Chemistry Reference
In-Depth Information
The space group
G
may be decomposed into cosets appropriate for either the
n
x
×
n
y
×
n
z
or 1
×
1
×
1 cells.
G
=
p
1
∪
p
2
∪
p
3
∪···
n
x
×
n
y
×
n
z
p
n
x
×
n
y
×
n
z
1
n
x
×
n
y
×
n
z
p
n
x
×
n
y
×
n
z
2
n
x
×
n
y
×
n
z
p
n
x
×
n
y
×
n
z
3
=
∪
∪
∪···
(29)
In Eq. (29), we have decomposed
G
into right cosets. For the full translation
subgroup, the choice between left and right cosets is irrelevant because
is a
normal subgroup of
G
, for which left and right cosets are identical. However,
n
x
×
n
y
×
n
z
might not be a normal subgroup of
G
, and the left and right cosets may
be distinct. In this case, decomposition into right cosets is the most convenient
choice because, according to Eq. (27), following the action of a coset representative
with any member of
n
x
×
n
y
×
n
z
leaves the
value
of the bond expression unchanged,
as explained in the discussion accompanying Eq. (27).
The application of
G
on a product of bond variables can be written using the
coset representatives of the enlarged
n
x
×
n
y
×
n
z
cell.
⎡
⎣
β
⎤
n
y
−
1
n
z
−
1
n
x
−
1
1
#(
G/
)
n
x
n
y
n
z
I
n
x
×
n
y
×
n
z
τ
x
τ
y
τ
z
⎦
(30)
=
p
β
(
b
r
b
s
...
)
rs...
u
=
0
v
=
0
w
=
0
∈
G/
The sum over coset representatives now includes translations that would be sym-
metry operations for the averaged X-ray crystallographic cell. In the enlarged cell,
these translations bring bond variables
b
r
into another one that may not have an
identical value because of H-bond disorder. Equation (30) is our main result for
non-primitive cells. It can be used to prove that any invariant for a smaller unit cell
is also an invariant for a larger unit cell [38]. This property enables us to parametrize
an expansion for the energy [Eq. (11)] of a large simulation cell using, say,
ab initio
calculations for smaller cells. Equation (30) also states that any new invariants in-
troduced as the unit cell is enlarged involve products of bonds separated by the
size of the enlarged cell. These interactions are more distant, and of less impor-
tance. This provides a natural hierarchy of approximations for decomposing the
dependence of tensorial physical properties on H-bond topology. The most local
and dominant effects would be captured by fitting to invariants at the level of the
small cell. If these effects are completely dominant, then physical properties for a
larger
n
x
×
n
z
cell would be accurately predicted in terms of invariants that
are from the smaller cell, summed over all portions of the
n
x
×
n
y
×
n
z
cell. Devi-
ations from this picture are used to parametrize physical properties in terms of the
invariants of still larger cells. This improved characterization could, in principle,
be tested at even larger levels until convergence is achieved.
n
y
×
