Chemistry Reference
In-Depth Information
perfect fcc solid. These symmetries mean that the integrals in reciprocal space
do not need to be evaluated using the entire BZ, instead they can just be eval-
uated in a reduced portion of the zone that can then be extended without
approximation to fill the entire BZ using symmetry. This reduced region in
k
space is called the
irreducible Brillouin zone
(IBZ). For very symmetric
materials such as the perfect fcc crystal, using just the IBZ greatly reduces
the numerical effort required to perform integrals in
k
space. For example,
for the 10
10
10 Monkhorst-Pack sampling of the BZ, only 35 distinct
points in
k
space lie within the IBZ for our current example (compared to
the 1000 that would be used if no symmetry at all was used in the calculation).
Table 3.2 lists the number of
k
points in the IBZ for each calculation.
Comparing these with the timings also listed in the table explains why pairs
of calculations with odd and even values of
M
took the same time—they
have the same number of distinct points to examine in the IBZ. This occurs
because in the Monkhorst-Pack approach using an odd value of
M
includes
some
k
points that lie on the boundaries of the IBZ (e.g., at the
G
point)
while even values of
M
only give
k
points inside the IBZ. An implication of
this observation is that when small numbers of
k
points are used, we can
often expect slightly better convergence with the same amount of compu-
tational effort by using even values of
M
than with odd values of
M
.
†
Of
course, it is always best to have demonstrated that your calculations are well
converged in terms of
k
points. If you have done this, the difference between
even and odd numbers of
k
points is of limited importance.
To show how helpful symmetry is in reducing the work required for a
DFT calculation, we have repeated some of the calculations from Table 3.2
for a four-atom supercell in which each atom was given a slight displacement
away from its fcc lattice position. These displacements were not large—they
only changed the nearest-neighbor spacings between atoms by
0.09
˚
,
but they removed all symmetry from the system. In this case, the number of
k
points in the IBZ is
M
3
+
2. The results from these calculations are listed in
Table 3.3. This table also lists
DE
, the energy difference between the
symmetric and nonsymmetric calculations.
Although the total computational time in the examples in Table 3.3 is most
closely related to the number of
k
points in the IBZ, the convergence of the
calculations in
k
space is related to the density of
k
points in the full BZ. If
we compare the entries in Table 3.3 with Fig. 3.2, we see that the calculation
for the nonsymmetric system with
M
8 is the only entry in the table that
might be considered moderately well converged. Further calculations with
larger numbers of
k
points would be desirable if a highly converged energy
for the nonsymmetric system was needed.
=
†
In some situations such as examining electronic structure, it can be important to include a
k
point at the
G
point.
