Bandgap of Si using different hybrid (DFT+HF) methods

 

In this tutorial we will calculate the band gap of the Si compound using 4 hybrid methods (PBE0,  B3LYP, HSE06 and HF).

We need the the 4 input files: POSCAR     INCAR    KPOINTS  POTCAR

 

PBE Calculation

POSCAR

fcc Si:
 3.9
 0.5 0.5 0.0
 0.0 0.5 0.5
 0.5 0.0 0.5
   1
cartesian
0 0 0

     Fcc Si lattice constant of 3.9  .
    1 atom per unit cell.

 

INCAR

## Better preconverge with PBE first!
## and use the WAVECAR file as inout for the DFT+HF calculation
## Leave this in
ISMEAR =  0
SIGMA  =  0.01
GGA    = PE

    Initial charge density form overlapping atoms.
    Energy cutoff of 240 eV from POTCAR file.
 

KPOINTS

k-points
 0
Monkhorst Pack
 11 11 11
 0  0  0

 

POTCAR

Download the pseudopotential file from this link POTCAR

The POTCAR contain the PBE functional as follows:

PAW_PBE Si 05Jan2001
 4.00000000000000000
 parameters from PSCTR are:
   VRHFIN =Si: s2p2
   LEXCH  = PE
   EATOM  =   103.0669 eV,    7.5752 Ry

   TITEL  = PAW_PBE Si 05Jan2001

 

NB: you can download the input files from  Si_hybrids_gap.tgz

 

Execution

~/Si-hybrid_gap/gap-pbe$> vasp_std    ---> for serial calculation

~/Si-hybrid_gap/gap-pbe$> mpirun -np 4 vasp_std    ---> for parallel calculation

 

~/Si-hybrid_gap/gap-pbe$> bandgap.py

 Gap, VBM, CBm
0.648 5.897 6.545

 To see how to use the script bandgap.py  click here

 PBE0 Calculation

 The PBE0 functional^[2][10] mixes the Perdew–Burke–Ernzerhof (PBE) exchange energy and Hartree–Fock exchange energy in a set 3:1 ratio, along with the full PBE correlation energy: 

 

 where ${\displaystyle E_{\text{x}}^{\text{HF}}}$ is the Hartree–Fock exact exchange functional, ${\displaystyle E_{\text{x}}^{\text{PBE}}}$ is the PBE exchange functional, and ${\displaystyle E_{\text{c}}^{\text{PBE}}}$ is the PBE correlation functional.^[11]

We need to modify only the INCAR as follows:

## Better preconverge with PBE first
## and use the WAVECAR file as inout for the DFT+HF calculation
   
## Leave this in
ISMEAR =  0
SIGMA  =  0.01
GGA    = PE

Execution

~/Si-hybrid_gap/gap-pbe0$> vasp_std    ---> for serial calculation

~/Si-hybrid_gap/gap-pbe0$> mpirun -np 4 vasp_std    ---> for parallel calculation

 

~/Si-hybrid_gap/gap-pbe0$> bandgap.py

Gap, VBM, CBm
1.88 5.223 7.103

 HSE06 Calculation

 The HSE (Heyd–Scuseria–Ernzerhof)^[12] exchange–correlation functional uses an error-function-screened Coulomb potential to calculate the exchange portion of the energy in order to improve computational efficiency, especially for metallic systems: 

 where ${\displaystyle a}$ is the mixing parameter, and ${\displaystyle \omega }$ is an adjustable parameter controlling the short-rangeness of the interaction. Standard values of ${\displaystyle a=1/4}$ and ${\displaystyle \omega =0.2}$ (usually referred to as HSE06) have been shown to give good results for most systems. The HSE exchange–correlation functional degenerates to the PBE0 hybrid functional for ${\displaystyle \omega =0}$. ${\displaystyle E_{\text{x}}^{\text{HF,SR}}(\omega )}$ is the short-range Hartree–Fock exact exchange functional, ${\displaystyle E_{\text{x}}^{\text{PBE,SR}}(\omega )}$ and ${\displaystyle E_{\text{x}}^{\text{PBE,LR}}(\omega )}$ are the short- and long-range components of the PBE exchange functional, and ${\displaystyle E_{\text{c}}^{\text{PBE}}(\omega )}$ is the PBE^[11] correlation functional.

We need to modify only the INCAR as follows:

## Better preconverge with PBE first!
## and use the WAVECAR file as inout for the DFT+HF calculation
## Selects the HSE06 hybrid function
LHFCALC = .TRUE. ; HFSCREEN = 0.2 ; 
ALGO = D ; TIME = 0.4 

## Leave this in
ISMEAR =  0
SIGMA  =  0.01
GGA    = PE

 

Execution

~/Si-hybrid_gap/gap-hee06$> vasp_std    ---> for serial calculation

~/Si-hybrid_gap/gap-he06$> mpirun -np 4 vasp_std    ---> for parallel calculation

 

~/Si-hybrid_gap/gap-he06$> bandgap.py

Gap, VBM, CBm
1.365 5.494 6.859

 B3LYP Calculation

The popular B3LYP (Becke,^[3] 3-parameter,^[4] Lee–Yang–Parr)^[5] exchange-correlation functional is 

where ${\displaystyle a=0.20}$, ${\displaystyle b=0.72}$, and ${\displaystyle c=0.81}$. ${\displaystyle E_{\text{x}}^{\text{B}}}$ is a generalized gradient approximation: the Becke 88 exchange functional^[6] and the correlation functional of Lee, Yang and Parr^[7] for B3LYP, and ${\displaystyle E_{\text{c}}^{\text{LSDA}}}$ is the VWN local spin density approximation to the correlation functional.^[8]

The three parameters defining B3LYP have been taken without modification from Becke's original fitting of the analogous B3PW91 functional to a set of atomization energies, ionization potentials, proton affinities, and total atomic energies.^[9]

 

We need to modify only the INCAR as follows:

## Better preconverge with PBE first!
## and use the WAVECAR file as inout for the DFT+HF calculation
## Selects the B3LYP hybrid function
LHFCALC = .TRUE. ; GGA = B3 ; AEXX = 0.2 ; AGGAX = 0.72 
AGGAC = 0.81 ; ALDAC = 0.19
ALGO = D ; TIME = 0.4 

## Leave this in
ISMEAR =  0
SIGMA  =  0.01
GGA    = PE

 

Execution

~/Si-hybrid_gap/gap-b3lyp$> vasp_std    ---> for serial calculation

~/Si-hybrid_gap/gap-b3lyp$> mpirun -np 4 vasp_std    ---> for parallel calculation

 

~/Si-hybrid_gap/gap-b3lyp$> bandgap.py

Gap, VBM, CBm
1.918 5.54 7.458

 HF Calculation

We need to modify only the INCAR as follows:

 

## Better preconverge with PBE first!
## and use the WAVECAR file as inout for the DFT+HF calculation
## Selects HF 
LHFCALC = .TRUE. ; AEXX = 1.0 ; ALDAC = 0.0 ; AGGAC = 0
ALGO = D ; TIME = 0.4 

## Leave this in
ISMEAR =  0
SIGMA  =  0.01
GGA    = PE

 

Execution

~/Si-hybrid_gap/gap-hf$> vasp_std    ---> for serial calculation

~/Si-hybrid_gap/gap-hf$> mpirun -np 4 vasp_std    ---> for parallel calculation

 

~/Si-hybrid_gap/gap-hf$> bandgap.py

Gap, VBM, CBm
7.579 3.522 11.101

 

Reference:  https://www.vasp.at/wiki/index.php/Bandgap_of_Si_using_different_DFT%2BHF_methods

https://en.wikipedia.org/wiki/Hybrid_functional