The k-point sampling grid

The points in k-space at which the Brillouin zone is to be sampled can be defined in one of three ways.

As a list

%BLOCK KPOINTS_LIST
units

$\begin{array}{cccc} \begin{displaymath} R_{1i} & R_{1j} & R_{1k} & R_{1w} ... ... & \vdots & \vdots & \vdots \\ R_{ni} & R_{nj} & R_{nk} & R_{nw} \end{array}$

%BLOCK KPOINTS_LIST

The first three numbers on a line are the co-ordinates of a k-point as fractions of the relevent reciprocal space lattice vector. The final number is a weight for the k-point. The weights of all n k-points must add up to 1.

Another way is to use a Monkhurst-Pack grid and there are two ways of doing this.

The first way is to specify the grid dimensions in the three directions of the reciprocal space lattice vectors

KPOINTS_MP_GRID $I_{i}$ $I_{j}$ $I_{k}$

So, for example, we could have a 2x2x2 grid or a 3x4x1 grid. The choice depends upon computational resources and whether a more detailed sampling of the electronic wave function is required in a given reciprocal space direction.

A second way of requesting a Monkhurst-Pack grid is to specify a minimum grid density.

KPOINTS_MP_SPACING $R$ [units]

Where $R$ is the maximum distance between any two k-points in the grid. The default [units] of the k-point spacing are Å$^{-1}$.

Another feature of the Monkhurst-Pack grid that the user can specify is the alignment of the grid with respect to the origin of the Brillouin zone. By including

KPOINT_MP_OFFSET $R_{i}$ $R_{j}$ $R_{k}$

the Monkhurst-Pack grid can be offset from the origin of the Brillouin zone. The entries $R_{i}$ $R_{j}$ $R_{k}$ are the fractions of the reciprocal space lattice vector that give the value of the offset in the three directions.