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# Equilibrium volume of Si in the RPA

## 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:

- a “standard” DFT groundstate calculation with a “dense” mesh of k-points.
- compute the Hartree-Fock energy using the DFT orbitals of Step 1. Needs WAVECAR file from step 1.
- a “standard” DFT groundstate calculation with “coarse” mesh of k-points.
- obtain DFT “virtual” orbitals (empty states). Needs WAVECAR file from step 3.
- 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.

**N.B.:**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* (see below):

./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:

ALGO = EIGENVAL ; NELM = 1 LWAVE = .FALSE. LHFCALC = .TRUE. AEXX = 1.0 ; ALDAC = 0.0 ; AGGAC = 0.0 NKRED = 2 ISMEAR = 0 ; SIGMA = 0.05 KPAR = 8 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.

- This is the mesh of k-points that will be used in the subsequent ACFDT calculation.

- 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) = -10.9079580568*eV (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).

## Running this example

The following bash-script `doall.sh`

will run through all of the aforementioned calculational steps (step 1-5) for a range of different lattice constants (*a=5.1-5.8* Å in steps of *0.1* Å)

# # To run VASP this script calls $vasp_std # (or posibly $vasp_gam and/or $vasp_ncl). # These variables can be defined by sourcing vaspcmd . vaspcmd 2> /dev/null # # When vaspcmd is not available and $vasp_std, # $vasp_gam, and/or $vasp_ncl are not set as environment # variables, you can specify them here [ -z "`echo $vasp_std`" ] && vasp_std="mpirun -np 8 /path-to-your-vasp/vasp_std" [ -z "`echo $vasp_gam`" ] && vasp_gam="mpirun -np 8 /path-to-your-vasp/vasp_gam" [ -z "`echo $vasp_ncl`" ] && vasp_ncl="mpirun -np 8 /path-to-your-vasp/vasp_ncl" # # The real work starts here # for i in 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 ; do cat >POSCAR <<! system Si $i 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 ! # start with a PBE calculation with a lot of k-points (needed for EXX) rm WAVECAR WAVEDER cp KPOINTS.12 KPOINTS cp INCAR.DFT INCAR $vasp_std cp OUTCAR OUTCAR.DFT.$i e1=`awk '/free energy/ {print $5}' OUTCAR` # get the HF energy with PBE orbitals cp INCAR.EXX INCAR $vasp_std e2=`awk '/free energy/ {print $5}' OUTCAR` cp OUTCAR OUTCAR.EXX.$i # now a PBE calculation with less k-points rm WAVECAR WAVEDER cp KPOINTS.6 KPOINTS cp INCAR.DFT INCAR $vasp_std # obtain virtual orbitals cp INCAR.DIAG INCAR $vasp_std cp OUTCAR OUTCAR.DIAG.$i cp WAVECAR WAVECAR.$i cp WAVEDER WAVEDER.$i ## for metals # cp INCAR.HFC INCAR # $vasp_std # # cp OUTCAR OUTCAR.HFC.$i # e3=`awk '/HF-correction/ {print $4}' OUTCAR` # RPA correlation cp INCAR.ACFDT INCAR $vasp_std cp OUTCAR OUTCAR.ACFDT.$i e4=`awk '/converged value/ {print $3}' OUTCAR` # echo $i $e1 $e2 $e3 $e4 >> summary echo $i $e1 $e2 $e4 >> summary done

To execute the aforementions script:

./doall.sh

- When everything is finished you can quickly visualize (with gnuplot) the total energy vs. lattice-constant curves for DFT and RPA by means of:

./plotall.sh