Chemistry Reference
In-Depth Information
so the translation subgroup is of order
N
x
N
y
N
z
[or obviously
N
x
N
y
foratwo-
dimensional lattice like square ice]. We use
x, y, z
to designate the crystal axes,
but nothing in our formalism requires that these axes be orthogonal.
Other invariants can be constructed similarly:
C
rs
α
I
rs
=
g
α
(
b
r
b
s
)
(14)
C
rst
α
I
rst
=
g
α
(
b
r
b
s
b
t
)
(15)
.
We take the normalization constant to be the inverse of the order of the group,
making the invariants intensive quantities.
1
#(
G
)
C
rs...
=
(16)
We refer to
I
r
as a first-order invariant,
I
rs
as a second-order invariant, and so
on. From the definition of invariants, it is obvious to see that
I
rs
=
I
sr
. More
generally, invariants with permuted subscripts are equivalent. When all bonds are
filled, all bond variables
b
r
=±
1. Therefore, we have
I
rr
=
constant
, as well as
I
rrstu
···
=
. We have previously shown that if a symmetry operation can bring
a single bond
b
r
into
I
stu
···
b
r
, the first-order invariant of
b
r
is identically zero [37].
More generally, if
g
α
(
b
r
)
−
b
s
,
I
r
and
I
s
are equivalent. Local constraints, for
example, ice rules, can cause further degeneracy.
Symmetry properties are manifested by a group of permutation operations map-
ping the set of vertices onto themselves. The space group of a crystal can be treated
as a finite group by invoking periodic boundary conditions. Consider a lattice with
=±
possibly nonorthogonal unit cell vectors
a
x
,a
y
,a
z
. Even though we label the cell
vectors with “
x, y, z
”, our expressions apply equally well to nonorthogonal basis
vectors. The full space group is designated as
G
.
, the crystallographic transla-
tional group, is generated by the elementary translation operators
τ
x
τ
y
τ
z
, where
τ
x
τ
y
τ
z
(
R
)
=
R
+
u
a
x
+
v
a
y
+
w
a
z
.
That is,
=
0
,
1
,...,N
z
−
1
(17)
We will always assume a large but finite crystal with periodic boundary conditions.
=
τ
x
τ
y
τ
z
|
u
=
0
,
1
,...,N
x
−
1
,v
=
0
,
1
,...,N
y
−
1
,w
τ
x
,τ
v
+
N
y
τ
u
+
N
x
x
τ
y
τ
w
+
N
z
z
τ
z
=
=
and
=
(18)
y
Hence,
becomes a finite group and #(
), the order of
,is
N
x
N
y
N
z
.
