For a periodic system, the k points appearing in the wave function belong to the first Brillouin zone, by virtue of the Bloch's theorem. In the case of samples with defects, which are by definition aperiodic, the cell that contains the defecs is periodically repeated, using periodic boundaries. The Bloch theorem can be therefore applied to this supercell, the dimension of the Brillouin zone being determined by the dimension of the supercell itself, i. e. the larger the supercell, the smaller the Brillouin zone should be.
To calculate the energy band, or the charge density for example, the sum over these k points has to be done. Therefore, choosing a sufficiently dense mesh of summation is crucial for the convergence of the results. Monkhorst and Pack  proposed a scheme where the k are distributed homogeneously in the Brillouin zone according to
Usually, total energies of different stuctures are compared. Therefore, if the two structures have the same unit cell, the same set of k-points should be used. Since only the difference in the energies of the two structures is required, possible errors from a non-converged k-point sampling may cancel out. The computational effort could therefore be reduced by using a carefully chosen and small k-point set.
An alternative method for choosing k-point mesh has been proposed by Chadi and Cohen , on the basis of ``shells'' analysis. This concept can be explained by considering the Bloch funtion for a specific band (see equation 7.7)
To find these special k-points, we first consider the 48
possible symmetry operations T on one k-point. Each new k-point generated leads to a new charge density
Therefore, we obtain from equation (7.32)