View source for 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:

# 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 {{TAG|WAVECAR}} file from step 1.
# a “standard” DFT groundstate calculation with “coarse” mesh of k-points.
# obtain DFT “virtual” orbitals (empty states). Needs {{TAG|WAVECAR}} file from step 3.
# the RPA correlation energy (ACFDT) calculation. Needs {{TAG|WAVECAR}} and {{TAG|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 {{TAG|INCAR}} file is used (INCAR.DFT):
 {{TAGBL|ISMEAR}} = 0 ; {{TAGBL|SIGMA}} = 0.05
 {{TAGBL|EDIFF}} = 1E-8

*The following {{TAG|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 ({{TAG|WAVECAR}}) of Step 1.

*The {{TAG|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

*{{TAG|NKRED}}=2 is used for the downsample the k-space representation of the Fock-potential to save time.

*Using {{TAG|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 {{TAG|INCAR}} file is used (INCAR.DFT):
 {{TAGBL|ISMEAR}} = 0 ; {{TAGBL|SIGMA}} = 0.05
 {{TAGBL|EDIFF}} = 1E-8

*The following coarse {{TAG|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 {{TAG|INCAR}} file is used in this step (INCAR.DIAG):
 {{TAGBL|ALGO}} = Exact
 {{TAGBL|NBANDS}} = 64
 {{TAGBL|NELM}} = 1
 {{TAGBL|LOPTICS}} = .TRUE.
 {{TAGBL|ISMEAR}} = 0 ; {{TAGBL|SIGMA}} = 0.05 

*In this step one needs to set {{TAG|LOPTICS}}=''.TRUE.'' so that VASP calculates the derivative of the orbitals w.r.t. the Bloch wavevector (stored in the {{TAG|WAVEDER}} file). These are needed to correctly describe the long-wavelength limit of the dielectric screening.
*We use exact diagonalization ({{TAG|ALGO}}=''Exact'') and keep 64 bands after diagonalization ({{TAG|NBANDS}}=64).
*This calculations needs the orbitals ({{TAG|WAVECAR}} file) written in Step 3. 


=== Step 5: calculate the RPA correlation energy (ACFDT) ===

*The following {{TAG|INCAR}} file is used in this step (INCAR.ACFDT):
 {{TAGBL|ALGO}} = ACFDT
 {{TAGBL|NBANDS}} = 64
 {{TAGBL|ISMEAR}} = 0 ; {{TAGBL|SIGMA}} = 0.05

*In OUTCAR.ACFDT.X.X one finds the RPA correlation energy, e.g.:
<pre>
        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
</pre>

*Take the “converged value”, in this case: ''EC(RPA) = -10.9079580568''eV (an approximate “infinite basis set” limit).

*This calculations needs the orbitals ({{TAG|WAVECAR}} file) and the derivative of the orbitals w.r.t. the Bloch wavevectors ({{TAG|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 <code>grep “free  energy”</code> in the OUTCAR.EXX.* file (there are two spaces between free and energy).


== Running this example ==

The following bash-script <code>doall.sh</code> will run through all of the aforementioned calculational steps (step 1-5) for a range of different lattice constants (''a=5.1-5.8'' &Aring; in steps of ''0.1'' &Aring;)

<pre>
#
# 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
</pre>

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

[[File:Fig ACFDT 1 a.png|408px]]

[[File:Fig ACFDT 1 b.png|400px]]

----

== Download ==
[[Media:Si ACFDT vol.tgz| Si_ACFDT_vol.tgz]]

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