Chemistry Reference
In-Depth Information
Figure 3.3
Fermi Dirac function [Eq. (3.10)] with
k
0
¼
1 and several values of
s
.
being integrated to be continuous by “smearing” out the discontinuity. An
example of a smearing function is the Fermi -Dirac function:
¼
þ
1
1
k
k
0
k
k
0
f
exp
:
(3
:
10)
s
s
Figure 3.3 shows the shape of this function for several values of
s
. It can be
seen from the figure that as
s !
0, the function approaches a step function
that changes discontinuously from 1 to 0 at
k k
0
. The idea of using a smear-
ing method to evaluate integrals is to replace any step functions with smooth
functions such as the Fermi -Dirac function since this defines a continuous
function that can be integrated using standard methods. Ideally, the result of
the calculation should be obtained using some method that extrapolates the
final result to the limit where the smearing is eliminated (i.e.,
s !
0 for the
Fermi -Dirac function).
One widely used smearing method was developed by Methfessel and
Paxton. Their method uses expressions for the smearing functions that are
more complicated than the simple example above but are still characterized
by a single parameter,
s
(see Further Reading).
3.1.5 Summary of
k
Space
Because making good choices about how reciprocal space is handled in DFT
calculations is so crucial to performing meaningful calculations, you should
