Chemistry Reference
In-Depth Information
1
2
S
10
C
i
3
C
1
Figure 33.
Three of the 30026 H-bond isomers of the (H
2
O)
20
dodecahedron [25]. Structure (1)
is a high-symmetry (
S
10
), and also high-energy structure. Structure (2) is of
C
i
symmetry, and structure
(3), a very stable structure, has no symmetry (
C
1
).
Graph invariants, constructed using the point group operations of a finite cluster,
provide a precise way to link scalar physical properties of clusters to their H-bond
topology.
C
r
α
I
r
=
g
α
(
b
r
)
(44)
C
rs
α
I
rs
=
g
α
(
b
r
b
s
)
(45)
C
rst
α
I
rst
=
g
α
(
b
r
b
s
b
t
)
(46)
···
In the above equations, the sums are over all elements
g
α
of the cluster point group,
and Eqs. (44)-(46) generate first-, second-, and third-order invariants, respectively,
for finite water clusters. In line with our experience for ice, we have never found
the need to go further than second-order invariants to describe water clusters.
The (H
2
O)
20
dodecahedron provides an example of the ability of graph invari-
ants to organize and describe scalar physical properties among H-bond isomers.
