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 [104] proposed a scheme
where the **k** are distributed homogeneously in the Brillouin zone
according to

(7.28) |

where are the reciprocal lattice vectors, and

(7.29) |

where

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 [105], on the basis of ``shells''
analysis. This concept can be explained by considering the Bloch
funtion for a specific band (see equation 7.7)

(7.30) |

The charge density for this specific state is therefore

(7.31) |

which can be rewritten as

where the prime in the second sum is over all

(7.33) |

One can see that the first term in equation (7.32) is the average charge density for the band considered. It could therefore be a reliable approximation to the total charge density, providing a good choice of

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)

The T operations on do not change the norm of the vectors. Therefore, the first term in equation (7.34) would be a good approximation to the charge density if the

(7.35) |

where