CRPA of SrVO3

Overview > bandgap of Si in GW > bandstructure of Si in GW (VASP2WANNIER90) > bandstructure of SrVO3 in GW > CRPA of SrVO3 > Equilibrium volume of Si in the RPA > List of tutorials

Task

Calculation of the Coulomb matrix elements $U_{ijkl}(\omega=0)$ in the constrained Random Phase Approximation (CRPA) of SrVO₃ between the Vanadium t_2g states.

Performing a CRPA calculation with VASP is a 3-step procedure: a DFT groundstate calculation, a calculation to obtain a number of virtual orbitals, and the actual CRPA calculation itself.

N.B.: This example involves quite a number of individual calculations. The easiest way to run this example is to execute:

./doall.sh

And compare the output of the different steps (DFT, GW, HSE) by:

./plotall.sh

In any case, one can consider the doall.sh script to be an overview of the steps described below.

DFT groundstate calculation

The first step is a conventional DFT (in this case PBE) groundstate calculation.

INCAR (see INCAR.DFT)

SYSTEM  = SrVO3                        # system name
NBANDS = 36                            # small number  of bands
ISMEAR = 0                             # Gaussian smearing
EDIFF = 1E-8                           # high precision for groundstate calculation
KPAR = 2                               # parallelization of k-points in two groups

Copy the aforementioned file to INCAR:

cp INCAR.DFT INCAR

The POSCAR file describes the structure of the system:

POSCAR

SrVO3
3.84652  #cubic fit for 6x6x6 k-points
 +1.0000000000  +0.0000000000  +0.0000000000 
 +0.0000000000  +1.0000000000  +0.0000000000 
 +0.0000000000  +0.0000000000  +1.0000000000 
Sr V O
 1 1 3
Direct
 +0.0000000000  +0.0000000000  +0.0000000000 
 +0.5000000000  +0.5000000000  +0.5000000000 
 +0.5000000000  +0.5000000000  +0.0000000000 
 +0.5000000000  +0.0000000000  +0.5000000000 
 +0.0000000000  +0.5000000000  +0.5000000000

This file remains unchanged in the following.

The KPOINTS file describes how the first Brillouin zone is sampled. In the first step we use a uniform k-point sampling:

KPOINTS

Automatically generated mesh
       0
Gamma
 4 4 4
 0 0 0

Mind: this is definitely not dense enough for a high-quality description of SrVO₃, but in the interest of speed we will live with it.

Run VASP. If all went well, one should obtain a WAVECAR file containing the PBE wavefunction.

Obtain DFT virtual orbitals and long-wave limit

Use following INCAR file to increase the number of virtual states and to determine the long-wave limit of the polarizability (stored in WAVEDER):

INCAR (see INCAR.DIAG)

SYSTEM = SrVO3                         # system name
ISMEAR = 0                             # Gaussian smearing
KPAR = 2                               # parallelization of k-points in two groups
ALGO = Exact                           # exact diagonalization
NELM = 1                               # one electronic step suffices, since WAVECAR from previous step is present
NBANDS = 96                            # need for a lot of bands in GW
LOPTICS = .TRUE.                       # we need d phi/ d k  for GW calculations for long-wave limit

Restart VASP. At this stage it is a good idea to make a safety copy of the WAVECAR and WAVEDER files since we will repeatedly need them in the calculations that follow:

cp WAVECAR WAVECAR.DIAG
cp WAVEDER WAVEDER.DIAG

CRPA Calculation

Calculate the CRPA interaction parameters for the t2g states by using the PBE wavefunction as input

cp WAVECAR.DIAG WAVECAR
cp WAVEDER.DIAG WAVEDER

Use following Wannier projection for the basis:

wannier90.win (see wannier90.win.CRPA)

num_wann =    3
num_bands=   96

# PBE energy window of t2g states (band 21-23)
dis_win_min = 6.4
dis_win_max = 9.0

begin projections
 V:dxy;dxz;dyz
end projections

Copy this file to wannier90.win

cp wannier90.win.CRPA wannier90.win

And use following input file as

INCAR (see INCAR.CRPA)

SYSTEM = SrVO3                         # system name
ISMEAR = 0                             # Gaussian smearing
NCSHMEM = 1                            # switch off shared memory for chi
ALGO = CRPA                            # Switch on CRPA
NBANDS = 96                            # CRPA needs many empty states
PRECFOCK = Fast                        # fast mode for FFTs
NTARGET_STATES = 1 2 3                 # exclude wannier states 1 - 3 in screening
LWRITE_WANPROJ = .TRUE.                # write wannier projection file

and run VASP. The CRPA interaction values for $\omega=0$ can be found in the OUTCAR file after following lines

screened Coulomb repulsion U_iijj between MLWFs:

including an averaged value:

screened Hubbard U =    3.3746   -0.0000

Make a copy of the output file

cp OUTCAR OUTCAR.CRPA

N.B.: The frequency point

\omega

can be set by OMEGAMAX in the INCAR. For instance to evaluate the CRPA interaction matrix at

\omega=10

eV, add

 OMEGAMAX = 10

to the INCAR and restart VASP. In contrast, adding following two lines to the INCAR

 OMEGAMAX = 10 
 NOMEGAR = 0

tells VASP to calculate the interaction on the imaginary frequency axis at $\omega=i 10$ . This can be used to evaluate $U$ at a specific Matsubara frequency point.

CRPA calculation on full imaginary frequency axis (optional)

To calculate the CRPA interaction for a set of imaginary frequency points use once again the PBE wavefunction as input

cp WAVECAR.DIAG WAVECAR
cp WAVEDER.DIAG WAVEDER

This step requires uses the WANPROJ file from previous step, no wannier90.win file is necessary.

Select the space-time CRPA algorithm with following file:

INCAR (see INCAR.CRPAR)

SYSTEM = SrVO3                         # system name
ISMEAR = 0                             # Gaussian smearing
NCSHMEM = 1                            # switch off shared memory for chi
ALGO = CRPAR                           # Switch on CRPA on imaginary axis
NBANDS = 96                            # CRPA needs many empty states
PRECFOCK = Fast                        # fast mode for FFTs
NTARGET_STATES = 1 2 3                 # exclude wannier states 1 - 3 in screening
NCRPA_BANDS = 21 22 23                 # remove bands 21-23 in screening, currently required for space-time algo
NOMEGA = 12                            # use 12 imaginary frequency points
NTAUPAR = 4                            # distribute 12 time points into 4 groups

Run VASP and make a copy of the output file

cp OUTCAR OUTCAR.CRPAR

The resulting interactions are written for every imaginary frequency point to the OUTCAR file. For instance, to extract the averaged on-site U interaction for each point enter following command

grep "screened Hubbard U" OUTCAR

resulting in following output

screened Hubbard U =    3.3798   -0.0000
screened Hubbard U =    3.4172   -0.0000
screened Hubbard U =    3.5169   -0.0000
screened Hubbard U =    3.7418   -0.0000
screened Hubbard U =    4.2069   -0.0000
screened Hubbard U =    5.0802   -0.0000
screened Hubbard U =    6.5456   -0.0000
screened Hubbard U =    8.6426   -0.0000
screened Hubbard U =   11.0815   -0.0000
screened Hubbard U =   13.3615   -0.0000
screened Hubbard U =   15.0636   -0.0000
screened Hubbard U =   16.0412   -0.0000

Here each line corresponds to the averaged Hubbard interaction of the imaginary frequency points listed in the OUTCAR:

 imag. energies (cos points) w= 
   0.225    0.722    1.376    2.338    3.849    6.312   10.422   17.424
  29.656   51.949   96.989  223.873

Note that first interaction value is roughly the same as the one obtained in previous step for $\omega=0$ , while the last entry at a high imaginary frequency point of $\omega=i 223.873$ approaches the bare Coulomb interaction:

bare Hubbard U =   16.3485    0.0000

Overview > bandgap of Si in GW > bandstructure of Si in GW (VASP2WANNIER90) > bandstructure of SrVO3 in GW > CRPA of SrVO3 > Equilibrium volume of Si in the RPA > List of tutorials

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