Chemistry Reference
In-Depth Information
12
N
N
ln
N
N
2
10
8
6
4
2
0
0
100,000
200,000
300,000
N
(number of graphs)
Figure 32.
The CPU time for enumeration of all symmetry-distinct H-bond topologies for a
(H
2
O)
20
dodecahedral water cluster is plotted against the number of graphs from which symmetry-
redundant configurations are eliminated [37]. Data is generated for a sequence of structures containing a
subset of the bonds, and then finally the full cluster. Total CPU time, including calculation of invariants
and sorting is plotted. Least-square fits clearly show that the computational cost scales as either
N
or
N
ln
N
.
P
used in Eqs. (40)-(43) must be taken as an average or effective number of groups.
The basic idea is confirmed, and evidence presented in Fig. 32 showing
N
ln
N
scaling in a realistic calculation.
V. WATER CLUSTERS
In common with the disordered phases of ice, there are families of water clusters
(H
2
O)
n
for which the oxygen atom positions are similar, and that differ in the
connectivity of the H-bonds. The 14 isomers of cubic (H
2
O)
8
shown in Fig. 3 are a
good example [25, 26]. There are 27, 96, 10, 70, and 194 H-bond topologies pos-
sible for the (H
2
O)
6
cage, book, prism, chair, and boat clusters, respectively [188].
All of the possible cage and prism topologies appear to correlate with actual local
minima on the potential energy surface of (H
2
O)
6
, while in other cases the fraction
of topologies that correspond to actual minima decreases with the openness (i.e.,
H-bond per water molecule) of the cluster. The beautiful (H
2
O)
20
dodecahedron
(Fig. 33) has received significant attention [189-196], despite the possibility that
it may nowhere exist as an isolated cluster. McDonald et al. [25] were the first
to determine that the (H
2
O)
20
dodecahedron possesses 30026 symmetry-distinct
H-bond isomers, and to elucidate their properties [37].
