GW band structure (Si)

Example to compute GW band structure

Here we provide a simple example on how to run Green/Weakcoupling on the example of a periodic silicon on 6x6x6 lattice. Prerequisites: you must have both the mbpt and the ac executables compiled, and the green-mbtools python package installed. Change to a new directory where you will keep your simulations, create a directory for the Si simulation, and create the file a.dat containing the following unit cell information:

0.0,  2.7155, 2.7155
2.7155, 0.0,  2.7155
2.7155, 2.7155, 0.0

Then create a file with the atom positions in the atom.dat:

Si 0.0  0.0  0.0
Si 1.35775 1.35775 1.35775

The remaining parameters will be specified on the command line.

We then obtain input parameters and the initial mean-field solution by running pySCF via the script:

python <source root>/green-mbpt/python/        \
  --a a.dat --atom atom.dat --nk 6                            \
  --basis gth-dzvp-molopt-sr --pseudo gth-pbe --xc PBE

Here we use the gth-dzvp-molopt-sr basis with the gth-pbe psudopotential and run DFT mean-field approximation with a PBE exchange correlation potential.

The primary goal of this example is to generate the band structure of Silicon. For that, we require the overlap matrix and one-electron Hamiltonian along the high-symmetry path in addition to the density functional solution. We set the high-symmetry path to WGXWLG to reproduce the results from [Phys. Rev. B 106, 235104]. To calculate these quantities, we run the script again, but this time with an additional --job sym_path flag:

python <source root>/green-mbpt/python/        \
  --a a.dat --atom atom.dat --nk 6                            \
  --basis gth-dzvp-molopt-sr --pseudo gth-pbe --xc PBE        \
  --job sym_path --high_symmetry_path WGXWLG  --high_symmetry_path_points 100

Note that it is possible to run both jobs at once using the flag --job init sym_path.

After that we will run the GW approximation

We then store bulk results into Si.h5 file. After the self-consistent simulation is finished band-path interpolation along the high-symmetry path will be evaluated and stored in Si_hs.h5.

To obtain the spectral function for silicon run

<install dir>/bin/ac.exe  --BETA 100 --grid_file ir/1e4.h5    \
   --input_file Si_hs.h5 --output_file ac.h5 --group G_tau_hs \
   --e_min -5.0 --e_max 5.0 --n_omega 4000 --eta 0.01 \
   --kind Nevanlinna

This will run Nevanlinna analytical continuation for the data obtained at the last iteration for all k-points. The output will be stored in the group G_tau_hs of the output HDF5 file ac.h5.

A plot for the band structure can then be obtained with the script:

python <source root>/green-mbpt/python/ \
        --input_file ac.h5 --output_dir bands

This will read the analytically continued data and plot it to an <output_dir>/bands.png file. In addition, it will create a plain-text data file for every k-point along the chosen path inside the <output_dir> directory. The expected band structure plot should look as following

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