The *k*-point sampling approximation is used by evaluating Equation
(3.18) with

where *v* is the unit cell volume,

where

and is the Fourier transform of ,
that is, the region of space which is occupied by electrons given by the
contribution of *k*-point *k*.
Here *k* runs over the a set of *k*-points and *i* labels the occupied
states and the integral over the Brillouin zone is replaced by a sum
over the discrete *k*-point set. This shows that a calculation to obtain
the charge density is required at each *k*-point in the set. To optimize
any total energy calculation it is necessary to compute the charge
density at a carefully chosen set of *k*-points which characterises the
shape of the reciprocal space of the cell. For example, the space group
operations of a given cell will map some *k*-point onto others - these
will have a related reciprocal space charge density[42],
therefore only a single computation is necessary.

This fact is used by Chadi and Cohen to generate a special set of
*k*-points at which the reciprocal space should be sampled. The *k*-point
set is generated as follows: picking two starting *k*-points
and satisfying certain uniqueness conditions[40] in
the reciprocal space of the cell with point group operations
then a new set of points can be generated by

The new set of points generated in this way can then be used in a similar process to generate larger sets.

This procedure does not generate *k*-points that are necessarily unique in
the irreducible part of the Brillouin zone. Through the symmetry
operations, T, the new set of *k*-points can be `folded back' into the
irreducible part. A normalised weighting factor can then be associated
with each point with their ratios indicating the number of times that
each point in the irreducible zone has been generated. The generation
procedure can be repeated until a sufficiently dense set of *k*-points is
achieved to give as small an error required by the approximation given
by Equation (3.27).

The Monkhorst-Pack scheme uses an alternate approach which gives an
identical set of *k*-points to Chadi and Cohen. It is based on generating
periodic functions based on the point group symmetries on a regular grid
of points in the Brillouin zone.

Thu Oct 31 19:32:00 GMT 1996