Number of k-points, and method for smearing

Read and understand section 7.4 before reading this section.

The number of k-points necessary for a calculation depends critically on the necessary precision and on the fact whether the system is metallic. Metallic systems require an order of magnitude more k-points than semiconducting and insulating systems. The number of k-points also depends on the smearing method in use; not all methods converge with similar speed. In addition the error is not transferable at all i.e. a $9\times 9\times 9$ leads to a completely different error for fcc, bcc and sc. Therefore absolute convergence with respect to the number of k-points is necessary. The only exception are commensurable super cells. If it is possible to use the same super cell for two calculations it is definitely a good idea to use the same k-point set for both calculations.

k-point mesh and smearing are closely connected. We repeat here the guidelines for ISMEAR already given in section 6.37:

• For semiconductors or insulators always use tetrahedron method (ISMEAR=-5), if the cell is too large to use tetrahedron method use ISMEAR=0.

• For relaxations in metals always use ISMEAR=1 and an appropriated SIGMA value (so that the entropy term is less than 1 meV per atom). Mind: Avoid to use ISMEAR$>$0 for semiconductors and insulators, it might result in problems.

• For the DOS and very accurate total energy calculations (no relaxation in metals) use the tetrahedron method (ISMEAR=-5).

Once again, if possible we recommend the tetrahedron method with Blöchl corrections (ISMEAR=-5), this method is fool proof and does not require any empirical parameters like the other methods. Especially for bulk materials we were able to get highly accurate results using this method.

Even with this scheme the number of k-points remains relatively large. For insulators 100 k-points/per atom in the full Brillouin zone are generally sufficient to reduce the energy error to less than 10 meV. Metals require approximately 1000 k-points/per atom for the same accuracy. For problematic cases (transition metals with a steep DOS at the Fermi-level) it might be necessary to increase the number of k-points up to 5000/per atom, which usually reduces the error to less than 1 meV per atom.

Mind: The number of k-points in the irreducible part of the Brillouin zone (IRBZ) might be much smaller. For fcc/bcc and sc a $11\times 11\times 11$ containing 1331 k-points is reduced to 56 k-points in the IRBZ. This is a relatively modest value compared with the values used in conjunction with LMTO packages using linear tetrahedron method.

Not in all cases it is possible to use the tetrahedron method, for instance if the number of k-points falls beneath 3, or if accurate forces are required. In this case use the method of Methfessel-Paxton with N=1 for metals and N=0 for semiconductors. SIGMA should be as large as possible, but the difference between the free energy and the total energy (i.e. the term

 entropy T*S

in the OUTCAR file) must be small (i.e. $<$ 1-2 meV/per atom). In this case the free energy and the energy one is really interested in $E(\sigma \to 0)$ are almost the same. The forces are also consistent with $E(\sigma \to 0)$.

Mind: A good check whether the entropy term causes any problems is to compare the entropy term for different situations. The entropy must be the same for all situations. One has a problem if the entropy is $100~$meV per atom at the surface but $10~$meV per atom for the bulk.

Comparing different k-points meshes:

It is necessary to be careful comparing different k-point meshes. Not always does the number of k-points in the IRBZ increase continuously with the mesh-size. This is for instance the case for fcc, where even grids centered not at the $\Gamma$-point (e.g. Monkhorst Pack $8\times 8\times 8 \to 60$) result in a larger number of k-points than odd divisions (e.g. $9\times 9\times 9 \to 35$). In fact the difference can be traced back to whether or whether not the $\Gamma$-point is included in the resulting k-point mesh. Meshes centered at $\Gamma$ (option 'G' in KPOINTS file or odd divisions, see Sec. 5.5.3) behave different than meshes without $\Gamma$ (option 'M' in the KPOINTS file and even divisions). The precision of the mesh is usually directly proportional to the number of k-points in the IRBZ, but not to the number of divisions. Some ambiguities can be avoided if even meshes (not centered at $\Gamma$) are not compared with odd meshes (meshes centered at $\Gamma$).

Some other considerations:

It is recommended to use even meshes (e.g. $8\times 8\times 8$) for up to $n=8$. From there on odd meshes are more efficient (e.g. $11\times 11\times 11$). However we have already stressed that the number of divisions is often totally unrelated to the total number of k-points and to the precision of the grid. Therefore a $8\times 8\times 8$ might be more accurate then a $9\times 9\times 9$ grid. For fcc a $8\times 8\times 8$ grid is approximately as precise as a $8\times 8\times 8$ mesh. Finally, for hexagonal cells the mesh should be shifted so that the $\Gamma$ point is always included i.e. a KPOINTS file

automatic mesh 
0
Gamma
  8   8   6   
  0.  0.  0.

is much more efficient than a KPOINTS file with ``Gamma'' replaced by ``Monkhorst'' (see also Ref. 5.5.3).