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
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.
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.