Bandgap of Si in GW

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Task

Calculation of the bandgap of Si using various flavours of GW.

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Mind: before you start doing GW calculations it might be beneficial to have a look at the examples on the description of dielectric properties.

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To do GW calculations we have to follow a 3-step procedure.

Step 1: DFT groundstate calculation

Everything starts with a standard DFT groundstate calculation (in this case PBE).

• INCAR (see INCAR.DFT)

ISMEAR =  0
SIGMA  =  0.05
EDIFF  = 1E-8

• KPOINTS (see KPOINTS.6)

6x6x6
 0
G
 6 6 6
 0 0 0

or to save some time use a "quick-and-dirty" setup (take KPOINTS.4):

4x4x4
 0
G
 4 4 4
 0 0 0

• POSCAR

system Si
5.430
0.5 0.5 0.0
0.0 0.5 0.5
0.5 0.0 0.5
2
cart
0.00 0.00 0.00
0.25 0.25 0.25

Step 2: obtain DFT virtual orbitals

To obtain a WAVECAR file with a reasonable number of virtual orbitals (50-100 per atom) we need to restart from the previous groundstate calculation with ALGO=Exact, and manually set the number of bands by means of the NBANDS-tag. To obtain the corresponding WAVEDER file we additionally specify LOPTICS=.TRUE..

• INCAR (see INCAR.DIAG)

ALGO = Exact
NBANDS  = 64
LOPTICS = .TRUE. ; CSHIFT = 0.1
NEDOS = 2000
# you might try
#LPEAD = .TRUE.
   
ISMEAR =  0
SIGMA  =  0.05
EDIFF  = 1E-8

Mind: make a copy of your WAVECAR and WAVEDER files, as we will repeatedly need them in the following. For instance

cp WAVECAR WAVECAR.DIAG
cp WAVEDER WAVEDER.DIAG

Step 3: the actual GW calculation

Restart from the WAVECAR and WAVEDER files of the previous calculation, with

• INCAR (see INCAR.GW)

# Frequency dependent dielectric tensor including
# local field effects within the RPA (default) or
# including changes in the DFT xc-potential (LRPA=.FALSE.).
# N.B.: beware one first has to have done a
# calculation with ALGO=Exact, LOPTICS=.TRUE.
# and a reasonable number of virtual states (see above)
ALGO = GW0 ; LSPECTRAL = .TRUE. ; NOMEGA = 50
   
# be sure to take the same number of bands as for
# the LOPTICS=.TRUE. calculation, otherwise the
# WAVEDER file is not read correctly
NBANDS = 64
     
# Add this to update the quasiparticle energies
# in the Green's function (GW0)
#NELM = 4
    
ISMEAR =  0
SIGMA  =  0.05
EDIFF  = 1E-8

At the bottom of the OUTCAR file you will find the quasi-particle (QP) energies.

 QP shifts <psi_nk| G(iteration)W_0 |psi_nk>: iteration 1
 for sc-GW calculations column KS-energies equals QP-energies in previous step
 and V_xc(KS)=  KS-energies - (<T + V_ion + V_H > + <T+V_H+V_ion>^1  + <V_x>^1)

 k-point   1 :       0.0000    0.0000    0.0000
  band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z            occupation

      1      -6.4888      -6.8243     -10.9766     -10.4570     -17.5189       0.6458       2.0000
      2       5.4800       5.1162     -11.8859     -11.4060     -12.7290       0.7580       2.0000
      3       5.4800       5.1162     -11.8859     -11.4060     -12.7290       0.7580       2.0000
      4       5.4800       5.1162     -11.8859     -11.4060     -12.7290       0.7580       2.0000
      5       8.0443       8.3296      -9.7235     -10.1038      -5.7364       0.7502       0.0000
      6       8.0443       8.3296      -9.7235     -10.1038      -5.7364       0.7502       0.0000
      7       8.0443       8.3296      -9.7235     -10.1038      -5.7364       0.7502       0.0000
      8       8.8407       9.2475     -10.5130     -11.0594      -6.0662       0.7445       0.0000

 k-point   2 :       0.1667    0.0000    0.0000
  band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z            occupation

      1      -6.1276      -6.4734     -11.0208     -10.4905     -17.3978       0.6521       2.0000
      2       3.0946       2.6991     -11.4452     -10.9063     -13.1354       0.7340       2.0000
      3       5.0279       4.6595     -11.7159     -11.2282     -12.6625       0.7552       2.0000
      4       5.0279       4.6595     -11.7159     -11.2282     -12.6625       0.7552       2.0000
      5       7.8309       8.1065      -9.8441     -10.2097      -5.8680       0.7539       0.0000
      6       8.6943       8.9816      -9.8669     -10.2533      -5.6768       0.7436       0.0000
      7       8.6943       8.9816      -9.8669     -10.2533      -5.6768       0.7436       0.0000
      8      10.9341      11.2678     -10.4716     -10.9278      -5.5632       0.7316       0.0000

     ..         ..           ..           ..           ..           ..           ..           .. 
     ..         ..           ..           ..           ..           ..           ..           .. 
     ..         ..           ..           ..           ..           ..           ..           .. 
     ..         ..           ..           ..           ..           ..           ..           .. 


 k-point  16 :      -0.3333    0.5000    0.1667
  band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z            occupation

      1      -2.2240      -2.5911     -11.4857     -10.9550     -16.0663       0.6917       2.0000
      2      -2.2240      -2.5911     -11.4857     -10.9550     -16.0663       0.6917       2.0000
      3       1.8279       1.3698     -10.7735     -10.1380     -12.9637       0.7209       2.0000
      4       1.8279       1.3698     -10.7735     -10.1380     -12.9637       0.7209       2.0000
      5       8.2346       8.4128      -9.3111      -9.5472      -5.1975       0.7546       0.0000
      6       8.2346       8.4128      -9.3111      -9.5472      -5.1975       0.7546       0.0000
      7      12.2605      12.5170      -9.7969     -10.1486      -4.3607       0.7294       0.0000
      8      12.2605      12.5170      -9.7969     -10.1486      -4.3607       0.7294       0.0000

To quickly find the QP-energy of the highest lying occupied state, try

./gap_GW.sh OUTCAR

Beyond the random-phase-approximation

To include local field effects beyond the random-phase-approximation in the description of the frequency dependent dielectric response function (local field effects in DFT) add the following line to your INCAR file:

LRPA = .FALSE.

and again restart from the WAVECAR and WAVEDER files from step 2.

Beyond G₀W₀: GW₀

The most usual step beyond single-shot GW (G₀W₀) is to iterate the quasi-particle energies in the Greens functions. This is the socalled GW₀ approximation. To have VASP do, for instance, 4 iterations of the QP-energies in G, add the following line to the INCAR file:

NELM = 4

and again restart from the WAVECAR and WAVEDER files from step 2.

To quickly find the QP-energy of the highest lying occupied state after 4 iterations of the QP energies in G, type:

./gap_GW.sh OUTCAR

Download

Si_GW_gap.tgz

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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