The first Brillouin zone can be mapped out by a continuous set of
points, , throughout that region of reciprocal
space (*k*-space). The occupied states at each *k*-point contribute to
the electronic potential of the bulk solid. Since the set is
dense, there are an infinite number of *k*-points in the Brillouin zone
at which the wavefunctions must be calculated. Therefore if a
continuum of plane wave basis sets were required, the basis set for any
calculation would still be infinite, no matter how small the plane wave
energy cut-off was chosen.

For this reason electronic states are only calculated at a set of
*k*-points determined by the shape of the Brillouin zone compared to
that of its irreducible part. The reason that this can be done is that
the electronic wavefunctions at *k*-points that are very close together
will almost be identical. It is therefore possible to represent the
electronic wavefunctions over a region of reciprocal space at a single
*k*-point. This approximation allows the electronic potential to be
calculated at a finite number of *k*-points and hence determine the
total energy of the solid.

The error incurred by this approximation can be made arbitrarily small by
choosing a sufficiently dense set of *k*-points. This is easily made
clear by the use of an example. Each *k*-point has energy eigenvalues
associated with it which is usually presented in the form of a band
structure - the band structure of GaAs in the zincblende structure along
several lines through the Brillouin zone is given in Figure
3.2[39].

**Figure 3.2:** The band structure of GaAs (zincblende) along several lines of
high symmetry.

At each *k*-point the energy eigenvalues are averaged, which gives a
measure of the energy associated with the electrons which occupy that
region of *k*-space. These average energies are given in Figure
3.3.

**Figure 3.3:** Averaged energy eigenvalues of GaAs corresponding to the above
band structure.

It can be seen that there is a large variation in energy over the
Brillouin zone. In this case it is found that the average band energy
is -4.8eV with a standard deviation of 1.7eV. This shows that there is
a large dispersion in the band energies of GaAs in the zincblende
structure. Therefore, in order to converge the total energies of the
structure to within 0.01eV it seems that *k*-points randomly
spread throughout the first Brillouin zone is necessary. Fortunately,
there are more efficient methods of choosing the *k*-point set in which
to sample the Brillouin zone.

Methods have been devised for obtaining very accurate approximations to
the electronic potential from a filled electronic band by calculating
the electronic wavefunctions at special sets of *k*-points. The two
most common methods are those of Chadi and Cohen[40] and
Monkhorst and Pack[41] which will be described below.

Thu Oct 31 19:32:00 GMT 1996