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## -point Sampling

By virtue of Bloch's theorem, any real-space integral over a periodic system with infinite extent can be replaced by an integral in reciprocal-space over the (finite) first Brillouin zone. However this still entails calculating the periodic functions at an infinite number of points in reciprocal space, which will be referred to as -points. This is a consequence of the infinite number of electrons. This problem can be overcome by exploiting the fact that electron wavefunctions do not change appreciably over a small distances in -space, therefore the integrations can be performed as summations over a finite, but sufficiently dense, mesh of -points. So, any integrated function , such as the density or total energy, can be computed as a discrete sum,

 (1.59)

where is the Fourier transform of , is the cell volume and are weighting factors. The number of -points required for a sufficiently accurate calculation must be ascertained by -point sampling - a procedure in which the total energy of the system is converged with respect to increases in the -point mesh density.

The positions of the -points within the Brillouin zone must be carefully selected since a judicious choice will result in an efficient description of a particular system, leading to quite significant computational savings. Different approaches for obtaining these optimal or special'' -point sets have been discussed in the past [21,22,23]. However the calculations performed in this work employ the Monkhorst-Pack method [24], whereby the -points are distributed homogeneously throughout space in rows and columns that follow the shape of the Brillouin zone, i.e.

 (1.60)

where are the reciprocal lattice vectors, and,
 (1.61)

where are the lengths of the reciprocal lattice vectors and characterises the number of special points in the set.

Typically, the point-group symmetry of the crystal is used to produce a smaller subset of the full special -point set, containing points located within the irreducible part of the Brillouin zone. The values of the weighting factors are adjusted according to this new -point set and the integrals (1.61) are calculated with this set. This results in a significant reduction in the computational expense since a smaller number of -points are used in the summations.

Next: Pseudopotentials Up: Plane-Wave Implementation of DFT Previous: Kohn-Sham Equations in Plane-Wave   Contents
Dr S J Clark
2003-05-04