Chemistry Reference
In-Depth Information
cell, would be more accurate. In the following paragraph, we illustrate that each
of the invariants appearing in Eq. (10) also appears as an invariant of the 4
4 cell
[as predicted in general by Eq. (30)], and so the
α
's of Eq. (10) determined for the
smaller cell provide information about the 4
×
×
4 cell.
Just like
I
4
×
4
1
a,
3
a
in Eq. (31), each of the graph invariants given below in
Eqs. (33)-(36) has a counterpart in among those of the 2
×
2 unit cell, specifi-
cally in Eqs. (6)-(9).
b
2
α
b
3
β
+
b
5
α
b
8
β
1
64
I
4
×
4
2
a,
3
a
=
b
1
α
b
4
β
+
b
6
α
b
7
β
+
(33)
α,β
=
a,b,c,d
⎧
⎨
1
32
I
4
×
4
1
a,
2
a
=
(
b
1
α
b
2
α
+
b
3
α
b
4
α
+
b
5
α
b
7
α
+
b
6
α
b
8
α
)
⎩
α
=
a,b,c,d
b
1
α
b
2
β
+
b
4
α
b
3
β
+
b
2
α
b
1
β
+
b
3
α
b
4
β
+
(
α,β
)
=
(
a,b
)
,
(
c,d
)
⎫
⎬
b
5
α
b
7
β
+
b
8
α
b
6
β
+
b
7
α
b
5
β
+
b
6
α
b
8
β
+
(34)
⎭
(
α,β
)
=
(
a,c
)
,
(
b,d
)
1
64
I
4
×
4
1
a,
5
a
=
(
b
1
α
b
5
α
−
b
3
α
b
5
α
−
b
1
α
b
6
α
+
b
2
α
b
6
α
α
=
a,b,c,d
+
b
3
α
b
6
α
−
b
4
α
b
6
α
+
b
3
α
b
7
α
−
b
3
α
b
8
α
+
b
4
α
b
8
α
)
b
1
α
b
8
β
b
1
α
b
7
β
+
+
b
8
α
b
1
β
−
b
2
α
b
8
β
−
b
8
α
b
2
β
−
b
7
α
b
1
β
−
(
α,β
)
=
(
a,c
)
,
(
b,d
)
b
5
α
b
4
β
+
b
7
α
b
4
β
+
b
4
α
b
5
β
−
b
2
α
b
5
β
−
b
5
α
b
2
β
−
b
4
α
b
7
β
−
(
α,β
)
=
(
a,b
)
,
(
c,d
)
⎬
b
2
α
b
7
β
+
b
7
α
b
2
β
+
(35)
⎭
(
α,β
)
=
(
a,d
)
,
(
c,b
)
b
1
α
+
b
8
α
(36)
32
α
1
I
4
×
4
b
2
α
+
b
3
α
+
b
4
α
+
b
5
α
+
b
6
α
+
b
7
α
+
1
a,
1
a
=
=
a,b,c,d
Each of the invariants listed so far for the 4
4 unit cell involves products of bonds
that lie sufficiently close to each other so that they also generate an invariant for
the smaller 2
×
2 cell, and their contribution to scalar physical properties can be
estimated from calculations for the smaller 2
×
×
2 cell. In other words, if the
α
's in
Eq. (10) were determined for the 2
2 cell, then an estimate for the properties of
the larger number of H-bond isomers of the 4
×
×
4 cell would be available.
