This tutorial can be downloaded link.

Intro Tutorial 9: GW Calculations with Hybrid Functionals

This tutorial describes how to carry out \(G_0W_0\) calculations with hybrid exchange-correlation functionals. The general steps are the same as those needed for \(G_0W_0\) calculations with semi-local functionals, introduced in Tutorial 1.

Exact exchange operator without approximation

We first perform the mean-field electronic structure calculation within DFT using the Quantum ESPRESSO code. Download the following files to your working directory:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/exx/pw.in
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/H_ONCV_PBE-1.2.upf
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/Si_ONCV_PBE-1.2.upf

Let’s inspect the pw.in file, input for pw.x.

[1]:

%%bash
cat pw.in

&control
calculation  = 'scf'
restart_mode = 'from_scratch'
pseudo_dir   = './'
outdir       = './'
prefix       = 'silane'
wf_collect   = .TRUE.
/
&system
ibrav           = 1
celldm(1)       = 20
nat             = 5
ntyp            = 2
ecutwfc         = 25.0
nbnd            = 10
assume_isolated = 'mp'
input_dft       = 'pbe0'
/
&electrons
diago_full_acc = .TRUE.
/
ATOMIC_SPECIES
Si 28.0855  Si_ONCV_PBE-1.2.upf
H  1.00794   H_ONCV_PBE-1.2.upf
ATOMIC_POSITIONS bohr
Si      10.000000   10.000000  10.000000
H       11.614581   11.614581  11.614581
H        8.385418    8.385418  11.614581
H        8.385418   11.614581   8.385418
H       11.614581    8.385418   8.385418
K_POINTS {gamma}

Compared to the pw.in file used in Tutorial 1, here we only have one additional keyword, input_dft = 'pbe0', which instructs the code to use the PBE0 hybrid functional.

We run pw.x on 2 cores.

[ ]:

%%bash
mpirun -n 2 pw.x -i pw.in > pw.out

The static dielectric screening is computed using the projective dielectric eigendecomposition (PDEP) technique. Download the following file to your working directory:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/exx/wstat.in

Let’s inspect the wstat.in file.

[2]:

%%bash
cat wstat.in

input_west:
    qe_prefix: silane
    west_prefix: silane
    outdir: ./

wstat_control:
    wstat_calculation: S
    n_pdep_eigen: 50

It’s identical to the wstat.in file used in Tutorial 1.

We run wstat.x on 2 cores.

[ ]:

%%bash
mpirun -n 2 wstat.x -i wstat.in > wstat.out

The GW electronic structure is computed by treating the frequency integration of the correlation part of the self-energy with the contour deformation technique and by computing the dielectric screening at multipole frequencies with Lanczos iterations. Download the following file to your working directory:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/exx/wfreq.in

Let’s inspect the wfreq.in file.

[3]:

%%bash
cat wfreq.in

input_west:
    qe_prefix: silane
    west_prefix: silane
    outdir: ./

wstat_control:
    wstat_calculation: S
    n_pdep_eigen: 50

wfreq_control:
    wfreq_calculation: XWGQ
    n_pdep_eigen_to_use: 50
    qp_bandrange: [1,5]
    n_refreq: 300
    ecut_refreq: 2.0

It’s identical to the wfreq.in file used in Tutorial 1.

We run wfreq.x on 2 cores.

[ ]:

%%bash
mpirun -n 2 wfreq.x -i wfreq.in > wfreq.out

We save the output file silane.wfreq.save/wfreq.json.

[ ]:

%%bash
mv silane.wfreq.save/wfreq.json wfreq_exx.json

If the reader does NOT have the computational resources to run the calculations, the output file can be directly downloaded as:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/exx/wfreq_exx.json

Adaptively compressed exchange (ACE) operator

Now let’s use the adaptively compressed exchange (ACE) operator to speed up the calculations. Download the following files to your working directory:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/exx/pw.in
wget -N -q https://west-code.org/doc/training/exx/nscf.in
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/H_ONCV_PBE-1.2.upf
wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/Si_ONCV_PBE-1.2.upf

We run pw.x on 2 cores, and make a copy of the save directory generated by pw.x.

[ ]:

%%bash
mpirun -n 2 pw.x -i pw.in > pw.out
cp -r silane.save silane_copy.save

We run a non-self-consistent calculation to construct the ACE operator with a bit more empty bands. The content of nscf.in is almost identical to that of pw.in, except that in nscf.in we request a non-self-consistent calculation with a larger value of nbnd.

We run pw.x again on 2 cores, and copy the file that stores the ACE operator to the directory saved in the previous self-consistent pw.x run.

[ ]:

%%bash
mpirun -n 2 pw.x -i nscf.in > nscf.out
cp silane.save/ace* silane_copy.save
rm -rf silane.save
mv silane_copy.save silane.save

By doing this, in the silane.save directory, we have the wavefunctions generated in the first, self-consistent pw.x run and the ACE operator generated in the second, non-self-consistent pw.x run. Now we do the GW calculation. Download the following file to your working directory:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/exx/wstat_ace.in
wget -N -q https://west-code.org/doc/training/exx/wfreq_ace.in

Let’s inspect the wstat_ace.in and wfreq_ace.in files.

[4]:

%%bash
cat wstat_ace.in
cat wfreq_ace.in

input_west:
    qe_prefix: silane
    west_prefix: silane
    outdir: ./

wstat_control:
    wstat_calculation: S
    n_pdep_eigen: 50
    n_exx_lowrank: 30
input_west:
    qe_prefix: silane
    west_prefix: silane
    outdir: ./

wstat_control:
    wstat_calculation: S
    n_pdep_eigen: 50
    n_exx_lowrank: 30

wfreq_control:
    wfreq_calculation: XWGQ
    n_pdep_eigen_to_use: 50
    qp_bandrange: [1,5]
    n_refreq: 300
    ecut_refreq: 2.0

The new keyword, n_exx_lowrank: 30, activates the use of the ACE operator constructed from 30 KS wavefunctions. In production runs, n_exx_lowrank and therefore nbnd used on the non-self-consistent pw.x run should be converged.

We run wstat.x and wfreq.x on 2 cores.

[ ]:

%%bash
mpirun -n 2 wstat.x -i wstat_ace.in > wstat_ace.out
mpirun -n 2 wfreq.x -i wfreq_ace.in > wfreq_ace.out

We save the output file silane.wfreq.save/wfreq.json.

[ ]:

%%bash
mv silane.wfreq.save/wfreq.json wfreq_ace.json

If the reader does NOT have the computational resources to run the calculations, the output file can be directly downloaded as:

[ ]:

%%bash
wget -N -q https://west-code.org/doc/training/ace/wfreq_ace.json

We compare the quasiparticle corrections obtained with and without using the ACE operator.

[5]:

import json
import numpy as np

def read_qp(filename):
    with open(filename) as f:
        data = json.load(f)

    return np.array(data['output']['Q']['K000001']['eqpSec'],dtype='f8')

qp_exx = read_qp('wfreq_exx.json')
qp_ace = read_qp('wfreq_ace.json')

print('QP energy [eV]')
print('State      EXX      ACE     Diff')
for i in range(5):
    print(f'    {i+1}  {qp_exx[i]:7.3f}  {qp_ace[i]:7.3f}  {qp_ace[i]-qp_exx[i]:7.3f}')

QP energy [eV]
State      EXX      ACE     Diff
    1  -17.363  -17.366   -0.003
    2  -12.365  -12.359    0.006
    3  -12.365  -12.359    0.006
    4  -12.367  -12.362    0.005
    5    0.383    0.380   -0.003

In this case, the difference between the EXX and ACE results is at most 6 meV.

In addition to \(G_0W_0\) calculations, the ACE operator can also be used in Bethe-Salpeter equation (BSE), quantum defect embedding theory (QDET), and time-dependent density functional theory (TDDFT) calculations with hybrid functionals.