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# Bandstructure of SrVO3 in GW

## Task

Calculation of the GW bandstructure of SrVO_{3} using VASP and WANNIER90.

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:

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 SrVO_{3}, 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.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

Also make a backup of the charge density for later:

cp CHGCAR CHGCAR.DIAG

### 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.DIAG WAVECAR cp WAVEDER.DIAG WAVEDER

The following INCAR file selects the 'single shot' GW calculation also known as G_{0}W_{0}:

- 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 SrVO_{3} using the DFT/Hartree-Fock hybrid functional HSE, without actually doing a full selfconsistent calculation, we will recalculate the one-electron energies and DOS (ALGO=Eigenval) using the HSE functional with DFT orbitals as input

cp WAVECAR.DIAG 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.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.

### Bandstructure with `wannier90`

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

- INCAR (see INCAR.WAN)

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 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.DIAG CHGCAR cp WAVECAR.DIAG 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 *selfconsistent* 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

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