Equilibrium volume of Si in the RPA

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Task

In this example you will calculate the equilibrium lattice constant of Si in the RPA (ACFDT).

The workflow of a RPA total energy calculations consists of five consecutive steps:

  • Step 1: a “standard” DFT groundstate calculation with a “dense” mesh of k-points.
  • Step 2: compute the Hartree-Fock energy using the DFT orbitals of Step 1. Needs WAVECAR file from step 1.
  • Step 3: a “standard” DFT groundstate calculation with “coarse” mesh of k-points.
  • Step 4: obtain DFT “virtual” orbitals (empty states). Needs WAVECAR file from step 3.
  • Step 5: the RPA correlation energy (ACFDT) calculation. Needs WAVECAR and WAVEDER files from step 4.

In case of metallic systems there is an additional step between Steps 4 and 5, that is beyond the scope of this example.

To compute the equilibrium lattice constant of Si we need to calculate the RPA total energy for a range of different lattice constants. All of the calculation steps are prepared automatically performed by the script doall.sh 

 ./doall.sh

This script will perform the following calculations for a range of different lattice constants:

Step 1: DFT groundstate calculation with a “dense” mesh of k-points

  • The following INCAR file is used (INCAR.DFT):
ISMEAR = 0 ; SIGMA = 0.05
EDIFF = 1E-8
  • The following KPOINTS file is used (KPOINTS.12):
12x12x12
 0
G
 12 12 12
  0  0  0


Step 2: Compute the Hartree-Fock energy using the DFT orbitals

  • To Compute the Hartree-Fock energy using DFT orbitals we need the (WAVECAR) of Step 1.
  • The INCAR file INCAR.EXX is used in this step:
 {{TAGBL|ALGO}} = EIGENVAL ; {{TAGBL|NELM}} = 1
 {{TAGBL|LWAVE}} = .FALSE.
 {{TAGBL|LHFCALC}} = .TRUE.
 {{TAGBL|AEXX}} = 1.0 ; {{TAGBL|ALDAC}} = 0.0 ; {{TAGBL|AGGAC}} = 0.0
 {{TAGBL|NKRED}} = 2
 {{TAGBL|ISMEAR}} = 0 ; {{TAGBL|SIGMA}} = 0.05
 {{TAGBL|KPAR}} = 8
 {{TAGBL|NBANDS}} = 4
  • NKRED=2 is used for the downsample the k-space representation of the Fock-potential to save time.
  • Using NBANDS=4 only occupied states are considered to save time.


Step 3: DFT groundstate calculation with a “coarse” mesh of k-points

  • Perform a DFT groundstate calculation with a “coarse” mesh of k-points.
  • The following INCAR file is used (INCAR.DFT):
ISMEAR = 0 ; SIGMA = 0.05
EDIFF = 1E-8
  • The following coarse KPOINTS file is used (KPOINTS.6):
6x6x6
 0
G
  6  6  6
  0  0  0


Step 4: Obtain DFT "virtual" orbitals (empty states)

  • Obtain DFT "virtual" orbitals (empty states).
  • The following INCAR file is used in this step (INCAR.DIAG):
ALGO = Exact
NBANDS = 64
NELM = 1
LOPTICS = .TRUE.
ISMEAR = 0 ; SIGMA = 0.05
  • In this step one needs to set LOPTICS=.TRUE. so that VASP calculates the derivative of the orbitals w.r.t. the Bloch wavevector (stored in the WAVEDER file). These are needed to correctly describe the long-wavelength limit of the dielectric screening.
  • We use exact diagonalization (ALGO=Exact) and keep 64 bands after diagonalization (NBANDS=64).
  • This calculations needs the orbitals (WAVECAR file) written in Step 3.


Step 5: calculate the RPA correlation energy (ACFDT)

  • The following INCAR file is used in this step (INCAR.ACFDT):
ALGO = ACFDT
NBANDS = 64
ISMEAR = 0 ; SIGMA = 0.05
  • In OUTCAR.ACFDT.X.X one finds the RPA correlation energy, e.g.:
        cutoff energy      smooth cutoff    RPA   correlation   Hartree contr. to MP2
 ---------------------------------------------------------------------------------
             163.563            130.851       -10.7869840331      -19.0268026572
             155.775            124.620       -10.7813600055      -19.0200457142
             148.357            118.685       -10.7744584182      -19.0118291822
             141.292            113.034       -10.7659931963      -19.0017871991
             134.564            107.651       -10.7555712745      -18.9894197881
             128.156            102.525       -10.7428704760      -18.9742991317
             122.054             97.643       -10.7273118140      -18.9556871679
             116.241             92.993       -10.7085991597      -18.9331679971
 linear regression
 converged value                              -10.9079580568      -19.1711146204
  • Take the “converged value”, in this case: EC(RPA) = -10.9079580568eV (an approximate “infinite basis set” limit).
  • This calculations needs the orbitals (WAVECAR file) and the derivative of the orbitals w.r.t. the Bloch wavevectors (WAVEDER file) written in Step 4.
  • The RPA total energy is calculated as the, E(RPA)=EC(RPA)+EXX, the sum of the RPA correlation energy of step 5 EC(RPA) and the Hartree fock energy EXX of step 2.
To get the Hartree fock energy grep “free energy” in the OUTCAR.EXX.* file (there are two spaces between free and energy).

POSCAR

system Si
  5.8
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

Calculation

The following script doall.sh will run through all of the aforementioned calculational steps for a range of different lattice constants:


  • The sample output for the total energy vs. lattice-constant curves for DFT and RPA should look like the following:

Download

Si_ACFDT_vol.tgz

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