf{r}_2) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\chi_1(\\textbf{r}_N) & \\chi_2(\\textbf{r}_N) & \\cdots & \\chi_N(\\textbf{r}_N) \\end{vmatrix}Ψ=N!​1​​χ1​(r1​)χ1​(r2​)⋮χ1​(rN​)​χ2​(r1​)χ2​(r2​)⋮χ2​(rN​)​⋯⋯⋱⋯​χN​(r1​)χN​(r2​)⋮χN​(rN​)​​ The above antisymmetrized product can describe electrons that move independently of each other while they experience an average (mean-field) Coulomb force.","s":"Slater determinant","u":"/espresso/theory/hartree-fock","h":"#slater-determinant","p":183},{"i":188,"t":"http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html","s":"Resources","u":"/espresso/theory/hartree-fock","h":"#resources","p":183},{"i":190,"t":"We want to calculate the electronic structure of real materials and their physical properties by ab-initio method. Electrons are microscopic particle, hence their dynamics is governed by the laws of quantum mechanics. Quantum particles are described by the wave function. λ⋅p=h\\lambda \\cdot p = hλ⋅p=h where hhh is the Plank constant. The Wavefunction of an electron in a potential filed (V)(V)(V) is calculated by solving the Schrödinger equation: −ℏ22m∇2Ψ(r,t)+V(r,t)=iℏ∂Ψ(r,t)∂t-\\frac{\\hbar^2}{2m} \\nabla^2 \\Psi(\\textbf{r}, t) + V(\\textbf{r}, t) = i\\hbar \\frac{\\partial\\Psi(\\textbf{r}, t)}{\\partial t}−2mℏ2​∇2Ψ(r,t)+V(r,t)=iℏ∂t∂Ψ(r,t)​ Fortunately, in most practical purposes, the potential field is not a function of time (t)(t)(t), or even if it is a function of time, they changes relatively slowly compared to the dynamics we are interested in. For example, the electrons inside a material are subjected to the Coulomb field of the nucleus. The nucleus is heavy and their motion is much slower than the motion of the electrons. In such situation, we can separate out the spatial and temporal parts of the wave function: Ψ(r,t)=ψ(r)f(t)\\Psi(\\textbf{r}, t) = \\psi(\\textbf{r}) f(t)Ψ(r,t)=ψ(r)f(t) That reduces our task to solving only time independent Schrödinger equation: [−ℏ2∇22m+v(r)]ψ(r)=ϵψ(r)\\left[-\\frac{\\hbar^2 \\nabla^2}{2m} + v(\\textbf{r})\\right] \\psi(\\textbf{r}) = \\epsilon \\psi(\\textbf{r})[−2mℏ2∇2​+v(r)]ψ(r)=ϵψ(r) Once we have the wavefunction, we can calculate the observables by taking the expectation values. ⟨ψi∣ψj⟩=δij\\braket{\\psi_i | \\psi_j} = \\delta_{ij}⟨ψi​∣ψj​⟩=δij​ ⟨ψi∣H^∣ψi⟩=ϵi\\braket{\\psi_i | \\hat{H} | \\psi_i} = \\epsilon_i⟨ψi​∣H^∣ψi​⟩=ϵi​ However, the challenge is to solve the Schrödinger equation as a real physical system is consists of a large number of atoms. The Schrödinger equation becomes coupled many-body equation. [−ℏ2m∑i=1N∇i2+∑i=1NV(ri)+∑i=1N∑j<iU(ri,rj)]ψ(r1,r2,...,rN)\\left[-\\frac{\\hbar}{2m} \\sum_{i=1}^N \\nabla_i^2 + \\sum_{i=1}^NV(\\textbf{r}_i) + \\sum_{i=1}^N \\sum_{j<i}U(\\textbf{r}_i, \\textbf{r}_j)\\right]\\psi(\\textbf{r}_1, \\textbf{r}_2, ..., \\textbf{r}_N)[−2mℏ​i=1∑N​∇i2​+i=1∑N​V(ri​)+i=1∑N​j<i∑​U(ri​,rj​)]ψ(r1​,r2​,...,rN​) =Eψ(r1,r2,...,rN)= E\\psi(\\textbf{r}_1, \\textbf{r}_2, ..., \\textbf{r}_N)=Eψ(r1​,r2​,...,rN​) With today's available computing power, it is far from feasible to solve the actual electronic wavefunction of a condensed matter system, where NNN is of the order of 102310^{23}1023.","s":"What problem are we trying to solve?","u":"/espresso/theory/problem-statement","h":"","p":189},{"i":193,"t":"Wannier functions are an alternative representation of Bloch states in terms of a localized basis set. Suppose we have NNN atoms each separated by lattice constant aaa in one dimension. Bloch states are indexed by the wave vector kkk. H∣k⟩=ϵk∣k⟩\\mathcal{H}\\ket{k} = \\epsilon_k \\ket{k}H∣k⟩=ϵk​∣k⟩ ⟨x∣k⟩=ψk(x)\\braket{x | k} = \\psi_k(x)⟨x∣k⟩=ψk​(x) From the Bloch theorem, we have ψk(x+a)=eikaψk(x)\\psi_k(x + a) = e^{ika}\\psi_k(x)ψk​(x+a)=eikaψk​(x) Now, we want to find an alternative representation in terms of Wannier basis ∣n⟩\\ket{n}∣n⟩, where the states are labeled using site index (n = 1, 2, ..., N) instead of quantum number kkk. Wannier basis set is complete and orthonormal. ∣k⟩=∑n=1Nank∣n⟩\\ket{k} = \\sum_{n=1}^N a_{nk} \\ket{n}∣k⟩=n=1∑N​ank​∣n⟩ ⟨x∣k⟩=∑nank⟨x∣n⟩\\braket{x | k} = \\sum_n a_{nk} \\braket{x | n}⟨x∣k⟩=n∑​ank​⟨x∣n⟩ ⇒ψk(x)=∑nankw(x−na)\\Rightarrow \\psi_k(x) = \\sum_n a_{nk} w(x - na)⇒ψk​(x)=n∑​ank​w(x−na) where www represents Wannier function. Apply translation operator on both sides: Taψk(x)=∑nankTaw(x−na)⇒ ψk(x+a)=∑nankw(x−(n−1)a)⇒ eikaψk(x)=∑na(n+1)kw(x+na)⇒ eika∑nankw(x−na)=∑na(n+1)kw(x+na)⇒ eika∑nankw(x−na)=∑na(n+1)kw(x+na)\\begin{aligned} &T_a \\psi_k(x) = \\sum_n a_{nk} T_a w(x - na) \\\\ \\Rightarrow~ & \\psi_k(x + a) = \\sum_n a_{nk} w\\bigl(x - (n-1)a\\bigr) \\\\ \\Rightarrow~ & e^{ika} \\psi_k(x) = \\sum_n a_{(n+1)k} w(x + na) \\\\ \\Rightarrow~ & e^{ika} \\sum_n a_{nk} w(x - na) = \\sum_n a_{(n+1)k} w(x + na) \\\\ \\Rightarrow~ & e^{ika} \\sum_n a_{nk} w(x - na) = \\sum_n a_{(n+1)k} w(x + na) \\\\ \\end{aligned}⇒ ⇒ ⇒ ⇒ ​Ta​ψk​(x)=n∑​ank​Ta​w(x−na)ψk​(x+a)=n∑​ank​w(x−(n−1)a)eikaψk​(x)=n∑​a(n+1)k​w(x+na)eikan∑​ank​w(x−na)=n∑​a(n+1)k​w(x+na)eikan∑​ank​w(x−na)=n∑​a(n+1)k​w(x+na)​ ∴ a(n+1)k=ankeika=a(n−1)kei2ka⋯=a0kei(n+1)ka\\begin{aligned} \\therefore ~ a_{(n+1)k} &= a_{nk} e^{ika} \\\\ &= a_{(n-1)k}e^{i2ka} \\\\ &\\qquad\\cdots \\\\ &= a_{0 k}e^{i(n+1)ka} \\end{aligned}∴ a(n+1)k​​=ank​eika=a(n−1)k​ei2ka⋯=a0k​ei(n+1)ka​ Since Bloch states are orthonormal: ⟨k∣k⟩=⟨k∣∑nank∣n⟩=∑mn⟨m∣amk∗ank∣n⟩=∑n∣ank∣2=Na0k2=1\\begin{aligned} \\braket{k | k} &= \\braket{k | \\sum_n a_{nk} | n} \\\\ &= \\sum_{mn} \\braket{m | a_{mk}^* a_{nk} | n} \\\\ &= \\sum_{n} | a_{nk} |^2 \\\\ &= N a_{0k}^2 \\\\ &= 1 \\end{aligned}⟨k∣k⟩​=⟨k∣n∑​ank​∣n⟩=mn∑​⟨m∣amk∗​ank​∣n⟩=n∑​∣ank​∣2=Na0k2​=1​ ⇒a0k=1N(up to a phase factor)\\Rightarrow a_{0k} = \\frac{1}{\\sqrt{N}} \\qquad\\text{(up to a phase factor)}⇒a0k​=N​1​(up to a phase factor) ∣k⟩=1N∑neikna∣n⟩\\ket{k} = \\frac{1}{\\sqrt{N}} \\sum_n e^{ikna} \\ket{n}∣k⟩=N​1​n∑​eikna∣n⟩ While Bloch states represent the eigenstates of the single-particle Hamiltonian, WF (in general) cannot be assigned a single eigen-value, instead WFs are obtained as liner combination of Bloch states corresponding to different energies. The Hamiltonian can now be written as: H=∑m∑n∣m⟩⟨m∣H∣n⟩⟨n∣=∑nϵn∣n⟩⟨n∣+∑m≠n(−tmn)∣m⟩⟨n∣\\begin{aligned} \\mathcal{H} &= \\sum_m \\sum_n \\ket{m} \\braket{m | \\mathcal{H} | n} \\bra{n} \\\\ &= \\sum_n \\epsilon_n \\ket{n} \\bra{n} + \\sum_{m\\neq n} (-t_{mn}) \\ket{m} \\bra{n} \\end{aligned}H​=m∑​n∑​∣m⟩⟨m∣H∣n⟩⟨n∣=n∑​ϵn​∣n⟩⟨n∣+m=n∑​(−tmn​)∣m⟩⟨n∣​ where ϵn\\epsilon_nϵn​ is onsite or diagonal term and tmnt_{mn}tmn​ (> 0) is hopping or off-diagonal term.","s":"Introduction","u":"/espresso/theory/wannier","h":"#introduction","p":191},{"i":195,"t":"The choice of Wannier function is not unique. One such option could be the set that maximizes localization. Two different sets of Wannier basis are connected via unitary transformation. MLWFs can be considered as a generalization of localized molecular orbitals (LMOs) to periodic systems.","s":"Maximally Localized Wannier Function","u":"/espresso/theory/wannier","h":"#maximally-localized-wannier-function","p":191},{"i":197,"t":"Introduction to Wannier Basis lecture by Vijay A. Singh Maximally localized generalized Wannier functions for composite energy bands, Marzari and Vanderbilt, Phys. Rev. B 56, 12847 (1997) Maximally localized Wannier functions for entangled energy bands, Souza, Marzari and Vanderbilt, Phys. Rev. B 65, 035109 (2001) Maximally localized Wannier functions: Theory and applications, Marzari et. al., Rev. Mod. Phys. 84, 1419 (2012) Introduction to Maximally Localized Wannier Functions, Ambrosetti and Silvestrelli, Reviews in Computational Chemistry, Ch. 6, pp. 327 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tutorial is result of my personal notes while trying (which I still do) to learn Density Functional Theory calculations myself. I am no expert in this subject. I am sharing this notes here, just in case it helps you getting started. I will cite numerous other resources that I am following. Hope you will find this tutorial helpful. The quantum espresso input files, jupyter notebooks (containing python code for visualizations), and other source files related to this tutorial can be found on GitHub: pranabdas/espresso. You may clone the repository to your local machine: git clone https://github.com/pranabdas/espresso.git","s":"Density Functional Theory using Quantum Espresso","u":"/espresso/","h":"","p":1},{"i":4,"t":"Lately, I decided to follow specific pattern for the filenames, but you can choose whatever works best for you. Note that all example files in this tutorial does not follow this convention yet. {program}.{calculation}.{system}.{in, out} {program}.{calculation}.{system_description}.{in, out} {pw, pp, ...}.{scf, bands, ...}.{silicon, al_slab}.{in, out} Example: pw.bands.silicon.in → input file for the bands calculation using PWscf program for silicon. For PWTK scripts, we will use .pwtk extension (e.g., silicon_vc_relax.pwtk).","s":"Filename conventions","u":"/espresso/","h":"#filename-conventions","p":1},{"i":6,"t":"1 Bohr = 0.529177249 Å 1 Rydberg (Ry) = 13.6056981 eV. Angstrom to Bohr converter: lattice constants are often provided in angstrom, you can use following utility to convert to Bohr. Å = 1.8897259886 Bohr. Copy","s":"Unit conversions","u":"/espresso/","h":"#unit-conversions","p":1},{"i":9,"t":"First we are going to relax the cell and choose appropriate lattice constant for our chosen pseudo potential. In case of metals, it is important to provide smearing parameters in the input file. src/al/al_vc_relax.in &CONTROL calculation= 'vc-relax', prefix= 'al', outdir= '/tmp/' pseudo_dir = '../pseudos/' etot_conv_thr= 1e-6, forc_conv_thr= 1e-5 / &SYSTEM ibrav= 2, celldm(1)= 7.652, nat= 1, ntyp= 1, ecutwfc = 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / &IONS / &CELL cell_dofree= 'ibrav' / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS (automatic) 10 10 10 0 0 0 We run pw.x to perform variable cell relaxation calculation: pw.x < al_vc_relax.in > al_vc_relax.out Now you may open the output file in vi editor and invoke search by pressing / and type Final enthalpy You will find the final lattice parameters below it.","s":"Variable cell relaxation","u":"/espresso/hands-on/aluminum","h":"#variable-cell-relaxation","p":7},{"i":11,"t":"We obtain relaxed lattice constant = 7.652 * 0.498611683 / 0.5 = 7.63075 Bohr. We will use this value for our next step, self consistent calculation. src/al/al_scf.in &CONTROL calculation= 'scf', restart_mode= 'from_scratch', prefix= 'al', outdir= '/tmp/', pseudo_dir= '../pseudos/' / &SYSTEM ibrav= 2, celldm(1) = 7.63075, nat= 1, ntyp= 1, ecutwfc= 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS (automatic) 10 10 10 0 0 0 We run our self consistent calculation: pw.x < al_scf.in > al_scf.out","s":"Self consistent field (SCF) calculation","u":"/espresso/hands-on/aluminum","h":"#self-consistent-field-scf-calculation","p":7},{"i":13,"t":"Inspect the output file, and let's proceed to next step non-self consistent calculation: src/al/al_nscf.in &CONTROL calculation= 'nscf', restart_mode= 'from_scratch', prefix= 'al', outdir= '/tmp/', pseudo_dir= '../pseudos/' / &SYSTEM ibrav= 2, celldm(1) = 7.63075, nat= 1, ntyp= 1, ecutwfc= 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS (automatic) 40 40 40 0 0 0 Note the changes in input file. The calculation changed to nscf and we are now using a higher number of k-points grid. pw.x < al_nscf.in > al_nscf.out","s":"Non-self consistent field calculation","u":"/espresso/hands-on/aluminum","h":"#non-self-consistent-field-calculation","p":7},{"i":15,"t":"Next we go ahead with our density of states calculation: src/al/al_dos.in &DOS prefix= 'al', outdir= '/tmp/', fildos= 'al_dos.dat', emin= -10, emax= 35 / We run dos.x with DOS inputs: dos.x < al_dos.in > al_dos.out Note from our al_nscf.out that our Fermi energy is at 7.9421 eV. We plot our density of states:","s":"Density of states","u":"/espresso/hands-on/aluminum","h":"#density-of-states","p":7},{"i":17,"t":"We prepare the input file the same as the case of our previous example silicon: src/al/al_bands.in &CONTROL calculation= 'bands', restart_mode= 'from_scratch', prefix= 'al', outdir= '/tmp/', pseudo_dir= '../pseudos/' / &SYSTEM ibrav= 2, celldm(1) = 7.63075, nat= 1, ntyp= 1, ecutwfc= 50, ecutrho= 500, occupations= 'smearing', smearing= 'gaussian', degauss= 0.01 / &ELECTRONS conv_thr= 1e-8 / ATOMIC_SPECIES Al 26.981539 Al.pbe-n-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS (alat) Al 0.00 0.00 0.00 K_POINTS {crystal_b} 5 00.000 0.500 00.000 20 !L 00.000 0.000 00.000 30 !G -0.500 0.000 -0.500 10 !X -0.375 0.250 -0.375 30 !U 00.000 0.000 00.000 20 !G Followed by run pw.x: pw.x < al_bands.in > al_bands.out Now we proceed with post-processing: src/al/al_bands_pp.in &BANDS prefix = 'al' outdir = '/tmp/' filband = 'al_bands.dat' / And run bands.x: bands.x < al_bands_pp.in > al_bands_pp.out We obtain the following bandstructure:","s":"Bandstructure calculation","u":"/espresso/hands-on/aluminum","h":"#bandstructure-calculation","p":7},{"i":19,"t":"Smearing is a technique used for suppressing unstable electron density in the calculation of metals. Such a problem occurs in metals (and semimetals) because the valence bands that cross Fermi level are partially occupied. Due to numerical accuracy, the electrons may occupy the unoccupied states during some iterations, making the algorithm unstable. In order to stablize the algorithm without using excessive number of k-points, smearing technique is used, which replaces the occupation number (either 0 or 1) is replaced by a smoothly varying function of energy. Such a smearing function could be Fermi Dirac distribution, instead of a step function (T = 0 K), we can use the finite temperature form. Below we will test the convergence using PWTK against the number of k-points, three different smearing functions (Gauss, Methfessel-Paxton, and Marzari-Vanderbilt), and for various smearing values. pwtk al.degauss.pwtk We see that the m-v and m-p broadening allow for faster and smother convergence while depending less on degauss value than Gaussian broadening. The number suffix next to the legend labels are number of uniform k-points in Monkhorst-Plank grid.","s":"Importance of smearing in convergence","u":"/espresso/hands-on/aluminum","h":"#importance-of-smearing-in-convergence","p":7},{"i":21,"t":"Before we can run bands calculation, we need to perform single-point self consistent field calculation. We have our input scf file with some new parameters: src/silicon/si_bands_scf.in &CONTROL calculation = 'scf', restart_mode = 'from_scratch', prefix = 'silicon', outdir = './tmp/' pseudo_dir = './pseudos/' verbosity = 'high' / &SYSTEM ibrav = 2, celldm(1) = 10.2076, nat = 2, ntyp = 1, ecutwfc = 50, ecutrho = 400, nbnd = 8, ! occupations = 'smearing', ! smearing = 'gaussian', ! degauss = 0.005 / &ELECTRONS conv_thr = 1e-8, mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 8 8 8 0 0 0 Run the scf calculation: pw.x < si_bands_scf.in > si_bands_scf.out Next step is our band calculation (non-self consistent field) calculation. The bands calculation is non self-consistent and reads/uses the ground state electron density, Hartree, exchange and correlation potentials obtained in the previous step (scf calculation). In case of non self-consistent calculation, the pw.x program determines the Kohn-Sham eigenfunction and eigenvalues without updating Kohn-Sham Hamiltonian at every iteration. We need to specify the k-points for which we want to calculate the eigenvalues. You may use the See-K-path tool by materials cloud to visualize the K-path. We can specify nbnd, by default it calculates half the number of valence electrons, i.e., only the occupied ground state bands. Usually we are interested also in the unoccupied bands above the Fermi energy. Number of occupied bands can be found in the scf output as number of Kohn-Sham states. Below is a sample input file for the band calculation: src/silicon/si_bands.in &control calculation = 'bands', restart_mode = 'from_scratch', prefix = 'silicon', outdir = './tmp/' pseudo_dir = './pseudos/' verbosity = 'high' / &system ibrav = 2, celldm(1) = 10.2076, nat = 2, ntyp = 1, ecutwfc = 50, ecutrho = 400, nbnd = 8 / &electrons conv_thr = 1e-8, mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 Si 0.25 0.25 0.25 K_POINTS {crystal_b} 5 0.0000 0.5000 0.0000 20 !L 0.0000 0.0000 0.0000 30 !G -0.500 0.0000 -0.500 10 !X -0.375 0.2500 -0.375 30 !U 0.0000 0.0000 0.0000 20 !G Run pw.x with bands calculation input file: pw.x < si_bands.in > si_bands.out After the bands calculation is performed, we need some postprocessing using bands.x utility in order to obtain the data in more usable format. Input file for bands.x postprocessing: src/silicon/si_bands_pp.in &BANDS prefix = 'silicon' outdir = './tmp/' filband = 'si_bands.dat' / Run bands.x from post processing (PP) module: bands.x < si_bands_pp.in > si_bands_pp.out Finally, we run plotband.x to visualize bandstructure. We can either run it interactively (as described below) or provide an input file. In order to run interactively, type plotband.x in your terminal. Input file > si_bands.dat Reading 8 bands at 91 k-points Range: -5.8300 16.3420eV Emin, Emax > -6, 16 high-symmetry point: 0.5000 0.5000 0.5000 x coordinate 0.0000 high-symmetry point: 0.0000 0.0000 0.0000 x coordinate 0.8660 high-symmetry point: 1.0000 0.0000 0.0000 x coordinate 1.8660 high-symmetry point: 1.0000 0.2500 0.2500 x coordinate 2.2196 high-symmetry point: 0.0000 0.0000 0.0000 x coordinate 3.2802 output file (gnuplot/xmgr) > si_bands.gnuplot bands in gnuplot/xmgr format written to file si_bands.gnuplot output file (ps) > si_bands.ps Efermi > 6.6416 deltaE, reference E (for tics) 4, 0 bands in PostScript format written to file si_bands.ps You will have si_bands.ps with band diagram. Alternatively, you can use your favorite plotting program to make the plots. Below is an example of using Python matplotlib. notebooks/si-bands.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline plt.rcParams[\"figure.dpi\"]=150 plt.rcParams[\"figure.facecolor\"]=\"white\" plt.rcParams[\"figure.figsize\"]=(8, 6) # load data data = np.loadtxt('../src/silicon/si_bands.dat.gnu') k = np.unique(data[:, 0]) bands = np.reshape(data[:, 1], (-1, len(k))) for band in range(len(bands)): plt.plot(k, bands[band, :], linewidth=1, alpha=0.5, color='k') plt.xlim(min(k), max(k)) # Fermi energy plt.axhline(6.6416, linestyle=(0, (5, 5)), linewidth=0.75, color='k', alpha=0.5) # High symmetry k-points (check bands_pp.out) plt.axvline(0.8660, linewidth=0.75, color='k', alpha=0.5) plt.axvline(1.8660, linewidth=0.75, color='k', alpha=0.5) plt.axvline(2.2196, linewidth=0.75, color='k', alpha=0.5) # text labels plt.xticks(ticks= [0, 0.8660, 1.8660, 2.2196, 3.2802], \\ labels=['L', '$\\Gamma$', 'X', 'U', '$\\Gamma$']) plt.ylabel(\"Energy (eV)\") plt.text(2.3, 5.6, 'Fermi energy', fontsize= small) plt.show() info The k values corresponding to high symmetry points (such as Γ\\GammaΓ, X, U, L) which we need to label in our band diagram, can be found in the post-processing output file (si_bands_pp.out). Bandgap value can be determined from the highest occupied, lowest unoccupied level values printed in scf calculation output.","s":"Bandstructure Calculation","u":"/espresso/hands-on/bands","h":"","p":20},{"i":23,"t":"Usually, band gaps computed using common exchange-correction functionals such as LDA or GGA are severely underestimated compared to actual experimental values. This discrepancy is mainly due to (1) approximations used in the exchange correction functional and (2) a derivative discontinuity term, originating from the density functional being discontinuous with the total number of electrons in the system. The second contribution is larger contributor to the error. It can be partly addressed by a variety of techniques such as the GW approximation. Strategies to improve band gap prediction at moderate to low computational cost now been developed by several groups, including Chan and Ceder (delta-sol)1, Heyd et al. (hybrid functionals)2, and Setyawan et al. (empirical fits)3.","s":"Note on bandgap","u":"/espresso/hands-on/bands","h":"#note-on-bandgap","p":20},{"i":25,"t":"https://docs.materialsproject.org/methodology/materials-methodology/electronic-structure#accuracy-of-band-structures See K-pat online tool M. Chan, G. Ceder, Efficient Band Gap Predictions for Solids, Physical Review Letters 19 (2010) https://doi.org/10.1103/PhysRevLett.105.196403↩ J. Heyd, J.E. Peralta, G.E. Scuseria, R.L. Martin, Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional, Journal of Chemical Physics 123 (2005) https://doi.org/10.1063/1.2085170↩ W. Setyawan, R.M. Gaume, S. Lam, R. Feigelson, S. Curtarolo, High-throughput combinatorial database of electronic band structures for inorganic scintillator materials., ACS Combinatorial Science. (2011) https://doi.org/10.1021/co200012w.↩","s":"Resources","u":"/espresso/hands-on/bands","h":"#resources","p":20},{"i":27,"t":"Topological insulators are a special class of material that is insulating in the bulk, however exhibit conducting states in the surface. Bi2Se3 is such a material. Spin orbit coupling and breaking of the inversion symmetry at the surface of the crystal is crucial to the existence of the Dirac surface state. Here we will calculate the bandstructure step by step: first for the bulk, next including SOC, and finally for the slab. Please check the respective input files. I followed the following steps: # SCF calculation for bulk mpirun -np 24 pw.x -i pw.scf.bi2se3_01.in > pw.scf.bi2se3_01.out # bands calculation for bulk mpirun -np 24 pw.x -i pw.bands.bi2se3_01.in > pw.bands.bi2se3_01.out # post processing for bulk bands mpirun -np 24 bands.x -i pp.bands.bi2se3_01.in > pp.bands.bi2se3_01.out # for bulk with SOC mpirun -np 24 pw.x -i pw.scf.bi2se3_02.in > pw.scf.bi2se3_02.out mpirun -np 24 pw.x -i pw.bands.bi2se3_02.in > pw.bands.bi2se3_02.out mpirun -np 24 bands.x -i pp.bands.bi2se3_02.in > pp.bands.bi2se3_02.out # slab calculation mpirun -np 24 pw.x -i pw.scf.bi2se3_03.in > pw.scf.bi2se3_03.out mpirun -np 24 pw.x -i pw.bands.bi2se3_03.in > pw.bands.bi2se3_03.out mpirun -np 24 bands.x -i pp.bands.bi2se3_03.in > pp.bands.bi2se3_03.out # DOS mpirun -np 24 pw.x -i pw.nscf.bi2se3_04.in > pw.nscf.bi2se3_04.out mpirun -np 24 dos.x -i pp.dos.bi2se3_04.in > pp.dos.bi2se3_04.out For the slab calculation the periodicity of the lattice was broken along the c-axis to artificially add 10 Å vacuum. In above calculation electronic spin was not considered (meaning the states are degenerate with spin up and down). If starting_magnetization is set to zero (or not given) the code makes a spin-orbit calculation without spin magnetization. It assumes that time reversal symmetry holds and it does not calculate the magnetization. The states are still two-component spinors but the total magnetization is zero. Notice that for the Dirac surface states the gap did not completely close at the Fermi energy. This is possibly due to finite size effect. We could repeat the calculation with larger vacuum, and see what happens. Also the Fermi energy estimation seems incorrect. In order to sample the Γ\\GammaΓ point for our DOS calculation, an odd k-grid mesh (25✕25✕5) was used. The signature of Dirac cone is evident from the DOS figure.","s":"Bandstructure of topological insulating Bi2Se3","u":"/espresso/hands-on/Bi2Se3","h":"","p":26},{"i":29,"t":"https://docs.quantumatk.com/tutorials/topological_insulator_bi2se3/","s":"Resources","u":"/espresso/hands-on/Bi2Se3","h":"#resources","p":26},{"i":32,"t":"We can automate the previous self consistent calculation by varying a certain parameter. Say we want to check the total energy of the system for various values of ecutwfc. We can do that by using pwtk script. src/silicon/si_scf_ecutoff.pwtk # load the pw.x input from file load_fromPWI pw.scf.silicon.in # open a file for writing resulting total energies set fid [open etot_vs_ecutwfc.dat w] # loop over different \"ecut\" values foreach ecut { 12 16 20 24 28 32 } { # name of I/O files: $name.in & $name.out set name si_scf_ecutwfc-$ecut # set the pw.x \"ecutwfc\" variable SYSTEM \"ecutwfc = $ecut\" # run the pw.x calculation runPW $name.in # extract the \"total energy\" and write it to file set Etot [::pwtk::pwo::totene $name.out] puts $fid \"$ecut $Etot\" } close $fid To run the above script: pwtk si_scf_ecutoff.pwtk Now we can plot the total energy with respect to ecutwfc. The data is in etot-vs-ecutwfc.dat We will use matplotlib to make the plots. Here is the python code for plotting: notebooks/si-plots.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline plt.rcParams[\"figure.dpi\"]=150 plt.rcParams[\"figure.facecolor\"]=\"white\" x, y = np.loadtxt('../src/silicon/etot-vs-ecutwfc.dat', delimiter=' ', unpack=True) plt.plot(x, y, \"o-\", markersize=5, label='Etot vs ecutwfc') plt.xlabel('ecutwfc (Ry)') plt.ylabel('Etot (Ry)') plt.legend(frameon=False) plt.show()","s":"Convergence with cutoff energy using PWTK","u":"/espresso/hands-on/convergence","h":"#convergence-with-cutoff-energy-using-pwtk","p":30},{"i":34,"t":"We can do the convergence test with various parameters. We can calculate the total energy of the system by varying various parameters. We will use the shell script to automate the process with different cutoff energy values. src/silicon/si_script.sh #!/bin/sh NAME=\"ecut\" for CUTOFF in 10 15 20 25 30 35 40 do cat > ${NAME}_${CUTOFF}.in << EOF &control calculation = 'scf', prefix = 'silicon' outdir = './tmp/' pseudo_dir = './pseudos/' / &system ibrav = 2, celldm(1) = 10.0, nat = 2, ntyp = 1, ecutwfc = $CUTOFF / &electrons mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 6 6 6 1 1 1 EOF pw.x < ${NAME}_${CUTOFF}.in > ${NAME}_${CUTOFF}.out echo ${NAME}_${CUTOFF} grep ! ${NAME}_${CUTOFF}.out done Make sure the file has executable permission for the user: chmod 700 si_script.sh Run the script file: ./si_script.sh # or sh si_script.sh We can plot the energy vs cutoff energy, and choose a reasonable value. caution Initially, I had problem in running the script in macOS. The problem occurred because the script file format was set to DOS. The file format can be checked in following way: Open the file in vi editor. vi si_script.sh Now in vi editor command mode (ESC key), type :set ff? This would tell you the file format. Now to change file format, use the command :set fileformat=unix","s":"Convergence test using UNIX shell script","u":"/espresso/hands-on/convergence","h":"#convergence-test-using-unix-shell-script","p":30},{"i":36,"t":"We can run similar convergence test against another parameter, and choose the best value of that particular parameter. Here we will try to calculate the number of k-points in the Monkhorst-Pack mesh. src/silicon/si_scf_kpoints.pwtk load_fromPWI pw.scf.silicon.in set fid [open etot-vs-kpoint.dat w] foreach k { 2 4 6 8 } { set name si_scf_kpoints-$k K_POINTS automatic \"$k $k $k 1 1 1\" runPW $name.in set Etot [::pwtk::pwo::totene $name.out] puts $fid \"$k $Etot\" } close $fid Run pwtk program: pwtk si_scf_kpoints.pwtk notebooks/silicon-scf.ipynb x, y = np.loadtxt('../src/silicon/etot-vs-kpoint.dat', delimiter=' ', unpack=True) plt.plot(x, y, \"o-\", markersize=5, label='Etot vs kpoints') plt.xlabel('# kpoints') plt.ylabel('Etot (Ry)') plt.legend(frameon=False) plt.show()","s":"Convergence test against the number of k-points","u":"/espresso/hands-on/convergence","h":"#convergence-test-against-the-number-of-k-points","p":30},{"i":38,"t":"Calculating total energy with respect to varying lattice constant. src/silicon/si_scf_alat.pwtk load_fromPWI pw.scf.silicon.in # please uncomment & insert value as determined in the \"ecutwfc\" exercise SYSTEM { ecutwfc = 30 } # please uncomment & insert values as determined in the \"kpoints\" exercise K_POINTS automatic { 6 6 6 1 1 1 } set fid [open etot-vs-alat.dat w] foreach alat { 9.7 9.8 9.9 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 } { set name si_scf_alat-$alat SYSTEM \"celldm(1) = $alat\" runPW $name.in set Etot [::pwtk::pwo::totene $name.out] puts $fid \"$alat $Etot\" } close $fid Run the above code: pwtk si_scf_alat.pwtk notebooks/silicon-scf.ipynb x, y = np.loadtxt('../src/silicon/etot-vs-alat.dat', delimiter=' ', unpack=True) plt.plot(x, y, \"o-\", markersize=5, label='Etot vs alat') plt.xlabel('alat (Bohr)') plt.ylabel('Etot (Ry)') plt.legend(frameon=False) plt.show()","s":"Convergence against lattice constant","u":"/espresso/hands-on/convergence","h":"#convergence-against-lattice-constant","p":30},{"i":40,"t":"CPU time is proportional to the number of plane waves used for the calculation. Number of plane wave is proportional to the (ecutwfc)3/2 CPU time is proportional to the number if inequivalent k-points CPU time increases as N3, where N is the number of atoms in the system.","s":"Note on CPU time","u":"/espresso/hands-on/convergence","h":"#note-on-cpu-time","p":30},{"i":42,"t":"Electronic structure for transition metals (with localized ddd or fff electrons) is not accurately described by standard DFT, and therefore the need for DFT+U formulation. &SYSTEM ... lda_plus_u = .TRUE. Hubbard_u(i) = 2.0 ... / Here i refers to the atomic index in the &ATOMIC_SPECIES card corresponding to each ntyp. We can specify Hubbard_u(i) corresponding to more than one atom in separate lines. There is also Ueff=U−JU_{eff} = U - JUeff​=U−J implementation in QE. JJJ represents on-site exchange interaction. Number of JJJ terms depends on the manifold of localized electrons. For ppp, we have 1; for ddd, we have 2; and for fff, we have 3 terms. ... lda_plus_u = .TRUE. lda_plus_u_kind = 1 Hubbard_u(i) = U Hubbard_J(k, i) = J_{ki} ... COMMON ERRORS If you add Hubbard_u for elements that is not implemented to have UUU term in QE, you might see a \"pseudopotential not yet inserted\" error.","s":"DFT+U calculation","u":"/espresso/hands-on/dft-u","h":"","p":41},{"i":44,"t":"Starting from Quantum Espresso version 7.1, there are changes to input syntax for DFT+U calculations. In the new version, instead of defining the necessary DFT+U parameters, now there is a new Hubbard card. &system ... - lda_plus_u = .true., - lda_plus_u_kind = 0, - U_projection_type = 'atomic', - Hubbard_U(1) = 4.6 - Hubbard_U(2) = 4.6 ... / + HUBBARD (ortho-atomic) + U Fe1-3d 4.6 + U Fe2-3d 4.6 Please refer to the qe-x.x/Doc/Hubbard_input.pdf for details.","s":"Changes to input syntax in v7.1","u":"/espresso/hands-on/dft-u","h":"#changes-to-input-syntax-in-v71","p":41},{"i":46,"t":"We will first perform the standard DFT calculation. Perform the SCF calculation: pw.x -in feo_scf.in > feo_scf.out Perform NSCF calculation with denser k-grid: pw.x -in feo_nscf.in > feo_nscf.out Perform P-DOS calculation: projwfc.x -in feo_projwfc.in > feo_projwfc.out This gives us metallic density of states. In practice we get insulating FeO.","s":"DFT calculation for FeO","u":"/espresso/hands-on/dft-u","h":"#dft-calculation-for-feo","p":41},{"i":48,"t":"src/FeO/feo_hp.in &inputhp prefix = 'FeO' outdir = './tmp/' nq1 = 1, nq2 = 1, nq3 = 1 / Perform a linear-response calculation using hp.x program: hp.x -in feo_hp.in > feo_hp.out Check the file FeO.Hubbard_parameters.dat. info We need to check the convergence against q-mesh (as well as k-mesh in SCF calculation). Here 1×1×11\\times 1\\times 11×1×1 mesh is used. Important: lda_plus_u must be set to .true. during the SCF calculation, UUU may be set to zero. We can update the obtained UUU value in our SCF calculation, and repeat linear response calculation until we have reached self consistency in UUU value. To go even further one can check the convergence of geometry during UUU updates. There is also inter-site Hubbard correction DFT+U+V calculation. The results could be more closer to hybrid functionals like GW. The VVV can also be calculated using Quantum Espresso hp.x code. Obtained value of UUU depends on pseudopotential, Hubbard manifold (whether atomic, ortho-atomic etc.). danger The above hp.x code is not suitable for closed cell systems (e.g., fully occupied d-shell element), in such cases this linear response method gives unrealistically large UUU value.","s":"Calculating Hubbard U","u":"/espresso/hands-on/dft-u","h":"#calculating-hubbard-u","p":41},{"i":49,"t":"We repeat the calculation after setting in the &SYSTEM card: Hubbard_U(1) = 4.6 Hubbard_U(2) = 4.6 We repeat the above calculation and plot the results. Now we find insulating ground state. info U_projection_type = 'ortho-atomic' might give more realistic result than the default 'atomic'. When performing DFT+UDFT+UDFT+U calculation, the ground state might get stuck in a local minimum, in such cases we need to provide starting_ns_eigenvalue to help calculation reach desired/actual ground state. Please see these slides by Dr. Iurii Timrov for a relevant example. tip Here we have plotted the lpdos (local density of states). If we want to know the contribution of dz2,dyz,dx2−z2d_{z^2}, d_{yz}, d_{x^2-z^2}dz2​,dyz​,dx2−z2​ ect., we can find them from the pdos columns. Also there arise important Lowdin charges information in the feo_projwfc.out file.","s":"DFT+U calculation","u":"/espresso/hands-on/dft-u","h":"#dftu-calculation","p":41},{"i":51,"t":"Hands-on DFT+U by Iurii Timrov and Matteo Cococcioni Hubbard parameter calculation","s":"Resources","u":"/espresso/hands-on/dft-u","h":"#resources","p":41},{"i":53,"t":"Electronic density of states is an important property of a material. ρ(E)dE\\rho(E)dEρ(E)dE = number of electronic states in the energy interval (E,E+dE)(E, E + dE)(E,E+dE) Before we can run the Density of States (DOS) calculation, we need Perform fixed-ion self consistent filed (scf) calculation. In plane-wave based DFT calculations the electronic density is expressed by functions of the form exp⁡(ik⋅r)\\exp (i \\textbf{k} \\cdot \\textbf{r})exp(ik⋅r) with energy given by E=ℏ2k2/2mE = \\hbar^2k^2/2mE=ℏ2k2/2m. Perform non-self consistent field (nscf) calculation with denser k-point grid. A large number of kkk points are required DOS calculation, as the accuracy of DOS depends on the integration in kkk space. Finally, the DOS can be determined by integrating the electron density in kkk space. I have created a new input file (si_scf_dos.in) which is very much the same as our previous scf input file except some parameters are modified. You can find all the input files in my GitHub repository. We used the lattice constant value that we obtained from the relaxation calculation. We should not directly use the experimental/real lattice constant values. Depending on the method and pseudo-potential, it might result stress in the system. We have increased the ecutwfc to have better precision. We run the scf calculation: pw.x < si_scf_dos.in > si_scf_dos.out Next, we have prepared the input file for the nscf calculation. Where is have added occupations in the &system card as tetrahedra (appropriate for DOS calculation). We have increased the number of k-points to 12 × 12 × 12 with automatic option. Also specify nosym = .TRUE. to avoid generation of additional k-points in low symmetry cases. outdir and prefix must be the same as in the scf step, some of the inputs and output are read from previous step. Here we can specify a larger number of nbnd to calculate unoccupied bands. Number of occupied bands can be found in the scf output as number of Kohn-Sham states. pw.x < si_nscf_dos.in > si_nscf_dos.out Now our final step is to calculate the density of states. The DOS input file as follows: src/silicon/si_dos.in &DOS prefix='silicon', outdir='./tmp/', fildos='si_dos.dat', emin=-9.0, emax=16.0 / We run: dos.x < si_dos.in > si_dos.out The DOS data in the si_dos.dat file that we specified in our input file. We can plot the DOS: notebooks/silicon-dos.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline # load data energy, dos, idos = np.loadtxt('../src/silicon/si_dos.dat', unpack=True) # make plot plt.figure(figsize = (12, 6)) plt.plot(energy, dos, linewidth=0.75, color='red') plt.yticks([]) plt.xlabel('Energy (eV)') plt.ylabel('DOS') plt.axvline(x=6.642, linewidth=0.5, color='k', linestyle=(0, (8, 10))) plt.xlim(-6, 16) plt.ylim(0, ) plt.fill_between(energy, 0, dos, where=(energy < 6.642), facecolor='red', alpha=0.25) plt.text(6, 1.7, 'Fermi energy', fontsize= med, rotation=90) plt.show() Important For a set of calculation, we must keep the prefix same. For example, the nscf or bands calculation uses the wavefunction calculated by the scf calculation. When performing different calculations, for example you change a parameter and want to see the changes, you must use different output folder or unique prefix for different calculations so that the outputs do not get mixed. tip Sometimes it is important to sample the Γ\\GammaΓ point for DOS calculation (e.g., the conducting bands cross the Fermi surface only at Γ\\GammaΓ point). In such cases, we need to use odd k-grid (e.g., 9✕9✕5).","s":"Density of States calculation","u":"/espresso/hands-on/dos","h":"","p":52},{"i":55,"t":"First we perform our self consistent field calculation: mpirun -np 4 pw.x -in pw.scf.silicon.in > pw.scf.silicon.out Next step we perform nscf calculation: src/silicon/pw.nscf.silicon_epsilon.in &CONTROL calculation = 'nscf', prefix = 'silicon', outdir = '/tmp/silicon/' pseudo_dir = '../pseudos/' verbosity = 'high' / &SYSTEM ibrav = 2, celldm(1) = 10.26, nat = 2, ntyp = 1, ecutwfc = 30 nbnd = 16 nosym = .true. noinv = .true. / &ELECTRONS mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 30 30 30 0 0 0 Especially, notice following changes: nbnd = 16 nosym = .true. noinv = .true. We turn off the automatic reduction of k-points that pw.x does by using crystal symmetries (nosym = .true. and noinv = .true.). This is because epsilon.x does not recognize crystal symmetries, therefore the entire list of k-points in the grid is needed. Secondly, we calculate a larger number of bands (16), since we are interested in interband transitions. The final step is to prepare the input file for epsilon.x: src/silicon/epsilon.si.in &inputpp outdir = \"/tmp/silicon/\" prefix = \"silicon\" calculation = \"eps\" / &energy_grid smeartype = \"gauss\" intersmear = 0.2 wmin = 0.0 wmax = 30.0 nw = 500 / The variables smeartype and intersmear define the numerical approximation used to represent the Dirac delta functions in the expression for ϵ2\\epsilon_2ϵ2​ given above. The variables wmin, wmax and nw define the energy grid for the dielectric function. All the energy variables are in eV. mpirun -np 1 epsilon.x < epsilon.si.in > epsilon.si.out We will see the results are saved in separate .dat files. We can plot the real (ϵ1\\epsilon_1ϵ1​) and imaginary (ϵ2\\epsilon_2ϵ2​) parts of dielectric constants: import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline plt.rcParams[\"figure.dpi\"]=150 plt.rcParams[\"figure.facecolor\"]=\"white\" data_r = np.loadtxt('../src/silicon/epsr_silicon.dat') data_i = np.loadtxt('../src/silicon/epsi_silicon.dat') energy_r, epsilon_r = data_r[:, 0], data_r[:, 2] energy_i, epsilon_i = data_i[:, 0], data_i[:, 2] plt.plot(energy_r, epsilon_r, lw=1, label=\"$\\\\epsilon_1$\") plt.plot(energy_i, epsilon_i, lw=1, label=\"$\\\\epsilon_2$\") plt.xlim(0, 15) plt.xlabel(\"Energy (eV)\") plt.ylabel(\"$\\\\epsilon_1~/~\\\\epsilon_2$\") plt.legend(frameon=False) plt.show() danger Ultra-soft pseudopotentials do not work with epsilon.x.","s":"Dielectric constant","u":"/espresso/hands-on/epsilon","h":"","p":54},{"i":57,"t":"I am following this example from the ICTP online school 2021. We will perform the SCF KS orbital calculations on magnetic (nspin=2) iron. Since the d-orbitals of Fe atom are localized/ hard, we will use ultra-soft pseudo potential (USPP). note If we have crystal structure with only one atom per unit cell, or only one type of atoms, the only possible ordering is ferromagnetic. In such cases, we need to form supercell with more number of atoms or label multiple atoms separately, so that their magnetic orientation could be different thus having the possibility of ferro- or antiferromagnetic final states. Run the SCF calculations for both ferro- and antiferromagnetic structures. Notice that for ferromagnetic, we have BCC structure with only one type of atom, while we use simple cubic structure for antiferromagnetic case with two different atomic labels. For antiferromagnetic calculation, we need to start with opposite initial spins. pw.x -i pw.scf.fe_fm.in > pw.scf.fe_fm.out pw.x -i pw.scf.fe_afm.in > pw.scf.fe_afm.out note In case of the AFM calculation, if we have started with FM (say, for both atom types starting_magnetization=0.6 ), the calculation would still converge to AFM state as it is the true ground state for this system, albeit it would take more iteration to converge. If a system has complex potential surface with local minima, it it possible to get different final state magnetization depending on the starting magnetization. In such cases, a stricter convergence criteria might help. info In case of ultrasoft pseudo potentials, the Quantum Espresso default of ecutrho 4 times of ecutoff is not sufficient. We need to set ecutrho 8 or even 12 times that of ecutoff. We must test the convergence for our set values.","s":"Magnetic system: bulk iron","u":"/espresso/hands-on/fe","h":"","p":56},{"i":59,"t":"Below is the PWTK script file: src/fe/fe_ecut.pwtk # load the pw.x input from file load_fromPWI fe_scf_fm.in # dual is the ratio ecutrho/ecutwfc foreach dual { 4 8 12 } { set fid [open etot-vs-ecutwfc.dual$dual.dat w] foreach ecutwfc [seq 25 5 50] { set name pw.Fe.scf.ecutwfc-$ecutwfc.dual-$dual SYSTEM \"ecutwfc = $ecutwfc ecutrho = $ecutwfc*$dual \" runPW $name.in set Etot [pwo_totene $name.out] puts $fid \"$ecutwfc $Etot\" } close $fid } Run the script: pwtk fe_ecut.pwtk","s":"Convergence test for USPP","u":"/espresso/hands-on/fe","h":"#convergence-test-for-uspp","p":56},{"i":61,"t":"PWTK script to calculate DOS and p-DOS: src/fe/fe_dos.pwtk load_fromPWI fe_scf_fm.in SYSTEM \" ecutwfc = 40 ecutrho = 320 \" set name Fe runPW pw.$name.scf.in CONTROL { calculation = 'nscf' } SYSTEM { occupations = 'tetrahedra' , degauss = , ! this is how variable is unset in PWTK } K_POINTS automatic { 12 12 12 1 1 1 } runPW pw.$name.nscf.in DOS \" fildos = '$name.dos.dat' Emin = 5.0 Emax = 20.0, DeltaE = 0.1 \" runDOS dos.$name.in PROJWFC \" filpdos = '$name.pdos.dat' Emin = 5.0 Emax = 20.0, DeltaE = 0.1 \" runPROJWFC projwfc.$name.in Below is the plots of total and projected density of states. Also see bandstructure of Fe with and without SOC.","s":"Density of states calculation","u":"/espresso/hands-on/fe","h":"#density-of-states-calculation","p":56},{"i":63,"t":"Paramagnetic materials have fluctuating magnetic moments that may not be properly described DFT. One approach is to model paramagnetic materials in DFT calculation by building a large supercell and assign randomly oriented magnetic moments. Also note that DFT assumes zero temperature, so it makes sense to perform FM or AFM calculation for magnetic systems.","s":"Paramagnetism","u":"/espresso/hands-on/fe","h":"#paramagnetism","p":56},{"i":65,"t":"We can use XCrySDen to visualize the orientation of magnetic moments. XCrySDen cannot directly read the Quantum Espresso output files for magnetic moment vectors, instead we need to create the input .xsf file with magnetic moments as force vector. You can also change the background color from black from the Palette Menu which is located in the left of File menu. src/fe/fe.xsf # this is a specification for crystal structure CRYSTAL # primitive lattice vectors (in Angstroms) PRIMVEC 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 # conventional lattice vectors (in Angstroms) CONVVEC 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 0.0000000000 0.0000000000 0.0000000000 2.8681404710 # First number stands for number of atoms in primitive cell # the second number is always 1 for PRIMCOORD coordinates # followed by atomic coordinates (in Angstroms) and forces: # AtNum X Y Z Fx Fy Fz PRIMCOORD 2 1 26 0.0000000000 0.0000000000 0.0000000000 0.00 0.00 0.01 26 1.4340702350 1.4340702350 1.4340702350 0.00 0.00 -0.01 Open the file from XCrySDen Menu: File → Open Structure → Open XSF. Then go to Display menu and select Forces. If you want to adjust scaling for the force vectors, go to Modify → Force Settings and set suitable Length factor.","s":"Visualizing magnetic moments","u":"/espresso/hands-on/fe","h":"#visualizing-magnetic-moments","p":56},{"i":67,"t":"Here we will calculate Fermi surface of copper. First step is to perform self- consistent field calculation. src/cu/pw.scf.cu.in &control calculation = \"scf\", prefix = \"cu\", pseudo_dir = \"../pseudos/\", outdir = \"/tmp/cu/\" / &system ibrav = 2, celldm(1) = 6.678, nat = 1, ntyp = 1, ecutwfc = 40, ecutrho = 300, occupations = \"smearing\", smearing = \"mp\", degauss = 0.01, nbnd = 16 / &electrons conv_thr = 1e-9, / ATOMIC_SPECIES Cu 63.546 Cu_ONCV_PBE-1.0.oncvpsp.upf ATOMIC_POSITIONS alat Cu 0.00 0.00 0.00 K_POINTS automatic 8 8 8 1 1 1 mpirun -np 4 pw.x -in pw.scf.cu.in > pw.scf.cu.out Next we perform bands calculation over dense uniform k-grid: src/cu/pw.bands.cu.in &control calculation = \"bands\", prefix = \"cu\", pseudo_dir = \"../pseudos/\", outdir = \"/tmp/cu/\" / &system ibrav = 2, celldm(1) = 6.678, nat = 1, ntyp = 1, ecutwfc = 40, ecutrho = 300, occupations = \"smearing\", smearing = \"mp\", degauss = 0.01, nbnd = 16 / &electrons conv_thr = 1e-9, / ATOMIC_SPECIES Cu 63.546 Cu_ONCV_PBE-1.0.oncvpsp.upf ATOMIC_POSITIONS alat Cu 0.00 0.00 0.00 K_POINTS automatic 30 30 30 0 0 0 mpirun -np 4 pw.x -in pw.bands.cu.in > pw.bands.cu.out Finally, we process the data with fs.x post processing tool. Below is the input file: src/cu/fs.cu.in &fermi outdir = \"/tmp/cu/\" prefix = \"cu\" / mpirun -np 4 fs.x -in fs.cu.in > fs.cu.out We can visualize the output file cu_fs.bxsf using xcrysdens program: xcrysden --bxsf cu_fs.bxsf","s":"Fermi Surface","u":"/espresso/hands-on/fermi-surface","h":"","p":66},{"i":69,"t":"Now that we have calculated the bandstructure of silicon (semiconductor) and aluminum (metal), let us proceed with a compound which has two different atoms. We follow the steps like before: First check the lattice constant with cell relaxation according to our chosen pseudo potential. We use that lattice constant in our next steps. Our lattice constant = 10.6867 * 0.508176602 / 0.5 = 10.861462. pw.x < pw.relax.GaAs.in > pw.relax.GaAs.out Perform self consistent field calculation: pw.x < pw.scf.GaAs.in > pw.scf.GaAs.out Give denser k-points and perform non-self consistent calculation. This step is only necessary if you need to obtain density of states. pw.x < pw.nscf.GaAs.in > pw.nscf.GaAs.out Perform bands calculation pw.x < pw.bands.GaAs.in > pw.bands.GaAs.out Post process the data and plot the bandstructure. bands.x < pp.bands.GaAs.in > pp.bands.GaAs.out If everything goes well, you will get the bandstructure as below: Warning Sometimes a calculation with the same inputs converges in one computer, while fails in another due to library configuration or even due to floating point approximations. The final output numbers will always vary slightly for different machines, or even among different runs in the same machine. Also check the Quantum Espresso version for reproducibility. Also see the bandstructure of GaAs with SOC.","s":"Bandstructure of GaAs","u":"/espresso/hands-on/GaAs","h":"","p":68},{"i":71,"t":"I am following this example from the ICTP online school 2021. Graphene is single layer of carbon atoms. First perform the self consistent field calculation to obtain the Kohn-Sham orbitals. Please check the input files in GitHub repository. Run pw.x: pw.x -i graphene_scf.in > graphene_scf.out Next increase the k-grid, and perform the non-self-consistent field calculation. pw.x -i graphene_nscf.in > graphene_nscf.out","s":"DOS and Bandstructure of Graphene","u":"/espresso/hands-on/graphene","h":"","p":70},{"i":73,"t":"dos.x -i graphene_dos.in > graphene_dos.out","s":"DOS calculation","u":"/espresso/hands-on/graphene","h":"#dos-calculation","p":70},{"i":75,"t":"First run the bands calculation for given k-path: pw.x -i graphene_bands.in > graphene_bands.out Followed by the postprocessing to collect the bands: bands.x -i graphene_bands_pp.in > graphene_bands_pp.out Make plots: notebooks/graphene.ipynb import numpy as np import matplotlib.pyplot as plt data = np.loadtxt('../src/graphene/graphene_bands.dat.gnu') k = np.unique(data[:, 0]) bands = np.reshape(data[:, 1], (-1, len(k))) for band in range(len(bands)): plt.plot(k, bands[band, :], linewidth=1, alpha=0.5, color='k') plt.xlim(min(k), max(k)) # Fermi energy plt.axhline(0.921, linestyle=(0, (8, 10)), linewidth=0.75, color='k', alpha=0.5) # High symmetry k-points (check bands_pp.out) plt.axvline(0.6667, linewidth=0.75, color='k', alpha=0.5) plt.axvline(1, linewidth=0.75, color='k', alpha=0.5) # text labels plt.xticks(ticks= [0, 0.6667, 1, 1.5774], labels=['$\\Gamma$', 'K', 'M', '$\\Gamma$']) plt.ylabel(\"Energy (eV)\") plt.show()","s":"Bandstructure calculation","u":"/espresso/hands-on/graphene","h":"#bandstructure-calculation","p":70},{"i":77,"t":"Here we will calculate k-resolved density of states for silicon. First we begin with self consistent field calculation. Here is the input: pw.x -inp si_scf.in > si_scf.out Followed by the bands calculation. Note that for bands calculation I have doubled the number of k-points compared to our previous bands calculation. pw.x -inp si_bands.in > si_bands.out Calculate the orbital projections with k-resolved information: src/silicon/si_projwfc.in &projwfc outdir = './tmp/' prefix = 'silicon' ngauss = 0 degauss = 0.036748 DeltaE = 0.005 kresolveddos = .true. filpdos = 'silicon.k' / projwfc.x -inp si_projwfc.in > si_projwfc.out This will give separate orbital projections, as well as total sum for k-resolved DOS. Make plots: notebooks/silicon-kpdos.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np import zipfile %matplotlib inline # data file was compressed to reduce file size zipobj = zipfile.ZipFile('../src/silicon/silicon.k.pdos_tot.zip', 'r') zipdata = zipobj.open('silicon.k.pdos_tot') data = np.loadtxt(zipdata) k = np.unique(data[:, 0]) # k values e = np.unique(data[:, 1]) # dos energy values dos = np.zeros([len(k), len(e)]) for i in range(len(data)): e_index = int(i % len(e)) k_index = int(data[i][0] - 1) dos[k_index, e_index] = data[i][2] plt.pcolormesh(k, e, dos.T, cmap='magma', shading='auto') # plt.ylim(-2, 10) plt.xticks([]) plt.ylabel('Energy (eV)') plt.xticks([]) plt.gcf().text(0.12, 0.06, 'L', fontsize=16, fontweight='normal') plt.gcf().text(0.29, 0.06, '$\\Gamma$', fontsize=16, fontweight='normal') plt.gcf().text(0.55, 0.06, 'X', fontsize=16, fontweight='normal') plt.gcf().text(0.63, 0.06, 'U', fontsize=16, fontweight='normal') plt.gcf().text(0.892, 0.06, '$\\Gamma$', fontsize=16, fontweight='normal') plt.axvline(21, c='yellow', lw=1, alpha=0.5) plt.axvline(51, c='yellow', lw=1, alpha=0.5) plt.axvline(61, c='yellow', lw=1, alpha=0.5) plt.show() info If you are using ibrav=0, you can calculate projwfc with lsym=.false. option. If we have contribution from multiple orbitals, we can sum desired projections using sumpdos.x program. For example: sumpdos.x *\\(Cl\\)*\\(p\\) > Cl_2p_tot.dat This way we can plot different orbital projections along with energy and k-resolution.","s":"k-resolved DOS","u":"/espresso/hands-on/kpdos","h":"","p":76},{"i":79,"t":"We prepare the input file pw_scf_ni.in and run the calculation: mpirun -np 8 pw.x -i pw_scf_ni.in > pw_scf_ni.out Prepare the input file for bands calculation pw_bands_ni.in with our desired k-path and run: mpirun -np 8 pw.x -i pw_bands_ni.in > pw_bands_ni.out Now we perform the bands.x calculation with spin_component=1 to process only the spin up bands: src/ni/bands_ni_up.in &BANDS outdir='./tmp/', prefix='ni', filband='ni_bands_up.dat', spin_component = 1, / Run the calculation: mpirun -np 8 bands.x -i bands_ni_up.in > bands_ni_up.out Similarly, we process the spin down bands spin_component=2 and plot them.","s":"Spin polarized bandstructure calculation for nickel","u":"/espresso/hands-on/ni","h":"","p":78},{"i":81,"t":"Here we continue with our Aluminum example. Often it is needed to know the contribution from each individual atoms and/or each of their orbital contributions. We can achieve that using projwfc.x code. First, we must perform the self consistent field calculation followed by the non-self consistent field calculation with denser k-points. pw.x < al_scf.in > al_scf.out pw.x < al_nscf.in > al_nscf.out Then we prepare the input file for projwfc.x: src/al/al_projwfc.in &PROJWFC prefix= 'al', outdir= '/tmp/', filpdos= 'al_pdos.dat' / Perform the calculation: projwfc.x < al_projwfc.in > al_projwfc.out Output data format: the DOS values are written in the file {filpdos}.pdos_atm#N(X)_wfc#M(l), where N is atom number, X is atom symbol, M is wfc number, and l=s,p,d,f one file for each atomic wavefunction read from pseudopotential file. The header of file looks like (for spin polarized calculations, we have separate up and down columns): E LDOS(E) PDOS_1(E) ... PDOS_{2l+1}(E) LDOS=∑m=12l+1PDOSm(E)LDOS = \\sum\\limits_{m=1}^{2l+1} PDOS_m (E)LDOS=m=1∑2l+1​PDOSm​(E) PDOSm(E)→PDOS_m (E) \\rightarrowPDOSm​(E)→ projected DOS on atomic wfc with component mmm. Orbital order: for l=1l=1l=1: pz (m=0)p_z~(m=0)pz​ (m=0) pxp_xpx​ (real combination of m=±1m=\\pm 1m=±1 with cosine) pyp_ypy​ (real combination of m=±1m=\\pm 1m=±1 with sine) for l=2l=2l=2: dz2 (m=0)d_{z^2}~(m=0)dz2​ (m=0) dzxd_{zx}dzx​ (real combination of m=±1m=\\pm 1m=±1 with cosine) dzyd_{zy}dzy​ (real combination of m=±1m=\\pm 1m=±1 with sine) dx2−y2d_{x^2-y^2}dx2−y2​ (real combination of m=±2m=\\pm 2m=±2 with cosine) dxyd_{xy}dxy​ (real combination of m=±2m=\\pm 2m=±2 with sine) Let's make our plots: src/notebooks/al-pdos.ipynb import matplotlib.pyplot as plt from matplotlib import rcParamsDefault import numpy as np %matplotlib inline # load data def data_loader(fname): import numpy as np data = np.loadtxt(fname) energy = data[:, 0] pdos = data[:, 2] return energy, pdos energy, pdos_s = data_loader('../src/al/al_pdos.dat.pdos_atm#1(Al)_wfc#1(s)') _, pdos_p = data_loader('../src/al/al_pdos.dat.pdos_atm#1(Al)_wfc#2(p)') _, pdos_tot = data_loader('../src/al/al_pdos.dat.pdos_tot') # make plots plt.figure(figsize = (8, 4)) plt.plot(energy, pdos_s, linewidth=0.75, color='#006699', label='s-orbital') plt.plot(energy, pdos_p, linewidth=0.75, color='r', label='p-orbital') plt.plot(energy, pdos_tot, linewidth=0.75, color='k', label='total') plt.yticks([]) plt.xlabel('Energy (eV)') plt.ylabel('DOS') plt.axvline(x= 7.9421, linewidth=0.5, color='k', linestyle=(0, (8, 10))) plt.xlim(-5, 27) plt.ylim(0, ) plt.fill_between(energy, 0, pdos_s, where=(energy < 7.9421), facecolor='#006699', alpha=0.25) plt.fill_between(energy, 0, pdos_p, where=(energy < 7.9421), facecolor='r', alpha=0.25) plt.fill_between(energy, 0, pdos_tot, where=(energy < 7.9421), facecolor='k', alpha=0.25) # plt.text(6.5, 0.52, 'Fermi energy', fontsize= small, rotation=90) plt.legend(frameon=False) plt.show() Here is how our projected density of states plot looks like: We can perform sums of specific atom or orbital contributions using sumpdos.x code if there are multiple sss or ppp orbitals: sumpdos.x *\\(Al\\)* > atom_Al_tot.dat sumpdos.x *\\(Al\\)*\\(s\\) > atom_Al_s.dat sumpdos.x *\\(Al\\)*\\(p\\) > atom_Al_p.dat","s":"Projected Density of States","u":"/espresso/hands-on/pdos","h":"","p":80},{"i":83,"t":"In Quantum Espresso, phonon dispersion is calculated using ph.x program, which is implementation of density functional perturbation theory (DFPT). Here are the steps for calculating phonon dispersion: (1) perform SCF calculation using pw.x src/GaAs-phonon/pw.scf.GaAs.in &control calculation = 'scf' prefix = 'GaAs' pseudo_dir = '../pseudos/' outdir = './tmp/' verbosity = 'high' wf_collect = .true. / &system ibrav = 2 celldm(1) = 10.861462 nat = 2 ntyp = 2 ecutwfc = 80 ecutrho = 640 / &electrons mixing_mode = 'plain' mixing_beta = 0.7 conv_thr = 1.0e-8 / ATOMIC_SPECIES Ga 69.723 Ga.pbe-dn-kjpaw_psl.1.0.0.UPF As 74.921595 As.nc.z_15.oncvpsp3.dojo.v4-std.upf ATOMIC_POSITIONS Ga 0.00 0.00 0.00 As 0.25 0.25 0.25 K_POINTS {automatic} 8 8 8 0 0 0 We perform the SCF calculation: mpirun -np 4 pw.x -i pw.scf.GaAs.in > pw.scf.GaAs.out info Usually higher energy cutoff values are used for phonon calculation to get better accuracy. In case of two dimensional systems, use assume_isolated = '2D' in the SYSTEM namelist to avoid negative or imaginary acoustic frequencies near Γ\\GammaΓ point. Read more here. (2) calculate the dynamical matrix on a uniform mesh of q-points using ph.x src/GaAs-phonon/ph.GaAs.in &INPUTPH outdir = './tmp/' prefix = 'GaAs' tr2_ph = 1d-14 ldisp = .true. ! recover = .true. nq1 = 6 nq2 = 6 nq3 = 6 fildyn = 'GaAs.dyn' / Run the calculation: mpirun -np 4 ph.x -i ph.GaAs.in > ph.GaAs.out The above calculation is computationally demanding. Our example calculation took about a whole day on a 2.6 GHz quad core processor. info You can restart an interrupted ph.x calculation with recover = .true. in the INPUTPH namelist. You can cleanly exit an ongoing calculation by creating an empty file with name {prefix}.EXIT. (3) perform inverse Fourier transform of the dynamical matrix to obtain inverse Fourier components in real space using q2r.x. Below is our input file: src/GaAs-phonon/q2r.GaAs.in &INPUT fildyn = 'GaAs.dyn' zasr = 'crystal' flfrc = 'GaAs.fc' / mpirun -np 4 q2r.x -i q2r.GaAs.in > q2r.GaAs.out (4) Finally, perform Fourier transformation of the real space components to get the dynamical matrix at any q by using matdyn.x. src/GaAs-phonon/matdyn.GaAs.in &INPUT asr = 'crystal' flfrc = 'GaAs.fc' flfrq = 'GaAs.freq' flvec = 'GaAs.modes' ! loto_2d = .true. q_in_band_form = .true. / 5 0.500 0.500 0.500 20 ! L 0.000 0.000 0.000 20 ! G 0.500 0.000 0.500 20 ! X 0.375 0.375 0.750 20 ! K 0.000 0.000 0.000 1 ! G mpirun -np 4 matdyn.x -i matdyn.GaAs.in > matdyn.GaAs.out We can now plot the phonon dispersion of GaAs: notebooks/GaAs-phonon.ipynb import numpy as np import matplotlib.pyplot as plt data = np.loadtxt(\"../src/GaAs-phonon/GaAs.freq.gp\") nbands = data.shape[1] - 1 for band in range(nbands): plt.plot(data[:, 0], data[:, band], linewidth=1, alpha=0.5, color='k') # High symmetry k-points (check matdyn.GaAs.in) plt.axvline(x=data[0, 0], linewidth=0.5, color='k', alpha=0.5) plt.axvline(x=data[20, 0], linewidth=0.5, color='k', alpha=0.5) plt.axvline(x=data[40, 0], linewidth=0.5, color='k', alpha=0.5) plt.axvline(x=data[60, 0], linewidth=0.5, color='k', alpha=0.5) plt.xticks(ticks= [0, data[20, 0], data[40, 0], data[60, 0], data[-1, 0]], \\ labels=['L', '$\\Gamma$', 'X', 'U,K', '$\\Gamma$']) plt.ylabel(\"Frequency (cm$^{-1}$)\") plt.xlim(data[0, 0], data[-1, 0]) plt.ylim(0, ) plt.show() tip We may need to lower the value of conv_thr in scf calculation for more accurate result.","s":"Phonon dispersion","u":"/espresso/hands-on/phonon","h":"","p":82},{"i":85,"t":"Input file for phonon DOS calculation: src/GaAs-phonon/matdyn.dos.GaAs.in &INPUT asr = 'crystal' flfrc = 'GaAs.fc' flfrq = 'GaAs.dos.freq' flvec = 'GaAs.dos.modes' dos = .true. fldos = 'GaAs.dos' nk1 = 25 nk2 = 25 nk3 = 25 / Plot phonon DOS: notebooks/GaAs-phonon.ipynb freq, dos, pdos_Ga, pdos_As = np.loadtxt(\"../src/GaAs-phonon/GaAs.dos\", unpack=True) plt.plot(freq, dos, c='k', lw=0.5, label='Total') plt.plot(freq, pdos_Ga, c='b', lw=0.5, label='Ga') plt.plot(freq, pdos_As, c='r', lw=0.5, label='As') plt.xlabel('$\\\\Omega~(cm^{-1}$)') plt.ylabel('Phonon DOS (state/cm$^{-1}/u.c.$)') plt.legend(frameon=False, loc='upper left') plt.xlim(freq[0], freq[-1]) plt.show()","s":"Phonon Density of States","u":"/espresso/hands-on/phonon","h":"#phonon-density-of-states","p":82},{"i":87,"t":"School on Electron-Phonon Physics from First Principles (2018) (Video lectures on YouTube) https://github.com/nguyen-group/QE-SSP","s":"Resources","u":"/espresso/hands-on/phonon","h":"#resources","p":82},{"i":89,"t":"We need to provide various important parameters for the self consistent calculation (solves the Kohn-Sham equation self-consistently) via an input file. In QE input files, there are NAMELISTS and INPUT_CARDS. NAMELISTS variables have default values, and new values can be provided as required for a specific calculation. The variables can be declared in any specific order. On the other hand, the variables in the INPUT_CARDS has always to be specified and in specific order. Logically independent INPUT_CARDS may be organized in any order. There are three mandatory NAMELISTS in PWscf: (1) &CONTROL: specifies the flux of computation, (2) &SYSTEM: specifies the system, and (3) &ELECTRONS: specifies the algorithms used to solve the Kohn-Sham equation. There are two other NAMELISTS: &IONS and &CELLS, which need to be specified depending on the calculation. Three INPUT_CARDS: ATOMIC_SPECIES, ATOMIC_POSITIONS, and K_POINTS in PWscf are mandatory. There are few others that must be provided in certain calculations. Below is our input file pw.scf.silicon.in for silicon in standard diamond (FCC) structure. The input files are typically named with .in prefix, while output files are named with .out prefix for their easier identification. The input parameters are organized in &namelists followed by their fields or cards. The &control, &system, and &electrons namelists are required. There are also optional &cell and &ions, you must provide them if your calculation require them. Most parameters in the namelists have default values (which may or may not suit your needs), however some variables you must always provide. Comment lines can be added with lines starting with a ! like in Fortran. Also, parameter names are not case-sensitive, i.e., &control and &CONTROL are the same. src/silicon/pw.scf.silicon.in &CONTROL ! we want to perform self consistent field calculation calculation = 'scf', ! prefix is reference to the output files prefix = 'silicon', ! output directory. Note that it is deprecated. outdir = './tmp/' ! directory for the pseudo potential directory pseudo_dir = '../pseudos/' ! verbosity high will give more details on the output file verbosity = 'high' / &SYSTEM ! Bravais lattice index, which is 2 for FCC structure ibrav = 2, ! Lattice constant in BOHR celldm(1) = 10.26, ! number of atoms in an unit cell nat = 2, ! number of different types of atom in the cell ntyp = 1, ! kinetic energy cutoff for wavefunction in Ry ecutwfc = 30 ! number of bands to calculate nbnd = 8 / &ELECTRONS ! Mixing factor used in the self-consistent method mixing_beta = 0.6 / ATOMIC_SPECIES Si 28.086 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.0 0.0 0.0 Si 0.25 0.25 0.25 K_POINTS (automatic) 6 6 6 0 0 0 I am using the pseudo potential file (Si.pz-vbc.UPF) downloaded from Quantum Espresso Website. You must read the PWscf user manual for in-depth understanding. Check the qe-x.x/PW/Doc/ folder under your installation directory. There is also another file INPUT_PW.html regarding the details of input parameters. PW stands for plane waves. Run pw.x in self consistent mode for silicon. pw.x < pw.scf.silicon.in > pw.scf.silicon.out # For parallel execution mpirun -np 4 pw.x -inp pw.scf.silicon.in > pw.scf.silicon.out note Note that I have added the executable path to my bash/zsh profile, otherwise you have to provide the full path where the pw.x executable is located. Now let’s look at the output file pw.scf.silicon.out and see how the convergence is reached: grep -e 'total energy' -e estimate pw.scf.silicon.out and you should see something like this: total energy = -15.85014573 Ry Harris-Foulkes estimate = -15.86899637 Ry estimated scf accuracy < 0.06093037 Ry total energy = -15.85194177 Ry Harris-Foulkes estimate = -15.85292281 Ry estimated scf accuracy < 0.00462014 Ry total energy = -15.85218359 Ry Harris-Foulkes estimate = -15.85220235 Ry estimated scf accuracy < 0.00011293 Ry ! total energy = -15.85219789 Ry Harris-Foulkes estimate = -15.85219831 Ry estimated scf accuracy < 0.00000099 Ry The total energy is the sum of the following terms: It is important to note that the absolute value of DFT total energy is not with respect to the vacuum reference, and depends on the chosen pseudopotential. The meaningful measure is the difference in total energy, where various offsets cancel out. note In the above calculation, if you check the output file pw.scf.silicon.out, you will find: highest occupied, lowest unoccupied level (eV): 6.2117 6.8442. Therefore, the bandgap is 0.6325 eV, which is an underestimation of actual bandgap (1.12 eV). Tips on convergence Reduce mixing_beta value, especially if there is an oscillation around the convergence energy. If it is a metallic system, use smearing and degauss. In this case, the SCF accuracy gradually goes down then suddenly increases (due to slight change in Fermi energy highest occupied/ lowest unoccupied levels change). Increase energy and charge density cutoffs (make sure they are sufficient). Certain pseudo potential files have issues, you may try with pseudo potentials from different libraries. Suggested values for the conv_thr: for energy and eigenvalues (scf calculation) 1.0d-7, for forces (relax calculation) 1.0d-8, for stress (vc-relax calculation) 1.0d-9 Ry. For certain calculation convergence might be very slow for the first iteration, one can start the calculation with a higher threshold, after few iterations reduce it and restart the calculation. There are several other important information is printed on the output file. Exchange correlation used in the calculation: Exchange-correlation= SLA PZ NOGX NOGC Where SLA → Slater exchange; PZ → Perdew-Zunger parametrization of the LDA; NOGX and NOGC indicates that density gradients are not taken into account. We can see the total number of plane waves (1067) uses in our calculation: Parallelization info -------------------- sticks: dense smooth PW G-vecs: dense smooth PW Min 108 108 34 1489 1489 266 Max 109 109 35 1492 1492 267 Sum 433 433 139 5961 5961 1067 Number of Kohn-Sham states: number of electrons = 8.00 number of Kohn-Sham states= 8 In our calculation we have specified the number of bands = 8. Otherwise, there would be 4 bands for 8 electrons in case of non spin-polarized systems.","s":"Self consistent field calculation for silicon","u":"/espresso/hands-on/scf","h":"","p":88},{"i":91,"t":"https://www.quantum-espresso.org/Doc/pw_user_guide/ Quantum Espresso Input Generator (can help crating QE input files)","s":"Resources","u":"/espresso/hands-on/scf","h":"#resources","p":88},{"i":93,"t":"There are two types of structural optimization calculations in Quantum espresso: (1) relax: where only the atomic positions are allowed to vary, and (2) vc-relax: which allows to vary both the atomic positions and lattice constants. src/silicon/si_relax.in &control calculation = 'vc-relax' prefix = 'silicon' outdir = './tmp/' pseudo_dir = './pseudos/' etot_conv_thr = 1e-5 forc_conv_thr = 1e-4 / &system ibrav=2, celldm(1) =14, nat=2, ntyp=1, ecutwfc=30 / &electrons conv_thr=1e-8 / &ions / &cell cell_dofree='ibrav' / ATOMIC_SPECIES Si 28.0855 Si.pz-vbc.UPF ATOMIC_POSITIONS (alat) Si 0.00 0.00 0.00 0 0 0 Si 0.25 0.25 0.25 0 0 0 K_POINTS (automatic) 6 6 6 1 1 1 Perform the plane wave calculation: pw.x -inp si_relax.in > si_relax.out This produces following output (see the si_relax.out for more details, look for \"Final enthalpy\"): Final enthalpy = -15.8536258868 Ry Begin final coordinates new unit-cell volume = 265.89380 a.u.^3 ( 39.40140 Ang^3 ) density = 2.36728 g/cm^3 CELL_PARAMETERS (alat= 14.00000000) -0.364556379 0.000000000 0.364556379 0.000000000 0.364556379 0.364556379 -0.364556379 0.364556379 0.000000000 ATOMIC_POSITIONS (alat) Si 0.0000000000 0.0000000000 0.0000000000 0 0 0 Si 0.1822781896 0.1822781896 0.1822781896 0 0 0 End final coordinates Lattice constant = 0.364556379 * 14 / 0.5 = 10.2076 Bohr.","s":"Structure optimization","u":"/espresso/hands-on/structure-optimization","h":"","p":92},{"i":95,"t":"In order to consider spin orbit coupling effect in our electronic structure calculation in quantum espresso, we need to use a full relativistic pseudo potential. Following settings are needed in the &SYSTEM card: &SYSTEM ... noncolin = .true. lspinorb = .true. ... /","s":"Spin-Orbit Coupling","u":"/espresso/hands-on/soc","h":"","p":94},{"i":97,"t":"In simple spin polarized calculation (nspin=2), the spin quantum number (up or down) is considered in the calculation. In non-collinear case, the spin has more degrees of freedom, and can be oriented in any direction. Non-collinear magnetism is quite common in nature, where the spins are not parallel (ferromagnetic) or anti-parallel (antiferromagnetic), rather they orient in spirals, helicoids, canted or disordered. Non-collinear magnetism can occur because of geometric frustration of magnetic interaction. It can also occur due to the magnetocrystalline anisotropy which is the result of interaction between the spin and lattice interaction. This relativistic effect comes via spin-orbit coupling. We can constrain the magnetic moment: &SYSTEM ... constrained_magnetization = 'atomic direction' ... / Starting magnetization can be specified by angle1 (angle with zzz axis) and angle2 (angle of projection in xyxyxy-plane and with xxx-axis). Also check the penalty function (λ\\lambdaλ). &SYSTEM ... angle1(i) = 0.0d0 angle2(i) = 0.0d0 lambda = 0.5 ... / i is the index of the atom in ATOMIC_SPECIES card.","s":"Non collinear spin","u":"/espresso/hands-on/soc","h":"#non-collinear-spin","p":94},{"i":99,"t":"Spin-orbit coupling calculations are often hard to converge. Use a smaller mixing_beta for such calculations. First perform a collinear calculation with non-relativistic pseudopotential, and then start from the obtained charge density to perform non-colinear spin orbit calculation. &ELECTRONS ... mixing_beta = 1.0000000000d-01 startingpot = 'file' / When starting with non-collinear calculation from an existing charge density file from a collinear lsda calculation, we need to set lforcet=.true.. It assumes previous density points in z direction, and rotates in the direction specified by angle1 (initial magnetization angle with zzz-axis in degrees), and angle2 (angle in degrees for projections in xyxyxy-plane and with xxx-axis). &SYSTEM ... angle1(i) = 0.0 angle2(i) = 0.0 lforcet = .true. / Also, make sure that energy and charge density cutoffs are sufficient. Certain pseudo potentials might have issues, try with pseudo potentials from a different library. In case of metallic systems, remember to apply smearing. Common Errors S matrix not positive definite: This error might appear due to numerical instability from overlapping atoms. Check atomic positions carefully. In one my calculations, this error was resolved after setting higher ecutrho. Simplified LDA+U not implemented with noncol magnetism, use lda_plus_u_kind=1.","s":"Strategy for convergence","u":"/espresso/hands-on/soc","h":"#strategy-for-convergence","p":94},{"i":101,"t":"src/fe/pw.scf.fe_soc.in &control calculation='scf' pseudo_dir = '../pseudos/', outdir='./tmp/' prefix='fe' / &system ibrav = 3, celldm(1) = 5.39, nat= 1, ntyp= 1, noncolin=.true., lspinorb=.true., starting_magnetization(1)=0.3, ecutwfc = 70, ecutrho = 850.0, occupations='smearing', smearing='marzari-vanderbilt', degauss=0.02 / &electrons diagonalization='david' conv_thr = 1.0e-8 mixing_beta = 0.7 / ATOMIC_SPECIES Fe 55.845 Fe.rel-pbe-spn-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS alat Fe 0.0 0.0 0.0 K_POINTS AUTOMATIC 14 14 14 1 1 1 Run the scf calculation: mpirun -np 8 pw.x -i pw.scf.fe_soc.in > pw.scf.fe_soc.out Prepare the input file for nscf bands calculation: src/fe/pw.bands.fe_soc.in &control calculation='bands' pseudo_dir = '../pseudos/', outdir='./tmp/' prefix='fe' / &system ibrav = 3, celldm(1) = 5.39, nat= 1, ntyp= 1, noncolin=.true., lspinorb=.true., starting_magnetization(1)=0.3, ecutwfc = 70, ecutrho = 850.0, occupations='smearing', smearing='marzari-vanderbilt', degauss=0.02 / &electrons diagonalization='david' conv_thr = 1.0e-8 mixing_beta = 0.7 / ATOMIC_SPECIES Fe 55.845 Fe.rel-pbe-spn-rrkjus_psl.1.0.0.UPF ATOMIC_POSITIONS alat Fe 0.0 0.0 0.0 K_POINTS tpiba_b 6 0.000 0.000 0.000 40 !gamma 0.000 1.000 0.000 40 !H 0.500 0.500 0.000 30 !N 0.000 0.000 0.000 30 !gamma 0.500 0.500 0.500 30 !P 0.000 1.000 0.000 1 !H Run the bands calculation: mpirun -np 8 pw.x -i pw.bands.fe_soc.in > pw.bands.fe_soc.out Finally post process the bandstructure data: src/fe/pp.bands.fe_soc.in &BANDS outdir='./tmp/', prefix='fe', filband='fe_bands_soc.dat', / In this case spin_component has been removed and we add lsigma(3)=.true. that instructs the program to compute the expectation value for the z component of the spin operator for each eigenfunction and save all values in the file fe.noncolin.data.3. All values in this case are either +1/2 or -1/2. mpirun -np 8 bands.x -i pp.bands.fe_soc.in > pp.bands.fe_soc.out","s":"Bandstructure of Fe with SOC","u":"/espresso/hands-on/soc","h":"#bandstructure-of-fe-with-soc","p":94},{"i":103,"t":"Please check the respective input files. mpirun -np 8 pw.x -i pw.scf.GaAs_soc.in > pw.scf.GaAs_soc.out mpirun -np 8 pw.x -i pw.bands.GaAs_soc.in > pw.bands.GaAs_soc.out mpirun -np 8 bands.x -i pp.bands.GaAs_soc.in > pp.bands.GaAs_soc.out","s":"SOC calculation for GaAs","u":"/espresso/hands-on/soc","h":"#soc-calculation-for-gaas","p":94},{"i":106,"t":"Perform scf calculation using Quantum Espresso pw.x QE_PATH=\"/workspaces/q-e-qe-7.2/bin\" mpirun -np 4 ${QE_PATH}/pw.x -i pw.scf.silicon.in > pw.scf.silicon.out Perform nscf calculation using pw.x. Instead of automatic k-grid, we need to provide explicit list of k-points. Such explicit list of k-points can be generated using perl script included in the Wannier package under utility. WANNIER_PATH=\"/workspaces/wannier90-3.1.0\" # directly append the k-points to the input file ${WANNIER_PATH}/utility/kmesh.pl 4 4 4 >> pw.nscf.silicon.in Run nscf calculation: mpirun -np 4 ${QE_PATH}/pw.x -i pw.nscf.silicon.in > pw.nscf.silicon.out Prepare input file for wannier90 (silicon.win). Here we need the k-points list without the weights: ${WANNIER_PATH}/utility/kmesh.pl 4 4 4 wan Generate nnkp input: # we can just provide the seedname or seedname.win ${WANNIER_PATH}/wannier90.x -pp silicon Create input file for pw2wan, and generate initial projections: mpirun -np 4 ${WANNIER_PATH}/pw2wannier90.x -i pw2wan.silicon.in > pw2 wan.silicon.out Run wannier calculation: mpirun -np 4 ${WANNIER_PATH}/wannier90.x silicon","s":"Obtain bandstructure of Silicon","u":"/espresso/hands-on/wannier","h":"#obtain-bandstructure-of-silicon","p":104},{"i":108,"t":"https://sites.google.com/view/hubbard-koopmans/program","s":"Resources","u":"/espresso/hands-on/wannier","h":"#resources","p":104},{"i":110,"t":"This work is licensed under a Creative Commons Attribution 4.0 International License. Any third party materials in this work are not included in the article’s Creative Commons license, and users will need to obtain permission from the respective license holder to reproduce such materials. You are free to: Share — copy and redistribute the material in any medium or format. Adapt — remix, transform, and build upon the material for any purpose, even commercially. Under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. Notices: No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material. To view full copy of this license, visit http://creativecommons.org/licenses/by/4.0/","s":"License","u":"/espresso/license","h":"","p":109},{"i":112,"t":"Quantum Espresso Tutorials ICTP Quantum Espresso workshop 2021 Ljubljana QE summer school 2019 MASTANI Summer School, IISER Pune 2014 (archive) Examples included in Quantum Espresso (qe-x.x/PW/examples/). QE mailing list archive Materials square blog Cornell PARADIM Summer School School on Electron-Phonon Physics from First Principles (2018)","s":"Resources","u":"/espresso/resources","h":"","p":111},{"i":114,"t":"QUANTUM ESPRESSO for quantum simulations of materials Advanced capabilities for materials modelling with QE Hubbard parameters from density-functional perturbation theory Self-consistent Hubbard parameters from DFPT","s":"Papers describing DFT implementations in QE","u":"/espresso/resources","h":"#papers-describing-dft-implementations-in-qe","p":111},{"i":116,"t":"A bird's–eye view of DFT Density Functional Theory: A Practical Introduction by Sholl and Steckel Materials Modelling using Density Functional Theory by Feliciano Giustino Electronic Structure: Basic Theory and Practical Method by Richard M. Martin Electronic Structure Calculations for Solids and Molecules by Jorge Kohanoff PhD Thesis of Dominik Bogdan Jochym","s":"Books","u":"/espresso/resources","h":"#books","p":111},{"i":119,"t":"Density functional theory (DFT) calculations are ab-initio meaning the calculation is done from the scratch based on given input parameters. We need to provide the crystal structure in order to calculate DFT. Crystal structures are widely available in Crystallographic Information File (.CIF) format. There are several databases where you can look for crystal structures. http://crystallography.net/cod/ https://materialsproject.org https://mpds.io/ https://icsd.fiz-karlsruhe.de/index.xhtml http://aflowlib.org/CrystalDatabase/ http://crystdb.nims.go.jp/crystdb/search-materials In Quantum Espresso, the structure information is provided by ibrav number, and corresponding celldm values or lattice constants and cosines of angle between the axes. It is also possible to set ibrav=0 and provide lattice vectors in CELL_PARAMETERS. danger When set ibrav=0, the lattice vectors must be provided with sufficiently large number of decimal accuracy, otherwise symmetry detection may fail and strange problems may arrise. ibrav numbers for different lattice types: ibrav Lattice type 1 Simple cubic 2 Face centered cubic 3,-3 Body centered cubic 4 Hexagonal 5 Trigonal with c as 3-fold axis -5 Trigonal with <111> as 3-fold axis 6 Simple tetragonal 7 Centered tetragonal 8 Simple orthorhombic 9,-9,91 One-face centered orthorhombic 10 Face centered orthorhombic 11 Body centered orthorhombic 12 Simple monoclinic, c unique -12 Simple monoclinic, b unique 13 One base centered monoclinic, c unique -13 One base centered monoclinic, b unique 14 Triclinic","s":"Structure databases","u":"/espresso/setup/crystal-structure","h":"#structure-databases","p":117},{"i":121,"t":"Vesta - https://jp-minerals.org/vesta/en/. It helps you visualize crystal structure, create and modify supercells, crystal structures, and many other useful functionalities. We can prepare our Quantum Espresso input file using cif2cell utility. If you do not have cif2cell installed, you can use pip to install: sudo pip3 install cif2cell You may need to add it to the path in your .bashrc manually: export PATH=\"/home/pranab/.local/lib/python3.8/site-packages/:$PATH\" Running cif2cell command: cif2cell file.cif -p quantum-espresso -o inputfile.in","s":"Useful tools:","u":"/espresso/setup/crystal-structure","h":"#useful-tools","p":117},{"i":123,"t":"You can explore the crystal structure, find out k-path and many more using Xcrysdens application - http://www.xcrysden.org For certain functionality, Xcrysdens requires basic calculator program. On Ubuntu/ Debian: sudo apt update sudo apt install bc xcrysden Manual installation: # install dependencies sudo apt install --no-install-recommends bc tk libglu1-mesa libtogl2 \\ libfftw3-3 libxmu6 imagemagick openbabel libgfortran5 # download the latest version of xcrysden and extract wget http://www.xcrysden.org/download/xcrysden-1.6.2-linux_x86_64-shared.tar.gz tar -zxvf xcrysden-1.6.2-linux_x86_64-shared.tar.gz # launch (provided you extracted under your home directory) ~/xcrysden-1.6.2-bin-shared/xcrysden If you are on WSL, you need to install X-server (X-ming for Windows) on the host and set export DISPLAY=:0 in your WSL instance.","s":"Xcrysdens","u":"/espresso/setup/crystal-structure","h":"#xcrysdens","p":117},{"i":125,"t":"You can generate PWscf input files using tools in this website as well https://www.materialscloud.org/work/tools/qeinputgenerator The same website also has a tool for k-path visualization and generation https://www.materialscloud.org/work/tools/seekpath","s":"QE Input generator","u":"/espresso/setup/crystal-structure","h":"#qe-input-generator","p":117},{"i":127,"t":"Supercell construction using Vesta","s":"Resources","u":"/espresso/setup/crystal-structure","h":"#resources","p":117},{"i":129,"t":"In order to perform computationally heavy calculations, we would require access to high performance computing facilities.","s":"High Performance Computing","u":"/espresso/setup/hpc","h":"","p":128},{"i":131,"t":"Connect to a login node via ssh: ssh {username}@atlas9.nus.edu.sg Secure copy files between local and remote machines: scp {username}@10.10.0.2:/remote/file.txt /local/directory scp local/file.txt {username}@10.10.0.2:/remote/directory Check disk usage: du -hs . du -hs /path/ Rsync to synchronize two folders: rsync -azhv --delete /source/my_project/ /destination/my_project","s":"Useful UNIX commands","u":"/espresso/setup/hpc","h":"#useful-unix-commands","p":128},{"i":133,"t":"Check your storage quota: hpc s PBS commands: hpc pbs summary Example scrips for job submissions: hpc pbs script parallel20 hpc pbs vasp List available modules: module avail Load a module: module load {module-name} Purge loaded modules: module purge Quantum Espresso is already installed in NUS HPC clusters. Here is a sample job script for NUS HPC clusters: scripts/pbs_job.sh #!/bin/bash #PBS -q parallel24 #PBS -l select=2:ncpus=24:mpiprocs=24:mem=96GB #PBS -j eo #PBS -N qe-project-xx source /etc/profile.d/rec_modules.sh module load espresso6.5-intel_18 ## module load espresso6.5-Centos6_Intel cd $PBS_O_WORKDIR; np=$( cat ${PBS_NODEFILE} |wc -l ); mpirun -np $np -f ${PBS_NODEFILE} pw.x -inp qe-scf.in > qe-scf.out info Notice that the lines beginning with #PBS are actually PBS commands, not comments. For comments, I am using ##. Query about a queue system: qstat -q Check status of a particular queue system: qstat -Qx parallel24 Submitting a job: qsub pbs_job.sh Check running jobs: qstat Details about a job: qstat -f {job-id} Stopping a job: qdel {job-id}","s":"Running jobs at NUS HPC","u":"/espresso/setup/hpc","h":"#running-jobs-at-nus-hpc","p":128},{"i":135,"t":"If you need to modify certain parameters while the program is running, e.g., you want to change the mixing_beta value because SCF accuracy is oscillation without any sign of convergence. Create an empty file named {prefix}.EXIT in the directory where you have the input file or in the outdir as set in the &CONTROL card of input file. touch {prefix}.EXIT That will stop the program on the next iteration, and save the state. In order to restart, set the restart_mode in &CONTROL card to 'restart' and re-run after necessary changes. You must re-submit the job with the same number of processors. &CONTROL ... restart_mode = 'restart' ... /","s":"Abort and restart a calculation","u":"/espresso/setup/hpc","h":"#abort-and-restart-a-calculation","p":128},{"i":137,"t":"If you need a newer or specific version of Quantum Espresso that is not installed in the NUS clusters or you have modified the source codes yourself, here are the steps that I followed to successfully compile. info Quantum Espresso project is primarily hosted on GitLab, and its mirror is maintained at GitHub. You may check their repository at GitLab for more up to date information. The releases via GitLab can be found under: https://gitlab.com/QEF/q-e/-/releases Download and decompress the source files. wget https://gitlab.com/QEF/q-e/-/archive/qe-7.2/q-e-qe-7.2.tar.gz tar -zxvf q-e-qe-7.2.tar.gz Load the necessary modules (applicable for NUS clusters, last checked in Jun 2022): module load xe_2018 module load fftw/3.3.7 Go to QE directory and run configure: cd q-e-qe-7.2 ./configure You will see output something like: ... BLAS_LIBS= -lmkl_intel_lp64 -lmkl_sequential -lmkl_core LAPACK_LIBS= FFT_LIBS= ... For me, the LAPACK_LIBS and FFT_LIBS libs were not automatically detected. We need to specify them manually. First, get the link libraries line specific to your version of MKL and other configurations from the Intel link advisor. For my case, the link line was: -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl We need to insert the link for BLAS_LIBS, LAPACK_LIBS, and SCALAPACK_LIBS. We also need to find out where is the FFTW lib located. In NUS HPC, we can use module avail command to see where a particular module is located, usually under /app1/modules/. Open make.inc and make the following changes: make.inc # ... CFLAGS = -O2 $(DFLAGS) $(IFLAGS) CFLAGS = -O3 $(DFLAGS) $(IFLAGS) F90FLAGS = $(FFLAGS) -nomodule -fpp $(FDFLAGS) $(CUDA_F90FLAGS) $(IFLAGS) $(MODFLAGS) # compiler flags with and without optimization for fortran-77 # the latter is NEEDED to properly compile dlamch.f, used by lapack - FFLAGS = -O2 -assume byterecl -g -traceback + FFLAGS = -O3 -assume byterecl -g -traceback FFLAGS_NOOPT = -O0 -assume byterecl -g -traceback # ... # If you have nothing better, use the local copy # BLAS_LIBS = $(TOPDIR)/LAPACK/libblas.a - BLAS_LIBS = -lmkl_intel_lp64 -lmkl_sequential -lmkl_core + BLAS_LIBS = -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl # If you have nothing better, use the local copy # LAPACK = liblapack # LAPACK_LIBS = $(TOPDIR)/external/lapack/liblapack.a - LAPACK = + LAPACK = liblapack - LAPACK_LIBS = + LAPACK_LIBS = -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl - SCALAPACK_LIBS = + SCALAPACK_LIBS = -L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl # nothing is needed here if the internal copy of FFTW is compiled # (needs -D__FFTW in DFLAGS) - FFT_LIBS = + FFT_LIBS = -L/app1/centos6.3/gnu/fftw/3.3.7/lib/ -lmpi # ... Now we are ready to compile: make -j8 all I am parallelizing with 8 processors to speed things up. You may add the q-e-qe-7.2/bin path to your .bashrc: echo 'export PATH=\"/home/svu/{username}/q-e-qe-7.2/bin:$PATH\"' >> ~/.bashrc And don't forget to load dependencies before calling QE executables. module load xe_2018 module load fftw/3.3.7 note If you are submitting job via PBS queue, you need to provide full path of the QE executables, e.g., /home/svu/{username}/q-e-qe-7.2/bin/pw.x. PBS system won't read your bash settings, neither the relative paths of your login node would apply.","s":"Compiling Quantum Espresso using Intel® Math Kernel Library (MKL)","u":"/espresso/setup/hpc","h":"#compiling-quantum-espresso-using-intel-math-kernel-library-mkl","p":128},{"i":139,"t":"If you need to install Intel oneAPI libraries yourself, following instructions might be useful. Please refer to Intel website for up to date information. Intel oneAPI Base Toolkit:​ wget https://registrationcenter-download.intel.com/akdlm/IRC_NAS/992857b9-624c-45de-9701-f6445d845359/l_BaseKit_p_2023.2.0.49397_offline.sh # requires gnu-awk sudo apt update && sudo apt install -y --no-install-recommends gawk gcc g++ # interactive cli installation sudo apt install -y --no-install-recommends ncurses-term sudo sh ./l_BaseKit_p_2023.2.0.49397_offline.sh -a --cli # list components included in oneAPI Base Toolkit sh ./l_BaseKit_p_2023.2.0.49397_offline.sh -a --list-components # install a subset of components with silent/unattended option sudo sh ./l_BaseKit_p_2023.2.0.49397_offline.sh -a --silent --eula accept --components intel.oneapi.lin.dpcpp-cpp-compiler:intel.oneapi.lin.mkl.devel note If you install oneAPI without sudo privilege, it will be installed under the user directory: /home/{username}/intel/oneapi/. After installation is completed, the setup script will print the installation location. HPC Toolkit​ wget https://registrationcenter-download.intel.com/akdlm/IRC_NAS/0722521a-34b5-4c41-af3f-d5d14e88248d/l_HPCKit_p_2023.2.0.49440_offline.sh sudo sh ./l_HPCKit_p_2023.2.0.49440_offline.sh -a --silent --eula accept Intel MKL library​ Installing individual components: wget https://registrationcenter-download.intel.com/akdlm/IRC_NAS/adb8a02c-4ee7-4882-97d6-a524150da358/l_onemkl_p_2023.2.0.49497_offline.sh sudo sh ./l_onemkl_p_2023.2.0.49497_offline.sh -a --silent --eula accept After installation, do not forget to source the environment variables before using: source /opt/intel/oneapi/setvars.sh Compile quantum espresso: wget https://gitlab.com/QEF/q-e/-/archive/qe-7.2/q-e-qe-7.2.tar.gz tar -zxvf q-e-qe-7.2.tar.gz rm q-e-qe-7.2.tar.gz cd q-e-qe-7.2 ./configure \\ F90=mpiifort \\ MPIF90=mpiifort \\ CC=mpicc CXX=icc \\ F77=mpiifort \\ FFLAGS=\"-O3 -assume byterecl -g -traceback\" \\ LAPACK_LIBS=\"-L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl\" \\ BLAS_LIBS=\"-L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl\" \\ SCALAPACK_LIBS=\"-L${MKLROOT}/lib/intel64 -lmkl_scalapack_lp64 -lmkl_intel_lp64 -lmkl_sequential -lmkl_core -lmkl_blacs_intelmpi_lp64 -lpthread -lm -ldl\" make -j4 all","s":"Installing Intel oneAPI libraries","u":"/espresso/setup/hpc","h":"#installing-intel-oneapi-libraries","p":128},{"i":141,"t":"Please check out the official documentation for more details. It requires cmake version 3.14 or later. apt update && apt install autoconf cmake gawk gcc g++ make I used following steps to successfully compile Quantum Espresso using 2023 versions of Intel libraries in Ubuntu 22.04 system: cd q-e-qe-7.2 mkdir build && cd build cmake -DCMAKE_C_COMPILER=mpiicc -DCMAKE_Fortran_COMPILER=mpiifort -DQE_ENABLE_SCALAPACK=ON .. make -j4 mv bin .. cd .. rm -rf build","s":"Compiling Quantum Espresso with CMake","u":"/espresso/setup/hpc","h":"#compiling-quantum-espresso-with-cmake","p":128},{"i":143,"t":"https://nusit.nus.edu.sg/services/getting-started/introductory-guide-for-new-hpc-users/ https://help.nscc.sg/pbspro-quickstartguide/ https://www.youtube.com/watch?v=doudMLEaq3w","s":"Resources","u":"/espresso/setup/hpc","h":"#resources","p":128},{"i":145,"t":"We can install Quantum Espresso on our personal laptops or desktops to run relatively less computationally intensive calculations. If we intend to perform computationally heavy tasks, we would need access to better computing resources with large number of CPU (or GPU) cores, memory, bandwidth, and disc IO. Throughout this tutorial, I will be using a Ubuntu system for smaller calculations while other computationally intensive calculations will be done in HPC clusters. Perhaps the easiest way to install Quantum Espresso is from the package manager of respective Linux distribution. This should work fine for us and this is recommended option. Following commands are for Ubuntu / Debian. First make sure your system is up-to-date. sudo apt update && sudo apt upgrade Install Quantum Espresso from apt repository: sudo apt install --no-install-recommends \\ libfftw3-dev \\ quantum-espresso If you want to compile from the source yourself, here are the installation steps for the Quantum Espresso version 7.2 in a Ubuntu (LTS 22.04) system. I will be compiling for single processor. First install the recommended libraries and dependencies: sudo apt install --no-install-recommends \\ autoconf \\ build-essential \\ ca-certificates \\ gfortran \\ libblas3 \\ libc6 \\ libfftw3-dev \\ libgcc-s1 \\ liblapack-dev \\ wget If you want to compile for parallel processing, you also need to install: sudo apt install --no-install-recommends \\ libopenmpi-dev \\ libscalapack-openmpi-dev \\ libelpa17 # use libelpa4 on Ubuntu 20.04 Download Quantum Espresso (latest version 7.2 at the time of writing): wget https://gitlab.com/QEF/q-e/-/archive/qe-7.2/q-e-qe-7.2.tar.gz Un-tar the source files: tar -zxvf q-e-qe-7.2.tar.gz Go to the qe directory and issue configure: cd q-e-qe-7.2 ./configure Here we can provide various configuration options. Read the manual in oder to properly understand. But in most cases we will be just fine with the defaults, it should detect the system configuration automatically, in case you don't get what you want, try the various configuration flags with configure. caution Note that certain programs/utilities bundled with Quantum Espresso might not work correctly in parallel compilation, so we may need serial compilation for those by ./configure --disable-parallel option in case parallel option is automatically detected. Finally, compile the source files and create the binary executables: # compile individual packages make pw # or compile everything make all # we can parallelize e.g., below command uses 4 CPUs make -j4 all Now, the binary files or their symbolic links (shortcuts) would be placed in the bin directory. It would be good idea to include the executable path to your .bashrc (or .zshrc or whatever shell you use) file: # use the correct path if it differs from mine echo 'export PATH=\"/root/q-e-qe-7.2/bin:$PATH\"' >> ~/.bashrc Finally, you may need to restart your terminal or source .bashrc. source ~/.bashrc You can compile the documentation by going to particular directory (e.g., PW or PP) and execute (you need to have LaTeX installed in your system): make doc If you want docs in PDF format, you can use latex commands to create them as well: pdflatex filename.tex We are now ready to run Quantum Espresso pw.x (or any other program) using mpirun by following command: pw.x -inp inputfile > outputfile # For parallel version mpirun -np 12 pw.x -inp inputfile > outputfile Where -np 12 specifies the number of processors. -inp stands for input file. Alternatively, we can use -i, or -in, or -input, or even standard input redirect <. But beware some systems may not interpret all the different options, I think safe option is to use -i. Once installation is completed, optionally we can run tests if everything went OK. Go to the test-suite directory and run make run-tests If all is well, we will see Passed messages and we are good to go. caution Note that the above installation steps may not be the most optimal way to run Quantum Espresso in your computer. There are multiple implementations of same library. For example, you can replace openmpi libraries with Intel MKL or MPICH implementations. Please do research yourself or ask help from someone who has knowledge about high performance computing.","s":"Quantum Espresso installation","u":"/espresso/setup/install","h":"","p":144},{"i":147,"t":"We will install a very hand scripting package PWscf Toolkit (PWTK). First we need to install following dependencies: sudo apt install tcl tcllib Download the file from - http://pwtk.ijs.si/download/pwtk-2.0.tar.gz wget \"http://pwtk.ijs.si/download/pwtk-2.0.tar.gz\" Above command will download and save the file to your current directory. Next we need to just un-tar (no need to compile): tar -zxvf pwtk-2.0.tar.gz Add the path (modify below as appropriate) to .bashrc: echo 'export PATH=\"/root/pwtk-2.0:$PATH\"' >> ~/.bashrc source ~/.bashrc","s":"Installing PWTK","u":"/espresso/setup/install","h":"#installing-pwtk","p":144},{"i":149,"t":"There are several ways you can run Jupyterlab in your computer.","s":"Jupyter notebooks","u":"/espresso/setup/jupyter","h":"","p":148},{"i":151,"t":"Install Python 3 in your computer # on ubuntu / debian apt install python3 python3-pip Install the required python packages on your computer pip3 install --upgrade -r requirements.txt # or pip3 install --upgrade numpy scipy matplotlib jupyterlab Run Jupyterlab jupyter-lab # or the classic jupyter notebook jupyter-notebook","s":"1. Install on your computer","u":"/espresso/setup/jupyter","h":"#1-install-on-your-computer","p":148},{"i":153,"t":"Install Python 3 and virtualenv on your computer pip3 install --upgrade virtualenv create virtual environment in the project directory cd qe-dft virtualenv venv activate virtual env source venv/bin/activate Install required python packages under virtualenv Launch Jupyterlab Once done, deactivate virtualenv deactivate","s":"2. Install python packages via virtualenv","u":"/espresso/setup/jupyter","h":"#2-install-python-packages-via-virtualenv","p":148},{"i":155,"t":"Install Docker Create an image with Python and the required packages installed # build using the Dockerfile included in my github repo: # https://github.com/pranabdas/espresso # (adjust the Dockerfile according to your needs) docker build -t espresso . Run a container with port forwarding docker run -it --rm -p 8888:8888 -v ${PWD}:/home espresso bash Launch Jupyterlab jupyter-lab","s":"3. Run on a container","u":"/espresso/setup/jupyter","h":"#3-run-on-a-container","p":148},{"i":157,"t":"In Quantum Espresso, pseudopotential replaces the actual electron-ion interaction. The pseudopotential describes the atomic nucleus and all the electrons except the outermost valence shell. The rapidly changing potential field near the atomic core is replaced by a smoother function that simulates the potential field far from the core very well. By doing so, it requires less number plane wave basis for wavefunction expansion. We can choose form various pseudopotential libraries. Choice of pseudopotential depends on the problem we are investigating, e.g., if there is a heavy element present in our system and we are interested in the spin-orbit coupling effects, we should choose a full relativistic pseudopotential. We need to be careful whether our chosen pseudopotential correctly reproduces physical properties. Various pseudopotential libraries: https://www.quantum-espresso.org/pseudopotentials https://www.materialscloud.org/discover/sssp/table/efficiency http://www.pseudo-dojo.org https://www.physics.rutgers.edu/gbrv/ https://nninc.cnf.cornell.edu http://www.quantum-simulation.org/potentials/ BLYP pseudopotentials SCAN pseudopotentials Pseudopotential naming conventions in PSLibrary: an example pseudopotential filename is O.rel-pbe-n-rrkjus_psl.1.0.0.UPF. O → denotes the atomic species rel → full relativistic (optional) pbe → exchange correlation functional n → non-linear core correction (optional) rrkus → pseudopotential type Exchange correlation functionals: Identifier Functional pz Perdew-Zunger (LDA) pbe Perdew-Burke-Ernzerhof (GGA) pw91 Perdew-Wang 91 (GGA) blyp Becke-Lee-Yang-Parr (GGA) Pseudopotential types: Identifier PP types ae all-electron rrkj Rappe-Rabe-Kaxiras-Joannopoulos (Norm conserving) rrkjus Rappe-Rabe-Kaxiras-Joannopoulos (Ultrasoft) kjpaw Kresse-Joubert (PAW) Ultra soft pseudopotentials are computationally efficient than the norm conserving pseudopotentials. You will find the recommended ecutwfc in the header of each pseudopotential file. If you choose an ultra-soft pseudopotential, you will need ecutrho about 8 times the value of ecutwfc. The default ecutrho is 4 times ecutwfc in Quantum Espresso code, which is a good choice for norm conserving pseudopotentials. You should check energy convergence against ecutwfc for your system. By using pseudopotential, we want to get rid of the core electrons that do not participate in the chemical properties of material. This is known also as rigid core approximation. Instead of accounting the nucleus and core electrons separately, we want to have a pseudopotential that interacts in a similar way with the valence electrons. info We can mix different types of pseudo potentials (e.g., norm conserving, ultra-soft, or PAW), but we cannot mix different exchange correlation functional (e.g., PBE and LDA). Exchange correlation functional can be read from the pseudopotential file or be provided via input_dft parameter in Quantum Espresso. \"sol\" in PBE-sol stands for solid. For bulk systems PBE-sol should be used, while PBE is appropriate for molecules. In case of 2D materials generally PBE is chosen, but one can check PBE-sol. Common error If you mix PBE with PBE-sol type, it results in Error: conflicting values for igcx. However, it is allowed to mix those two types of pseudo. We can set desired exchange correlation functional via input_dft instead of reading from the pseudopotential file.","s":"Pseudo potentials","u":"/espresso/setup/pseudo-potential","h":"","p":156},{"i":159,"t":"Naming convention for PP files","s":"Resources","u":"/espresso/setup/pseudo-potential","h":"#resources","p":156},{"i":161,"t":"Density functional theory (DFT) approaches the many-body problem by focusing on the electronic density which is a function of three spatial coordinates instead of finding the wave functions. DFT tries to minimize the energy of a system (ground state) in a self consistent way, and it is very successful in calculating the electronic structure of solid state systems. info A functional is a function whose argument is itself a function. f(x)f(x)f(x) is a function of the variable xxx while F[f]F[f]F[f] is a functional of the function fff. y=f(x)y = f(x)y=f(x) fff is a function, it takes a number xxx as input and output yyy is also a number. y=F[f(x)]y = F[f(x)]y=F[f(x)] FFF is a functional it takes function f(x)f(x)f(x) as input and output yyy is a number.","s":"Introduction to Density Functional Theory","u":"/espresso/theory/dft","h":"","p":160},{"i":163,"t":"The ground state density n(r)n(\\textbf{r})n(r) determines the external potential energy v(r)v(\\textbf{r})v(r) to within a trivial additive constant. So what Hohenberg-Kohn theorem says, may not sound very trivial. Schrödinger equation says how we can get the wavefunction from a given potential. Once solved the wavefunction (which could be difficult), we can determine the density or any other properties. Now Hohenberg and Kohn theorem says the opposite is also true. For a given density, the potential can be uniquely determined. For non-degenerate ground states, two different Hamiltonian cannot have the same ground-state electron density. It is possible to define the ground-state energy as a function of electronic density.","s":"Hohenberg-Kohn Theorem 1","u":"/espresso/theory/dft","h":"#hohenberg-kohn-theorem-1","p":160},{"i":165,"t":"Total energy of the system E(n)E(n)E(n) is minimal when n(r)n(\\textbf{r})n(r) is the actual ground-state density, among all possible electron densities. The ground state energy can therefore be found by minimizing E(n)E(n)E(n) instead of solving for the many-electron wavefunction. However, note that HK theorems do not tell us how the energy depends on the electron density. In reality, apart from some special cases, the exact E(n)E(n)E(n) is unknown and only approximate functionals are used. The essence of the HK theorem is that the non-degenerate ground-state wave function is a unique functional of the ground-state density: Ψ0(r1,r2,…,rN)=Ψ[n0(r)]\\Psi_0(\\textbf{r}_1, \\textbf{r}_2, \\dots, \\textbf{r}_N) = \\Psi[n_0(\\textbf{r})]Ψ0​(r1​,r2​,…,rN​)=Ψ[n0​(r)]","s":"Hohenberg-Kohn Theorem 2","u":"/espresso/theory/dft","h":"#hohenberg-kohn-theorem-2","p":160},{"i":167,"t":"For any system of NNN interacting electrons in a given external potential vext(r)v_{ext} (\\textbf{r})vext​(r), there is a virtual system of NNN non-interacting electrons with exactly the same density as the interacting one. The non-interacting electrons subjected to a different external (single particle) potential. [−ℏ2∇22m+vs(r)]ψi(r)=ϵiψi(r)\\left[-\\frac{\\hbar^2 \\nabla^2}{2m} + v_s(\\textbf{r}) \\right] \\psi_i(\\textbf{r}) = \\epsilon_i \\psi_i(\\textbf{r})[−2mℏ2∇2​+vs​(r)]ψi​(r)=ϵi​ψi​(r) vs(r)=vext(r)+e2∫d3r′n(r)∣r−r′∣+vxc(r;[n])v_s(\\textbf{r}) = v_{ext}(\\textbf{r}) + e^2 \\int d^3r' \\frac{n(\\textbf{r})}{|\\textbf{r} - \\textbf{r}'|} + v_{xc}(\\textbf{r}; [n])vs​(r)=vext​(r)+e2∫d3r′∣r−r′∣n(r)​+vxc​(r;[n]) n(r)=∑ifi∣ψi(r)∣2n(\\textbf{r}) = \\sum_i f_i |\\psi_i (\\textbf{r})|^2n(r)=i∑​fi​∣ψi​(r)∣2 where fif_ifi​ is the occupation factor of electrons (0≤fi≤20 \\le f_i \\le 20≤fi​≤2). The KS equation looks like single particle Schrödinger equation, however e2∫d3r′n(r)∣r−r′∣e^2 \\int d^3r' \\frac{n(\\textbf{r})}{|\\textbf{r} - \\textbf{r}'|}e2∫d3r′∣r−r′∣n(r)​ (the Hartree energy due to electrostatic interaction of electronic cloud) and vxc(r;[n])v_{xc} (\\textbf{r}; [n])vxc​(r;[n]) (exchange-correlation potential, reminiscence from Hartree-Fock theory, it includes all the remaining/unknown energy corrections) terms depend on n(r)n(\\textbf{r})n(r) i.e., on ψi\\psi_iψi​ which in turn depends on vextv_{ext}vext​. Therefore the problem is non-linear. It is usually solved computationally by starting from a trial potential and iterate to self-consistency. Also note that we have not included the kinetic energy term for the nucleus. This is because the nuclear mass is about three orders of magnitude heavier than the electronic mass (M≫mM \\gg mM≫m), so essentially electronic dynamics is much faster than the nuclear dynamics (see Born-Oppenheimer approximation). Now we are left with the task of solving a non-interacting Hamiltonian. info vext(r)v_{ext}(\\textbf{r})vext​(r) includes the potential energy due to nuclear field, and external electric and magnetic fields if present.","s":"Kohn-Sham hypothesis","u":"/espresso/theory/dft","h":"#kohn-sham-hypothesis","p":160},{"i":170,"t":"Energy functional is a function of the local charge density: Exc=∫n(r)ϵxc(n(r))drE_{xc} = \\int n(\\textbf{r}) \\epsilon_{xc}(n(\\textbf{r})) d\\textbf{r}Exc​=∫n(r)ϵxc​(n(r))dr vxc(r)=ϵxc(n(r))+n(r)dϵxc(n)dn∣n=n(r)v_{xc}(\\textbf{r}) = \\epsilon_{xc}(n(\\textbf{r})) + n(\\textbf{r})\\frac{d\\epsilon_{xc}(n)}{dn}\\bigg\\rvert_{n=n(\\textbf{r})}vxc​(r)=ϵxc​(n(r))+n(r)dndϵxc​(n)​​n=n(r)​ where ϵxc(n)\\epsilon_{xc}(n)ϵxc​(n) is obtained for the homogeneous electron gas of density nnn (using Quantum Monte Carlo techniques) and fitted to some analytic form.","s":"Local Density Approximation (LDA)","u":"/espresso/theory/dft","h":"#local-density-approximation-lda","p":160},{"i":172,"t":"These are a family of functionals that depends on the local density and the local gradient of the density: Exc=∫n(r)ϵGGA(n(r),∣∇n(r)∣)drE_{xc} = \\int n(\\textbf{r}) \\epsilon_{GGA}(n(\\textbf{r}), |\\nabla n(\\textbf{r})|) d\\textbf{r}Exc​=∫n(r)ϵGGA​(n(r),∣∇n(r)∣)dr There are many flavor of this functional. There are also more advanced functionals: Meta-GGA (e.g., SCAN), hybrids (e.g., B3LYP), nonlocal functionals for van der Waals forces, Grimme's DFT+D (a semi-empirical correction to GGA). They usually produces more accurate result, but computationally more expensive and sometimes numerically unstable.","s":"Generalized Gradient Approximation (GGA)","u":"/espresso/theory/dft","h":"#generalized-gradient-approximation-gga","p":160},{"i":174,"t":"We can write our Schrödinger in Dirac Bra-Ket notation: H^∣ψ⟩=E∣ψ⟩\\hat{H} \\ket{\\psi} = E\\ket{\\psi}H^∣ψ⟩=E∣ψ⟩ we are going to solve non-interacting single particle Hamiltonian in terms of known basis functions (plane waves) with unknown coefficients. We start with an initial guess for the electron density n(r)n(\\textbf{r})n(r), and construct a pseudo potential for the nuclear potential. In turn, we have the Hamiltonian. Solve for ψi(r)\\psi_i(\\textbf{r})ψi​(r), subsequently n(r)n(\\textbf{r})n(r), and iterate until self consistency is achieved. Self consistency loop in DFT calculation. The above screenshot was taken from lecture slide of Professor Ralph Gevauer from ICTP MAX School 2021. The potential due to the ions is replaced by the pseudo potentials which removes the oscillations near the atomic core (reducing number of required plane wave basis vectors) and simulates the exact behavior elsewhere. The pseudo potential is also different for different exchange correlation functional, and it is specified in the pseudo potential file. If a system had more than one type of atom, always choose the pseudo potentials with same exchange correlation (e.g., PBE). It is important to note that DFT is calculations are not exact solution to the real systems because exact functional (vxcv_{xc}vxc​) we need to solve the Kohn-Sham equation is not known. Therefore, we have to compare the results with experimental observations. The Kohn-Sham wavefunction of orbitals is not an approximation to the exact wavefunction. Rather it is precisely defined property of any electronic system, which is uniquely determined by the density. The in-exactness of DFT results come from the fact that we do not know the exact correlation functional that truly describes real systems.","s":"Algorithmic implementation","u":"/espresso/theory/dft","h":"#algorithmic-implementation","p":160},{"i":176,"t":"The wavefunctions are expanded in terms of a basis set. In quantum espresso, the the basis function is plane waves. There exists other DFT codes that use localized basis function as well. Plane waves are simpler but generally requires much large number of them compared to other localized basis sets. ψi(r)=∑α=1Nbciαfα(r)\\psi_i(\\textbf{r}) = \\sum_{\\alpha = 1} ^{N_b} c_{i\\alpha} f_{\\alpha}(\\textbf{r})ψi​(r)=α=1∑Nb​​ciα​fα​(r) Where NbN_bNb​ is the size basis set. Then the eigenvalue equation becomes: ∑βHαβciβ=ϵiciα\\sum_{\\beta} \\rm{H}_{\\alpha\\beta} c_{i\\beta} = \\epsilon_i c_{i\\alpha}β∑​Hαβ​ciβ​=ϵi​ciα​ ⇒(H11...H1b.........Hb1...Hbb)(c1...cb)=ϵi(c1...cb)\\Rightarrow \\begin{pmatrix} H_{11} & ... & H_{1b} \\\\ ... & ... & ... \\\\ H_{b1} & ... & H_{bb} \\end{pmatrix} \\begin{pmatrix} c_1 \\\\ ... \\\\ c_b \\end{pmatrix} = \\epsilon_i \\begin{pmatrix} c_1 \\\\ ... \\\\ c_b \\end{pmatrix}⇒​H11​...Hb1​​.........​H1b​...Hbb​​​​c1​...cb​​​=ϵi​​c1​...cb​​​ This is a linear algebra problem, solving the above involves diagonalization of (Nb×NbN_b \\times N_bNb​×Nb​) matrix which gives us corresponding eigenvalue and eigenfunction. Apart from plane waves, various localized basis set could be used, e.g., Linear Combination of Atomic Orbitals (LCAO), Gaussian-type Orbitals (GTO), Linearized Muffin-Tin Orbitals (LMTO). Once could also consider mixed basis sets, such as the Linearized Augmented Plane Waves (LAPW). Localized sets are smaller in size, they can be used for both finite and periodic systems, however they are difficult to use/calculate. In case of plane waves, we need larger basis set, and requires periodicity. Need to construct supercell for finite systems. Use of pseudopotential reduces the number of required plane waves.","s":"Plane-wave expansion","u":"/espresso/theory/dft","h":"#plane-wave-expansion","p":160},{"i":178,"t":"Finding the ground state: E[Φ]=⟨Φ∣H^∣Φ⟩⟨Φ∣Φ⟩E[\\Phi] = \\frac{\\braket{\\Phi | \\hat H | \\Phi}}{\\braket{\\Phi|\\Phi}}E[Φ]=⟨Φ∣Φ⟩⟨Φ∣H^∣Φ⟩​ E[Φ]≥E0E[\\Phi] \\ge E_0E[Φ]≥E0​","s":"Variational Principle","u":"/espresso/theory/dft","h":"#variational-principle","p":160},{"i":180,"t":"ψk(r)=eik⋅ruk(r)\\psi_k(r) = e^{i \\textbf{k} \\cdot \\textbf{r}} u_k(\\textbf{r})ψk​(r)=eik⋅ruk​(r) uk(r)=uk(r+R)u_k(\\textbf{r}) = u_k(\\textbf{r} + \\textbf{R})uk​(r)=uk​(r+R) R\\textbf{R}R is lattice vector. Fourier expansion: uk(r)=1Ω∑Gck,GeiG⋅ru_k(\\textbf{r}) = \\frac{1}{\\Omega} \\sum_G c_{\\textbf{k,G}} e^{i \\textbf{G} \\cdot \\textbf{r}}uk​(r)=Ω1​G∑​ck,G​eiG⋅r G\\textbf{G}G is reciprocal lattice vector. ψk(r)=1Ω∑Gck,Gei(k + G)⋅r\\psi_k(\\textbf{r}) = \\frac{1}{\\Omega} \\sum_G c_{\\textbf{k,G}} e^{i (\\textbf{k + G}) \\cdot \\textbf{r}}ψk​(r)=Ω1​G∑​ck,G​ei(k + G)⋅r Contribution from higher Fourier components are small, we can limit the sum at finite ∣k + G∣|\\textbf{k + G}|∣k + G∣ ℏ2∣k + G∣2m≤Ecutoff\\frac{\\hbar^2 |\\textbf{k + G}|}{2m} \\le E_{\\text{cutoff}}2mℏ2∣k + G∣​≤Ecutoff​ The charge density can be obtained from: n(r)=∑kψk∗(r)ψk(r)n(\\textbf{r}) = \\sum_k \\psi_k^*(\\textbf{r}) \\psi_k(\\textbf{r})n(r)=k∑​ψk∗​(r)ψk​(r) We need two sets of basis vectors: one to store the wavefunctions, and another for the charge density. info We need about 4 times the cutoff for the charge density compared to the cutoff for the wavefunction. In case of ultrasoft pseudo potentials, we require a lower cutoff for energy, therefore ecutrho might require 8 or 12 times higher than the ecutwfc.","s":"Bloch theorem","u":"/espresso/theory/dft","h":"#bloch-theorem","p":160},{"i":182,"t":"MIT Course Quantum Espresso Tutorials Introduction to DFT by Paolo Giannozzi http://compmatphys.epotentia.com","s":"Resources","u":"/espresso/theory/dft","h":"#resources","p":160},{"i":184,"t":"Hatree-Fock theory is foundational to many subsequent electronic structure theories. It is an independent particle model or mean filed theory. Consider we have two non-interacting electrons. In that case, the Hamiltonian would be separable, and the total wavefunction Ψ(r1,r2)\\Psi(\\textbf{r}_1, \\textbf{r}_2)Ψ(r1​,r2​) would be product of the individual wave function. Now if we consider two electrons are forming a single system, then there are two issues. (1) We can no longer ignore the electron-electron interaction. (2) The wavefunction describing fermions must be antisymmetric with respect to the interchange of any set of space-spin coordinates. A simple Hartree product fails to satisfy that condition: ΨHP(r1,r2,⋯ ,rN)=ϕ1(r1)ϕ2(r2)⋯ϕN(rN)\\Psi_{HP}(\\textbf{r}_1, \\textbf{r}_2, \\cdots, \\textbf{r}_N) = \\phi_1(\\textbf{r}_1) \\phi_2(\\textbf{r}_2) \\cdots \\phi_N(\\textbf{r}_N)ΨHP​(r1​,r2​,⋯,rN​)=ϕ1​(r1​)ϕ2​(r2​)⋯ϕN​(rN​) In order to satisfy the antisymmetry condition, for our two electron system we can formulate a total wavefunction of the form: Ψ(r1,r2)=12[χ1(r1)χ2(r2)−χ1(r2)χ2(r1)]\\Psi(\\textbf{r}_1, \\textbf{r}_2) = \\frac{1}{\\sqrt{2}} [\\chi_1(\\textbf{r}_1) \\chi_2(\\textbf{r}_2) - \\chi_1(\\textbf{r}_2)\\chi_2(\\textbf{r}_1)]Ψ(r1​,r2​)=2​1​[χ1​(r1​)χ2​(r2​)−χ1​(r2​)χ2​(r1​)]","s":"Hartree-Fock Theory","u":"/espresso/theory/hartree-fock","h":"","p":183},{"i":186,"t":"The above equation can be written as: Ψ(r1,r2)=12∣χ1(r1)χ2(r1)χ1(r2)χ2(r2)∣\\Psi(\\textbf{r}_1, \\textbf{r}_2) = \\frac{1}{\\sqrt{2}} \\begin{vmatrix} \\chi_1(\\textbf{r}_1) & \\chi_2(\\textbf{r}_1) \\\\ \\chi_1(\\textbf{r}_2) & \\chi_2(\\textbf{r}_2) \\end{vmatrix}Ψ(r1​,r2​)=2​1​​χ1​(r1​)χ1​(r2​)​χ2​(r1​)χ2​(r2​)​​ Now what happens if we have more than two electrons? We can generalize the above determinant form to NNN electrons: Ψ=1N!∣χ1(r1)χ2(r1)⋯χN(r1)χ1(r2)χ2(r2)⋯χN(r2)⋮⋮⋱⋮χ1(rN)χ2(rN)⋯χN(rN)∣\\Psi = \\frac{1}{\\sqrt{N!}} \\begin{vmatrix} \\chi_1(\\textbf{r}_1) & \\chi_2(\\textbf{r}_1) & \\cdots & \\chi_N(\\textbf{r}_1) \\\\ \\chi_1(\\textbf{r}_2) & \\chi_2(\\textbf{r}_2) & \\cdots & \\chi_N(\\textbf{r}_2) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\chi_1(\\textbf{r}_N) & \\chi_2(\\textbf{r}_N) & \\cdots & \\chi_N(\\textbf{r}_N) \\end{vmatrix}Ψ=N!​1​​χ1​(r1​)χ1​(r2​)⋮χ1​(rN​)​χ2​(r1​)χ2​(r2​)⋮χ2​(rN​)​⋯⋯⋱⋯​χN​(r1​)χN​(r2​)⋮χN​(rN​)​​ The above antisymmetrized product can describe electrons that move independently of each other while they experience an average (mean-field) Coulomb force.","s":"Slater determinant","u":"/espresso/theory/hartree-fock","h":"#slater-determinant","p":183},{"i":188,"t":"http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html","s":"Resources","u":"/espresso/theory/hartree-fock","h":"#resources","p":183},{"i":190,"t":"We want to calculate the electronic structure of real materials and their physical properties by ab-initio method. Electrons are microscopic particle, hence their dynamics is governed by the laws of quantum mechanics. Quantum particles are described by the wave function. λ⋅p=h\\lambda \\cdot p = hλ⋅p=h where hhh is the Plank constant. The Wavefunction of an electron in a potential filed (V)(V)(V) is calculated by solving the Schrödinger equation: −ℏ22m∇2Ψ(r,t)+V(r,t)=iℏ∂Ψ(r,t)∂t-\\frac{\\hbar^2}{2m} \\nabla^2 \\Psi(\\textbf{r}, t) + V(\\textbf{r}, t) = i\\hbar \\frac{\\partial\\Psi(\\textbf{r}, t)}{\\partial t}−2mℏ2​∇2Ψ(r,t)+V(r,t)=iℏ∂t∂Ψ(r,t)​ Fortunately, in most practical purposes, the potential field is not a function of time (t)(t)(t), or even if it is a function of time, they changes relatively slowly compared to the dynamics we are interested in. For example, the electrons inside a material are subjected to the Coulomb field of the nucleus. The nucleus is heavy and their motion is much slower than the motion of the electrons. In such situation, we can separate out the spatial and temporal parts of the wave function: Ψ(r,t)=ψ(r)f(t)\\Psi(\\textbf{r}, t) = \\psi(\\textbf{r}) f(t)Ψ(r,t)=ψ(r)f(t) That reduces our task to solving only time independent Schrödinger equation: [−ℏ2∇22m+v(r)]ψ(r)=ϵψ(r)\\left[-\\frac{\\hbar^2 \\nabla^2}{2m} + v(\\textbf{r})\\right] \\psi(\\textbf{r}) = \\epsilon \\psi(\\textbf{r})[−2mℏ2∇2​+v(r)]ψ(r)=ϵψ(r) Once we have the wavefunction, we can calculate the observables by taking the expectation values. ⟨ψi∣ψj⟩=δij\\braket{\\psi_i | \\psi_j} = \\delta_{ij}⟨ψi​∣ψj​⟩=δij​ ⟨ψi∣H^∣ψi⟩=ϵi\\braket{\\psi_i | \\hat{H} | \\psi_i} = \\epsilon_i⟨ψi​∣H^∣ψi​⟩=ϵi​ However, the challenge is to solve the Schrödinger equation as a real physical system is consists of a large number of atoms. The Schrödinger equation becomes coupled many-body equation. [−ℏ2m∑i=1N∇i2+∑i=1NV(ri)+∑i=1N∑j<iU(ri,rj)]ψ(r1,r2,...,rN)\\left[-\\frac{\\hbar}{2m} \\sum_{i=1}^N \\nabla_i^2 + \\sum_{i=1}^NV(\\textbf{r}_i) + \\sum_{i=1}^N \\sum_{j<i}U(\\textbf{r}_i, \\textbf{r}_j)\\right]\\psi(\\textbf{r}_1, \\textbf{r}_2, ..., \\textbf{r}_N)[−2mℏ​i=1∑N​∇i2​+i=1∑N​V(ri​)+i=1∑N​j<i∑​U(ri​,rj​)]ψ(r1​,r2​,...,rN​) =Eψ(r1,r2,...,rN)= E\\psi(\\textbf{r}_1, \\textbf{r}_2, ..., \\textbf{r}_N)=Eψ(r1​,r2​,...,rN​) With today's available computing power, it is far from feasible to solve the actual electronic wavefunction of a condensed matter system, where NNN is of the order of 102310^{23}1023.","s":"What problem are we trying to solve?","u":"/espresso/theory/problem-statement","h":"","p":189},{"i":193,"t":"Wannier functions are an alternative representation of Bloch states in terms of a localized basis set. Suppose we have NNN atoms each separated by lattice constant aaa in one dimension. Bloch states are indexed by the wave vector kkk. H∣k⟩=ϵk∣k⟩\\mathcal{H}\\ket{k} = \\epsilon_k \\ket{k}H∣k⟩=ϵk​∣k⟩ ⟨x∣k⟩=ψk(x)\\braket{x | k} = \\psi_k(x)⟨x∣k⟩=ψk​(x) From the Bloch theorem, we have ψk(x+a)=eikaψk(x)\\psi_k(x + a) = e^{ika}\\psi_k(x)ψk​(x+a)=eikaψk​(x) Now, we want to find an alternative representation in terms of Wannier basis ∣n⟩\\ket{n}∣n⟩, where the states are labeled using site index (n = 1, 2, ..., N) instead of quantum number kkk. Wannier basis set is complete and orthonormal. ∣k⟩=∑n=1Nank∣n⟩\\ket{k} = \\sum_{n=1}^N a_{nk} \\ket{n}∣k⟩=n=1∑N​ank​∣n⟩ ⟨x∣k⟩=∑nank⟨x∣n⟩\\braket{x | k} = \\sum_n a_{nk} \\braket{x | n}⟨x∣k⟩=n∑​ank​⟨x∣n⟩ ⇒ψk(x)=∑nankw(x−na)\\Rightarrow \\psi_k(x) = \\sum_n a_{nk} w(x - na)⇒ψk​(x)=n∑​ank​w(x−na) where www represents Wannier function. Apply translation operator on both sides: Taψk(x)=∑nankTaw(x−na)⇒ ψk(x+a)=∑nankw(x−(n−1)a)⇒ eikaψk(x)=∑na(n+1)kw(x+na)⇒ eika∑nankw(x−na)=∑na(n+1)kw(x+na)⇒ eika∑nankw(x−na)=∑na(n+1)kw(x+na)\\begin{aligned} &T_a \\psi_k(x) = \\sum_n a_{nk} T_a w(x - na) \\\\ \\Rightarrow~ & \\psi_k(x + a) = \\sum_n a_{nk} w\\bigl(x - (n-1)a\\bigr) \\\\ \\Rightarrow~ & e^{ika} \\psi_k(x) = \\sum_n a_{(n+1)k} w(x + na) \\\\ \\Rightarrow~ & e^{ika} \\sum_n a_{nk} w(x - na) = \\sum_n a_{(n+1)k} w(x + na) \\\\ \\Rightarrow~ & e^{ika} \\sum_n a_{nk} w(x - na) = \\sum_n a_{(n+1)k} w(x + na) \\\\ \\end{aligned}⇒ ⇒ ⇒ ⇒ ​Ta​ψk​(x)=n∑​ank​Ta​w(x−na)ψk​(x+a)=n∑​ank​w(x−(n−1)a)eikaψk​(x)=n∑​a(n+1)k​w(x+na)eikan∑​ank​w(x−na)=n∑​a(n+1)k​w(x+na)eikan∑​ank​w(x−na)=n∑​a(n+1)k​w(x+na)​ ∴ a(n+1)k=ankeika=a(n−1)kei2ka⋯=a0kei(n+1)ka\\begin{aligned} \\therefore ~ a_{(n+1)k} &= a_{nk} e^{ika} \\\\ &= a_{(n-1)k}e^{i2ka} \\\\ &\\qquad\\cdots \\\\ &= a_{0 k}e^{i(n+1)ka} \\end{aligned}∴ a(n+1)k​​=ank​eika=a(n−1)k​ei2ka⋯=a0k​ei(n+1)ka​ Since Bloch states are orthonormal: ⟨k∣k⟩=⟨k∣∑nank∣n⟩=∑mn⟨m∣amk∗ank∣n⟩=∑n∣ank∣2=Na0k2=1\\begin{aligned} \\braket{k | k} &= \\braket{k | \\sum_n a_{nk} | n} \\\\ &= \\sum_{mn} \\braket{m | a_{mk}^* a_{nk} | n} \\\\ &= \\sum_{n} | a_{nk} |^2 \\\\ &= N a_{0k}^2 \\\\ &= 1 \\end{aligned}⟨k∣k⟩​=⟨k∣n∑​ank​∣n⟩=mn∑​⟨m∣amk∗​ank​∣n⟩=n∑​∣ank​∣2=Na0k2​=1​ ⇒a0k=1N(up to a phase factor)\\Rightarrow a_{0k} = \\frac{1}{\\sqrt{N}} \\qquad\\text{(up to a phase factor)}⇒a0k​=N​1​(up to a phase factor) ∣k⟩=1N∑neikna∣n⟩\\ket{k} = \\frac{1}{\\sqrt{N}} \\sum_n e^{ikna} \\ket{n}∣k⟩=N​1​n∑​eikna∣n⟩ While Bloch states represent the eigenstates of the single-particle Hamiltonian, WF (in general) cannot be assigned a single eigen-value, instead WFs are obtained as liner combination of Bloch states corresponding to different energies. The Hamiltonian can now be written as: H=∑m∑n∣m⟩⟨m∣H∣n⟩⟨n∣=∑nϵn∣n⟩⟨n∣+∑m≠n(−tmn)∣m⟩⟨n∣\\begin{aligned} \\mathcal{H} &= \\sum_m \\sum_n \\ket{m} \\braket{m | \\mathcal{H} | n} \\bra{n} \\\\ &= \\sum_n \\epsilon_n \\ket{n} \\bra{n} + \\sum_{m\\neq n} (-t_{mn}) \\ket{m} \\bra{n} \\end{aligned}H​=m∑​n∑​∣m⟩⟨m∣H∣n⟩⟨n∣=n∑​ϵn​∣n⟩⟨n∣+m=n∑​(−tmn​)∣m⟩⟨n∣​ where ϵn\\epsilon_nϵn​ is onsite or diagonal term and tmnt_{mn}tmn​ (> 0) is hopping or off-diagonal term.","s":"Introduction","u":"/espresso/theory/wannier","h":"#introduction","p":191},{"i":195,"t":"The choice of Wannier function is not unique. One such option could be the set that maximizes localization. Two different sets of Wannier basis are connected via unitary transformation. MLWFs can be considered as a generalization of localized molecular orbitals (LMOs) to periodic systems.","s":"Maximally Localized Wannier Function","u":"/espresso/theory/wannier","h":"#maximally-localized-wannier-function","p":191},{"i":197,"t":"Introduction to Wannier Basis lecture by Vijay A. Singh Maximally localized generalized Wannier functions for composite energy bands, Marzari and Vanderbilt, Phys. Rev. B 56, 12847 (1997) Maximally localized Wannier functions for entangled energy bands, Souza, Marzari and Vanderbilt, Phys. Rev. B 65, 035109 (2001) Maximally localized Wannier functions: Theory and applications, Marzari et. al., Rev. Mod. Phys. 84, 1419 (2012) Introduction to Maximally Localized Wannier Functions, Ambrosetti and Silvestrelli, Reviews in Computational Chemistry, Ch. 6, pp. 327 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