Reciprocal Space Numerical Integration (KSpace)

The k-space integration can be controlled via the KSpace block key. Two different k-space integration methods are available: the Regular Grid (default) and the Tetrahedron Method.

Regular K-Space grid

By default BAND uses a regular grid to sample the Brillouin zone (BZ).

Automatic mode

The simplest way to adjust the quality of the k-space integration is via the Quality key:

  Quality [Basic|Normal|Good|VeryGood|Excellent]

(Default: Normal) With the automatic grid, the program will look at the size of a lattice vector. A big vector in real space, means a small one in reciprocal space. Thinking in real space lattice vectors the following intervals will be distinguished: 0-5 Bohr, 5-10 Bohr, 10-20 Bohr, 20-50 Bohr, and beyond. Here is the table explaining how many points will be used along a lattice vector.

range Basic Normal Good VeryGood Excellent
0-5 5 9 13 17 21
5-10 3 5 9 13 17
10-20 1 3 5 9 13
20-50 1 1 3 5 9
50-... 1 1 1 3 5

By preferring odd-numbered values we can use a quadratic interpolation method, and have the \(\Gamma\) point in the grid. It is then reasonable to assume a decaying error when going to a better quality setting.

User-defined regular grid

It is possible to manually define the number of k-space points along each reciprocal lattice vector:

  Grid n1 {n2} {n3}

For 1D periodic systems you should specify only n1, for 2D systems n1 and n2, and for 3D systems n1, n2 and n3.

Tetrahedron Method

The tetrahedron method can be useful when especially high symmetry points in the BZ are needed to capture the correct physics of the system, graphene being a notable example.

KSpace integer &
   Grid Symmetric

The parameter for numerical integration over the BZ is an integer value.

  • 1: absolutely minimal (only the \(\Gamma\) point is used)
  • 2/4/...: linear tetrahedron method
  • 3/5/...: quadratic tetrahedron method

The linear tetrahedron method is usually inferior to the quadratic tetrahedron method. Try 3 for a reasonable result, or 5 for higher precision.

General Remark: The tetrahedron method samples the irreducible wedge of the first BZ, whereas the regular grid samples the whole, first BZ. As a rule of thumb you need to choose roughly twice the value for the regular grid. For example kspace 2 compares to grid 4 4 4, kspace 3 to grid 5 5 5, etc.. Sticking to this rule the number of unique k-points will be roughly similar.