* Crystalline silicon [#zd2bcf39]
This tutorial explains how to perform convergence study with respect to the cutoff energy and the number of k-points, and how to obtain the equilibrium cell parameters.
** Convergence study [#s0ba5546]
Let us investigate the convergence of the total energy with respect to the cutoff energy.
Here lattice constant of 10.2347 Bohr and 4 x 4 x 4 Monkhorst-Pack k-point mesh are used.
Normconserving pseudopotential (pot.Si_pbe1) is used and the cutoff energy for the augmentation charge is 4 times larger than that for the wave functions.
- Input file
0 0 0 0 0 0 : INPUT_CTRL(1:6) (DUMMY)
5.00 10.00 1 2 2 : GMAX GMAXP NTYP NATM NATM2
227 2 : num_space_group, type
10.2347 10.2347 10.2347 90.0 90.0 90.0 : A B C ALPHA BETA GAMMA
4 4 4 2 2 2 : N1 N2 N3 M1 M2 M3
0 0 : NCORD, NINV
0.00 0.00 0.00 1 1 1 : CPS(1,1:3) IWEI IMDTYP ITYP
0.25 0.25 0.25 1 1 1 : CPS(2,1:3) IWEI IMDTYP ITYP
14 0.5000 28.09 6 1 0.0 : ATOMN ALFA AMION ILOC IVAN ZETA1
0 0 0 0 0 : ICOND INIPOS INIVEL ININOS INIACC
0 1 : IPRE IPRI
200 200 0 84200.00 0 : NMD1 NMD2 LAST_ITER CPUMAX IFSTOP
6 1 : WAY_MIX MIX_WHAT
0 20 0.60 : ITER_START KBXMIX MIX_ALPHA
0.20 0.30 0.20 0.20 0.20 : DTIM1 DTIM2 DTIM3 DTIM4 DTIM
300.00 4 1 0.50D-09 : DTIO IMDALG IEXPL EDELTA
-0.0020 0.50D+03 0 : WIDTH FORCCR ISTRESS
ggapbe 1 : xctype,kspin
2.00 : DETSTM
101 : NBZTYP
0 0 0 : NKX NKY NKZ (DUMMY)
0 0 0 : NKX2 NKY2 NKZ2 (DUMMY)
8 : KEG
1 : NEXTST
0 : (DUMMY)
2 : IMSD
0 : EVALUATE_EKO_DIFF
0 : NPDOSAO
0 0.00 : SM_N DOPING
Using the above input file as a template, we obtain the following total energy as a function of the cutoff energy
#GMAX ETOT (Ha)
3.00 -7.83436760
4.00 -7.87325631
5.00 -7.87585304
6.00 -7.87874066
7.00 -7.87922178
8.00 -7.87928813
9.00 -7.87935331
10.00 -7.87935680
** Cell optimization [#ze71a96c]
In the current version of STATE, the stress tensor is not yet implemented (there was an implementation in the ancestor of STATE) and thus, the cell parameters should be optimized manually.
The equilibrium lattice parameter is obtained by calculating the total energy as a function of cell parameter and by fitting to a certain function.
In this example, we use the cutoff wave vector (cutoff energy) of 6 (36 Ry), 4x4x4 Monkhorst-Pack k-point mesh, and obtain the following total energy as a function of _volume_.
#volume(Bohr^3) ftot(Ha)
0.269223E+03 -7.87884234
0.270011E+03 -7.87890143
0.270802E+03 -7.87895926
0.271593E+03 -7.87900043
0.272387E+03 -7.87904967
0.273182E+03 -7.87907649
0.273978E+03 -7.87910231
0.274776E+03 -7.87911557
0.275576E+03 -7.87912945
0.276377E+03 -7.87912776
0.277179E+03 -7.87912853
0.277984E+03 -7.87911256
0.278789E+03 -7.87910050
0.279597E+03 -7.87907182
0.280406E+03 -7.87905614
0.281216E+03 -7.87901302
0.282028E+03 -7.87897140
0.282842E+03 -7.87891793
0.283657E+03 -7.87886603
0.284473E+03 -7.87879889
0.285292E+03 -7.87873581
This total energy is fittted to the Murnaghan equation of state using a gnuplot script "eosfit" as attached, to obtain the equilbrium volume of 276.381913141324 (Bohr^3) (40.9555250048756 Angstrom^3) , corresponding to the lattice constant of 10.34 Bohr (5.47 Angstrom).
For more critical assessment, the convergence of the lattice constant with respect to the cutoff energy and the number of k-points are studied.
* Exercise [#j7c79e4e]
- Perform a convergence study of the equilibrium lattice constant on the cutoff energy and the numer of k-points.
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