Bandgap of Si in GW
Description: calculation of the bandgap of Si using various flavours of GW.
To do GW calculations we have to follow a 3-step procedure.
Step 1: a DFT groundstate calculation
Everything starts with a standard DFT groundstate calculation (in this case PBE).
- INCAR
ISMEAR = 0 SIGMA = 0.05 GGA = PE
- KPOINTS
6x6x6 0 G 6 6 6 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
ALGO = Exact NBANDS = 64 LOPTICS = .TRUE. NEDOS = 2000 ## you might try #LPEAD = .TRUE. ISMEAR = 0 SIGMA = 0.05 GGA = PE
Mind: make a copy of your WAVECAR and WAVEDER files, as we will repeatedly need them in the following. For instance
cp WAVECAR WAVECAR.LOPTICS cp WAVEDER WAVEDER.LOPTICS
Step 3: the actual GW calculation
Restart from the WAVECAR and WAVEDER files of the previous calculation, with
- INCAR
## 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 and LOPTICS=.TRUE. ## and a reasonable number of virtual states (see above) ALGO = GW0 ; LSPECTRAL = .TRUE. ; NOMEGA = 50 #LRPA = .FALSE. ## 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
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
grep " 4 " OUTCAR | sort -n -k 3 | tail -1 | awk '{print $3}'
and for the lowest lying unoccupied state,
grep " 5 " OUTCAR | sort -n -k 3 | head -1 | awk '{print $3}'
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, try
grep " 4 " OUTCAR | tail -16 | sort -n -k 3 | tail -1 | awk '{print $3}'
and for the corresponding lowest lying unoccupied state,
grep " 5 " OUTCAR | tail -16 | sort -n -k 3 | head -1 | awk '{print $3}'
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