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# CRPA of SrVO3

The following tutorial describes how to perform CRPA calculations, which is available as of VASP 6.

Calculation of the Coulomb matrix elements ${\displaystyle U_{{ijkl}}(\omega =0)}$ in the constrained Random Phase Approximation (CRPA) of SrVO3 between the Vanadium t2g 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


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.

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:

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:

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):

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:

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

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 ${\displaystyle \omega =0}$ can be found in the OUTCAR:

spin components:  1  1, frequency:    0.0000    0.0000

screened Coulomb repulsion U_iijj between MLWFs:
1         2         3
1    3.3746    2.3385    2.3385
2    2.3385    3.3746    2.3385
3    2.3385    2.3385    3.3746

screened Coulomb repulsion U_ijji between MLWFs:
1         2         3
1    3.3746    0.4429    0.4429
2    0.4429    3.3746    0.4429
3    0.4429    0.4429    3.3746

screened Coulomb repulsion U_ijij between MLWFs:
1         2         3
1    3.3746    0.4429    0.4429
2    0.4429    3.3746    0.4429
3    0.4429    0.4429    3.3746

averaged interaction parameter
screened Hubbard U =    3.3746   -0.0000
screened Hubbard u =    2.3385    0.0000
screened Hubbard J =    0.4429   -0.0000

N.B.: The frequency point ${\displaystyle \omega }$ can be set by OMEGAMAX in the INCAR. For instance to evaluate the CRPA interaction matrix at ${\displaystyle \omega =10}$ 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 ${\displaystyle \omega =i10}$ . This can be used to evaluate ${\displaystyle U}$ at a specific Matsubara frequency point.

In addition, the bare Coulomb interaction matrix is calculated for a high VCUTOFF and low energy cutoff ENCUTGW and written in that order to the OUTCAR file. Look for the lines similar to:

spin components:  1  1

bare Coulomb repulsion V_iijj between MLWFs:
1         2         3
1   16.3485   15.0984   15.0984
2   15.0984   16.3485   15.0984
3   15.0984   15.0984   16.3485

bare Coulomb repulsion V_ijji between MLWFs:
1         2         3
1   16.3485    0.5351    0.5351
2    0.5351   16.3485    0.5351
3    0.5351    0.5351   16.3485

bare Coulomb repulsion V_ijij between MLWFs:
1         2         3
1   16.3485    0.5351    0.5351
2    0.5351   16.3485    0.5351
3    0.5351    0.5351   16.3485

averaged bare interaction
bare Hubbard U =   16.3485   -0.0000
bare Hubbard u =   15.0984   -0.0000
bare Hubbard J =    0.5351    0.0000


### CRPA calculation with spacetime Algorithm

Note that the same frequency grid is used as for ALGO=RPA (RPA correlation energy calculation) and can not be changed directly. To calculate the CRPA interaction for a set of automatically chosen imaginary frequency points use once again the PBE wavefunction as input

cp WAVECAR.DIAG WAVECAR
cp WAVEDER.DIAG WAVEDER


Currently, 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:

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


After a successful run, the interaction values are written to the OUTCAR file. Here is the result for the first frequency point:

spin components:  1  1, frequency:    0.0000    0.2248

screened Coulomb repulsion U_iijj between MLWFs:
1         2         3
1    3.3798    2.3436    2.3436
2    2.3436    3.3798    2.3436
3    2.3436    2.3436    3.3798

screened Coulomb repulsion U_ijji between MLWFs:
1         2         3
1    3.3798    0.4433    0.4433
2    0.4433    3.3798    0.4433
3    0.4433    0.4433    3.3798

screened Coulomb repulsion U_ijij between MLWFs:
1         2         3
1    3.3798    0.4433    0.4433
2    0.4433    3.3798    0.4433
3    0.4433    0.4433    3.3798

averaged interaction parameter
screened Hubbard U =    3.3798   -0.0000
screened Hubbard u =    2.3436    0.0000
screened Hubbard J =    0.4433   -0.0000

N.B.: The number of frequency points should be large enough to guarantee for an accurate Fourier transformation of the CRPA polarizability matrix from the imaginary time to imaginary frequency domain. See ACFDT/RPA calculations for more information.