Crystalline Aluminum

This tutorial explains how to perform the convergence study with respect to the number of k-points with the smearing and tetrahedron methods. This also shows that the total energy depends significantly on the smearing function and width.

Convergence with respect to the number of k-points

We present the convergence of the total energy with respect to the number of k-points with the Methefeesel-Paxton smearing (default) and tetrahedron methods. Use the keyword SMEARING to choose the smearing function to treat the Fermi level for metallic systems. A typical input file for aluminum in the fcc structure is:

WF_OPT DAV
NTYP   1
NATM   1
TYPE   2
NSPG   225
GMAX   4.00
KPOINT_MESH  4  4  4
MIX_ALPHA    0.6
SMEARING MP
WIDTH    0.0200
EDELTA   1.000D-10
NEG      6
CELL    7.59670000   7.59670000   7.59670000  90.00000000  90.00000000  90.00000000
&ATOMIC_SPECIES
 Al  26.980000 pot.Al_pbe1
&END
&ATOMIC_COORDINATES CRYSTAL
      0.000000000000      0.000000000000      0.000000000000    1    1    1
&END

Use BZINT TETRA to use the tetrahedron method (we haven’t tested if both BZINT and SMEARING exist in the input file).

Total energy as a function of k-point mesh for smearing method (Hermite-Gaussian) and tetrahedron method are obtained as follows:

Smearing method:

#K-point mesh Etot(Hartree)
 4  4  4      -2.06181805
 6  6  6      -2.07427411
 8  8  8      -2.07161451
10 10 10      -2.07135505
12 12 12      -2.07212484
14 14 14      -2.07162406
16 16 16      -2.07195589
18 18 18      -2.07179717
20 20 20      -2.07182881
22 22 22      -2.07185750
24 24 24      -2.07182100

Tetrahedron method:

#K-point mesh Etot(Hartree)
 4  4  4      -2.07385026
 6  6  6      -2.07267588
 8  8  8      -2.07220663
10 10 10      -2.07168189
12 12 12      -2.07202169
14 14 14      -2.07166456
16 16 16      -2.07184510
18 18 18      -2.07184080
20 20 20      -2.07180786
22 22 22      -2.07186279
24 24 24      -2.07180814
../_images/etot_al_gmax4_kpoint_nonshifted.png

Warning

The k-point shift should be switched off for the tetrahedron method.

Convergence with respect to the smearing width

Total (free) energy of the metallic system is sensitive to the smearing width, in particular, with the Gaussian and Fermi-Dirac function. We demonstrate the smearing width dependence of the total energy, following the seminal work by de Gironcoli [1].

We calculate the total energy as a function of smearing width by using different smearing function (Fermi-Dirac FD, Gaussian GA, Hermite-Gaussian of the order one of Methfessel-Paxton MP, and cold smearing of Marzari-Vanderbilt MV), as shown below. A cutoff wave vector of 4 and a non-shifted 12x12x12 k-point grid are used.

../_images/etot_al_gmax4_k12x12x12_sigma.png

We can see that the total energy depends significantly on the smearing width with Fermi-Dirac and Gaussian, whereas the total energy is not sensitive to the width with Hermite-Gaussian and cold smearing.