Bandstructure of SrVO3 in GW

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Task

Calculation of the GW bandstructure of SrVO₃ using VASP and WANNIER90.

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Performing a GW calculation with VASP is a 3-step procedure: a DFT groundstate calculation, a calculation to obtain a number of virtual orbitals, and the actual GW calculation itself. In this example we will also see how the results of the GW calculation may be postprocessed with WANNIER90 to obtain the dispersion of the bands along the usual high symmetry directions in reciprocal space.

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

The POSCAR can be visualized with p4v or VESTA and 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 (see KPOINTS.BULK)

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 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 = 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.PBE
cp WAVEDER WAVEDER.PBE

Also make a backup of the charge density for later:

cp CHGCAR CHGCAR.PBE

The dielectric function

As a bonus, VASP determines the frequency dependent dielectric function in the independent-particle (IP) picture and writes the result to the OUTCAR and vasprun.xml files. In the OUTCAR you should search for

 frequency dependent IMAGINARY DIELECTRIC FUNCTION (independent particle, no local field effects)

and

 frequency dependent      REAL DIELECTRIC FUNCTION (independent particle, no local field effects)

GW Step

The actual GW calculation requires a set of one-electron energies and eigenstates. In this case we use the PBE solution obtained from previous step:

cp WAVECAR.PBE WAVECAR
cp WAVEDER.PBE WAVEDER

The following INCAR file selects the 'single shot' GW calculation also known as G₀W₀:

• INCAR (see INCAR.GW0)

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
cp vasprun.xml vasprun.GW0.xml

HSE hybrid functional

To illustrate the kind of results one would obtain for SrVO₃ using the DFT/Hartree-Fock hybrid functional HSE, without actually doing a full self-consistent calculation, we will recalculate the one-electron energies and DOS (ALGO=Eigenval) using the HSE functional with DFT orbitals as input

cp WAVECAR.PBE WAVECAR

Use the following INCAR file:

• INCAR (see INCAR.HSE)

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

Post-processing: Density of states and Bandstructure 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

• INCAR (see INCAR.DOS)

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.PBE 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_comp.sh DOSCAR.???

This requires gnuplot to be installed.

Bandstructure with wannier90

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

• INCAR (see INCAR.WAN.XXX)

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

# As of vasp-6.4.3 define wannier90 interpolation from INCAR as follows:
# For older versions copy wannier90.win.XXX to wannier90.win and use INCAR.WAN as input for vasp
NUM_WANN = 3
WANNIER90_WIN = "

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

begin projections
V:dxy;dxz;dyz
end projections

# plot bands
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
"

Mind: Prior VASP.6.4.3 a proper wannier90.win.XXX file should be used instead.

Use the corresponding INCAR.WAN.XXX file as input for wannier90

cp INCAR.WAN.XXX INCAR

where XXX=PBE, GW0 or HSE.

Use the corresponding WAVECAR.XXX file as input

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).

The preferred way to calculate the PBE bandstructure

Provided one has a self-consistent charge density (CHGCAR) file of sufficient quality (generated using a regular grid of k-points of sufficient density) one may read this charge density and keep it fixed (ICHARG=11). For density functional calculations this charge density completely defines the Hamiltonian and using this Hamiltonian one may non-selfconsistently determine the orbitals and corresponding eigenenergies at arbitrary k-points. This is a very convenient way to calculate the bandstructure.

First we copy the self-consistent charge density of one of our previous calculations:

 cp CHGCAR.PBE CHGCAR
 cp WAVECAR.PBE WAVECAR

The bandstructure is conventionally plotted along lines of high symmetry in the 1st Brillouin zone. The easiest way to specify these is by means of the so-called linemode:

• KPOINTS (see KPOINTS.BSTR)

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

N.B.: using these k-points for a self-consistent calculation (ICHARG<10) would be a very bad idea since such and irregular sampling of the 1st Brillouin zone will not yield sensible charge densities.

Use the following INCAR file:

• INCAR (see INCAR.BSTR)

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                            # read the charge density from the CHGCAR file and keep it fixed
LORBIT = 11                            # compute lm-decomposed states
EMIN = -20 ; EMAX = 20                 # smallest/largest energy included in calculation
NEDOS = 1000                           # sampling points for DOS

N.B.: Mind that this approach works only for density functional calculations (e.g. PBE or LDA) and is not applicable to orbital dependent functionals (like hybrid functionals) or in case of GW calculations.

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

./plotbands.sh

Download

SrVO3_GW_band.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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