Chemistry Reference
In-Depth Information
TABLE 3.3 Results from Computing the Total Energy of the
Variant of fcc Cu with Broken Symmetry
a
E/
atom
(eV)
DE/
atom
(eV)
No. of
k
Points in IBZ
M
t
M
=t
1
1
1.8148
0.009
1
1.0
2
3.0900
0.010
4
2.1
3
3.6272
0.008
14
5.6
4
3.6969
0.009
32
12.3
5
3.7210
0.009
63
21.9
6
3.7446
0.010
108
40.1
7
3.7577
0.010
172
57.5
8
3.7569
0.010
256
86.8
a
Described in the text with
M M Mk
points generated using the
Monkhorst Pack method.
You may have noticed that in Table 3.3 the column of energy differences,
DE
, appears to converge more rapidly with the number of
k
points than the total
energy
E.
This is useful because the energy difference between the two con-
figurations is considerably more physically interesting than their absolute ener-
gies. How does this happen? For any particular set of
k
points, there is some
systematic difference between our numerically evaluated integrals for a par-
ticular atomic configuration and the true values of the same integrals. If we
compare two configurations of atoms that are structurally similar, then it is
reasonable to expect that this systematic numerical error is also similar. This
means that the energy difference between the two states can be expected to
cancel out at least a portion of this systematic error, leading to calculated
energy differences that are more accurate than the total energies from which
they are derived. It is important to appreciate that this heuristic argument
relies on the two configurations of atoms being “similar enough” that the sys-
tematic error from using a finite number of
k
points for each system is similar.
For the example in Table 3.3, we deliberately chose two configurations that
differ only by small perturbations of atom positions in the supercell, so it is
reasonable that this argument applies. It would be far less reasonable, however,
to expect this argument to apply if we were comparing two significantly differ-
ent crystal structures for a material.
There are many examples where it is useful to use supercells that do not
have the same length along each lattice vector. As a somewhat artificial
example, imagine we wanted to perform our calculations for bulk Cu using
a supercell that had lattice vectors
a
1
¼ a
(1,0,0),
a
2
¼ a
(0,1,0),
a
3
¼ a
(0,0,4)
:
(3
:
9)
