CRPA of SrVO3

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

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.

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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.

• INCAR (see INCAR.DFT)

SYSTEM  = SrVO3    # system name
NBANDS = 36        # small number of bands
ISMEAR = -1        # Fermi smearing
SIGMA = 0.1        # electronic temperature in eV (1eV ~ 11604K)
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.PBE)

SYSTEM = SrVO3          # system name
ISMEAR = -1             # Fermi smearing
SIGMA = 0.1             # electronic temperature in eV (1eV ~ 11604K)
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
LFINITE_TEMPERATURE = T # compute all optical matrix elements (only required for CRPAR)

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.PBE
cp WAVEDER WAVEDER.PBE

cRPA Calculation

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

cp WAVECAR.PBE WAVECAR
cp WAVEDER.PBE WAVEDER

And use following input file as

• INCAR (see INCAR.CRPA) and run vasp

SYSTEM = SrVO3            # system name
ISMEAR = -1               # Fermi smearing
SIGMA = 0.1               # electronic temperature in eV (1eV ~ 11604K)
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
#LFINITE_TEMPERATURE = T  # T>0 formalism can be avoided here, because U computed only at omega=0

Warning: As of version 6.2.0 run this tutorial successfully by adding following lines to INCAR. For older version copy wannier90.win.CRPA to wannier90.win and omit following lines in INCAR.

NUM_WANN = 3 
WANNIER90_WIN = "
num_bands=   96

# because bands 21, 22, 23 do not cross with other bands
# one can exclude all other bands in wannierization
# and omit the definition of an energy window like so
exclude_bands = 1-20, 24-96

begin projections
 V:dxy;dxz;dyz
end projections
"

The cRPA interaction values for $\omega=0$ 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    3.3459    2.3455    2.3455
   2    2.3455    3.3459    2.3455
   3    2.3455    2.3455    3.3459

screened Coulomb repulsion U_ijji between MLWFs:
        1         2         3
   1    3.3459    0.4281    0.4281
   2    0.4281    3.3459    0.4281
   3    0.4281    0.4281    3.3459

screened Coulomb repulsion U_ijij between MLWFs:
        1         2         3
   1    3.3459    0.4281    0.4281
   2    0.4281    3.3459    0.4281
   3    0.4281    0.4281    3.3459

averaged interaction parameter
screened Hubbard U =    3.3459    0.0000
screened Hubbard u =    2.3455    0.0000
screened Hubbard J =    0.4281   -0.0000

The full interaction matrix is written to UIJKL.

Mind: 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.

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

Similar to the effectively screened interaction the full output is written to VIJKL.

cRPA calculation on Matsubara axis

Mind: Available as of VASP.6.5.2.

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.PBE WAVECAR
cp WAVEDER.PBE WAVEDER

Currently, this step requires the WANPROJ file from previous step, no wannier90.win file is necessary. You can also, delete WANPROJ and define a wannier projection as in the previous step in the INCAR.

Select the space-time cRPA algorithm with following file:

• INCAR (see INCAR.CRPAR)

SYSTEM = SrVO3             # system name
LFINITE_TEMPERATURE = T    # use finite temperature formalism 
ISMEAR = -1                # required for finite temperature algorithm
SIGMA = 0.1                # electron temperature in eV (1 eV ~ 11000 K)
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 = 8                 # use 8 imaginary frequency points
NOMEGA_DUMP = 0            # write WFULLxxxx.tmp files at omega=0, used for off-centre Coulomb integrals in next step

Run VASP and make a copy of the output file

cp OUTCAR OUTCAR.CRPAR

After a successful run, the interaction values at NOMEGA+1 frequencies are written to the OUTCAR file, where the first point is always $\omega=0$:

 spin components:  1  1, frequency:    0.0000    0.0000

screened Coulomb repulsion U_iijj between MLWFs:
        1         2         3
   1    3.3450    2.3447    2.3447
   2    2.3447    3.3450    2.3447
   3    2.3447    2.3447    3.3450

...

 spin components:  1  1, frequency:    0.0000  109.6955

screened Coulomb repulsion U_iijj between MLWFs:
        1         2         3
   1   15.2510   14.0759   14.0759
   2   14.0759   15.2510   14.0759
   3   14.0759   14.0759   15.2510

The complete matrix at zero frequency is also written to UIJKL, while the result at the first frequency point of the minimax grid is found in UIJKL.1 and so on.

Optional: Analytic continuation

To obtain the effective interaction on the real frequency axis from the imaginary axis (stored in UIJKL.*) following python code can be used in a jupyter notebook:

import numpy as np 
from scipy.interpolate import AAA
import matplotlib.pyplot as plt
# results stored in NOMEGA dimensional array 
nomega=24
u = np.zeros( nomega, dtype='complex' )  
# one-site indices 
idx=[0,40,80 ]
# store one-site bare interaction
v = np.sum( np.loadtxt('VIJKL').T[4][idx] )/len(idx)
for i in range(nomega):
    raw = np.loadtxt( 'UIJKL.{omega:d}'.format(omega=i+1) ) 
    u[i] = np.sum( raw.T[4][idx] + 1j * raw.T[5][idx] ) / len(idx)
# extract omega points 
!grep "omega =" UIJKL.{?,??} | awk '{print $5}' > omegas.dat
omegas=np.loadtxt( 'omegas.dat' )*1j
# use AAA algorithm for analytic continuation 
u_cont = AAA(omegas,u )
# plot real part of U and bare interaction V 
z = np.linspace( 0, 200, num=1000)
fig, ax = plt.subplots()
ax.plot( z, u_cont(z).real, '-', color='r', label='U')
ax.axhline(y=v, color='b', linestyle='-', label='V')
ax.legend()
# add low-frequency regime as an inset 
zlow=np.linspace( 0, 20, num=1000)
inset_ax = ax.inset_axes([0.35, 0.1, 0.6, 0.6])  # [left, bottom, width, height]
inset_ax.plot(zlow, u_cont(zlow), color='red')
plt.xlabel('$\omega$ [eV]')
plt.show()

Tip: Increase NOMEGA points to resolve more details on the real frequency axis.

Off-centre Coulomb integrals

Every cRPA job writes the effectively screened Coulomb kernel (in reciprocal space) at zero frequency to WFULLxxxx.tmp files. These files can be read in and off-centre Coulomb integrals can be evaluated using following input:

ALGO = 2E4WA                           # Compute off-centre Coulomb integrals
ISMEAR = -1                            # Fermi smearing             
SIGMA = 0.1                            # electronic temperature
NBANDS = 96                            # use same number of bands as stored in WAVECAR
ALGO = 2E4WA                           # Compute off-centre Coulomb integrals
PRECFOCK = Fast                        # fast mode for FFTs
NTARGET_STATES = 1 2 3                 # Wannier states for which Coulomb integrals are evaluated

The bare off-centre Coulomb integrals are written to VRijkl, while the effectively screened ones are found in URijkl:

# File generated by VASP contains Coulomb matrix elements
# U_ijkl = [ij|kl] 
#  I   J   K   L          RE(U_IJKL)          IM(U_IJKL)
# R:    1  0.000000  0.000000  0.000000
   1   1   1   1        3.3450226866        0.0000000000
   2   1   1   1        0.0000058776        0.0000006717
   3   1   1   1        0.0000026927       -0.0000003292
...
   3   3   3   3        3.3450230976        0.0000000000
# R:    2  0.000000  0.000000  1.000000
   1   1   1   1        0.7321605022       -0.0001554484
   2   1   1   1        0.0000021144        0.0000002295
...

Mind: Available as of VASP.6.5.2.

Downloads

CRPA_of_SrVO3.zip

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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