Bandstructure and CRPA of SrVO3

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DFT, GW and HSE Calculation

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. Copy the aforementioned file to KPOINTS:

cp KPOINTS.BULK KPOINTS

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

Obtain DFT virtual states 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

Also make a backup of the charge density for later:

cp CHGCAR CHGCAR.DIAG

GW Calculation

The actual GW calculation uses the PBE solution as a basis set:

cp WAVECAR.DIAG WAVECAR 
cp WAVEDER.DIAG WAVEDER

The following INCAR file selects a so-called 'single shot' GW calculation:

SYSTEM = SrVO3                         # system name      
ISMEAR = 0                             # Gaussian smearing  
KPAR = 2                               # parallelization of k-points in two groups  
ALGO = GW0                             # GW with iteration in G, W kept on DFT level
NELM = 1                               # one electronic step suffices, since WAVECAR from previous step is present
NBANDS = 96                            # need for a lot of bands in GW
PRECFOCK = Fast                        # fast mode for FFTs
ENCUTGW = 100                          # small energy cutoff for response function suffices for this tutorial
NOMEGA = 200                           # large number of real frequency points for Hilbert transforms of W and self-energy

Restarting VASP will overwrite the present WAVECAR and vasprun.xml file. Make a copy them for later.

cp WAVECAR WAVECAR.GW0

HSE Calculation

To accelerate the electronic convergence it is wise to start the HSE calculation from the PBE solution:

cp WAVECAR.DIAG WAVECAR 
cp WAVEDER.DIAG WAVEDER

The HSE wavefunction can be obtained with following control file:

SYSTEM = SrVO3                         # system name                  
ISMEAR = 0                             # Gaussian smearing            
KPAR = 2                               # parallelization of k-points in two groups  
ALGO = Eigenval                        # calulate eigenvalues 
NELM = 1                               # one electronic step suffices, since WAVECAR from previous step is present
NBANDS = 48                            # small number of bands suffice
PRECFOCK = Fast                        # fast mode for FFTs
LHFCALC = .TRUE.                       # switch on Hartree-Fock routines to calculate exact exchange
HFSCREEN = 0.2                         # HSE06 screening parameter

Restart VASP and make a copy of the wavefunction for post-processing

cp WAVECAR WAVECAR.HSE

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 can be found in the OUTCAR file after following lines

screened Coulomb repulsion U_iijj between MLWFs:

including an averaged value:

screened Hubbard U =    3.4503   -0.0000

Make a copy of the output file

cp OUTCAR OUTCAR.CRPA

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:

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. Make a copy of the output file

cp OUTCAR OUTCAR.CRPAR

Post-processing: Density of states and Band structure for PBE, GW and HSE

Density of States

The DOS of the PBE, GW and HSE solution can be calculated in a post-processing step with

SYSTEM = SrVO3                         # system name                  
ISMEAR = -5                            # Bloechl's tetrahedron method (requires at least 3x3x3 k-points)             
ALGO = NONE                            # no electronic changes required 
NELM = 1                               # one electronic step suffices, since WAVECAR from previous step is present
NBANDS = 48                            # number of bands used 
EMIN = -20 ; EMAX = 20                 # smallest/largest energy included in calculation
NEDOS = 1000                           # sampling points for DOS
LORBIT = 11                            # calculate l-m decomposed DOS
LWAVE = .FALSE.                        # do not overwrite WAVECAR
LCHARG = .FALSE.                       # do not overwrite CHGCAR

and requires the apropriate WAVECAR file from one of the previous steps. Copy

cp WAVECAR.DIAG WAVECAR

or

cp WAVECAR.GW0 WAVECAR

or

cp WAVECAR.HSE WAVECAR 

and restart VASP. The density of states is written to DOSCAR, make a copy of this file

cp DOSCAR DOSCAR.XXX

where XXX is either PBE, GW0 or HSE. Visualize the projected DOS for the V-t2g, V-eg and O-p states with the scriptfile

./plotdos.sh DOSCAR.*

This requires gnuplot to be installed.

Band structure with wannier90

The band structure can be calculated via Wannier interpolation using wannier90 in the library mode

SYSTEM = SrVO3                         # system name                  
ISMEAR = 0                             # Gaussian smearing 
ALGO = NONE                            # no electronic changes required 
NELM = 1                               # one electronic step suffices, since WAVECAR from previous step is present
NBANDS = 48                            # number of bands used 
LWAVE = .FALSE.                        # do not overwrite WAVECAR
LCHARG = .FALSE.                       # do not overwrite CHGCAR
LWANNIER90_RUN = .TRUE.                # run wannier90 in library mode

Use the corresponding wannier90.win.XXX file as input for wannier90

cp wannier90.win.XXX wannier90.win

where XXX=PBE, GW0 or HSE and looks similar to

bands_plot = true

begin kpoint_path
R  0.50000000  0.50000000  0.50000000  G  0.00000000  0.00000000  0.00000000
G  0.00000000  0.00000000  0.00000000  X  0.50000000  0.00000000  0.00000000
X  0.50000000  0.00000000  0.00000000  M  0.50000000  0.50000000  0.00000000
M  0.50000000  0.50000000  0.00000000  G  0.00000000  0.00000000  0.00000000
end kpoint_path 

# number of wannier states
num_wann =    3 

# number of bloch bands 
num_bands=   96

# GW energy window for t2g states
dis_win_min = 7.4
dis_win_max = 9.95

begin projections
V:dxy;dxz;dyz
end projections

Use the corresponding WAVECAR.XXX file as imput

cp WAVECAR.XXX WAVECAR

and restart VASP. If all went well, the Vanadium t2g band dispersion thus obtained, may conveniently be visualized with gnuplot:

gnuplot -persist ./wannier90_band.gnu
N.B.: Most modern versions of gnuplot will respond with an error message unless you remove the first line of wannier90_band.gnu (some deprecated syntax issue).

Alternative way to calculate the PBE band structure

VASP allows to interpolate the PBE band structure from the PBE charge density

 cp CHGCAR.DIAG CHGCAR
 cp WAVECAR.DIAG WAVECAR

by adapting the KPOINTS file as follows:

Auto
15
Linemode
reciprocal
0.50000000  0.50000000  0.50000000   !R
0.00000000  0.00000000  0.00000000   !G 

0.00000000  0.00000000  0.00000000   !G
0.50000000  0.00000000  0.00000000   !X

0.50000000  0.00000000  0.00000000   !X
0.50000000  0.50000000  0.00000000   !M 

0.50000000  0.50000000  0.00000000   !M
0.00000000  0.00000000  0.00000000   !G

The following INCAR file tells VASP to interpolate the band structure:

System = SrVO3                         # system name                  
ISMEAR = 0                             # Gaussian smearing 
EDIFF = 1E-7                           # tight convergence criterion
NBANDS = 36                            # 36 bands are sufficient 
LWAVE = .FALSE.                        # do not overwrite WAVECAR
LCHARG = .FALSE.                       # do not overwrite CHGCAR
ICHARG = 11                            # use CHGCAR file for interpolation
LORBIT = 11                            # compute lm-decomposed states
EMIN = -20 ; EMAX = 20                 # smallest/largest energy included in calculation
NEDOS = 1000                           # sampling points for DOS

This PBE band structure and the Wannier-interpolated structures of the PBE, HSE and GW calculation can be compared via

./plotbands.sh

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