Chemistry Reference
In-Depth Information
Figure 4.
The full molecular geometry of a cubic water cluster with
D
2
d
symmetry is shown in
(a). In (b) and (c) are two representations of the H-bond topology in terms of directed graphs. In (b)
representation, the depiction of the vertices are faithful to the molecular geometry. The same H-bond
topology is captured by the graph in (c).
behavior. There are 14 symmetry-distinct ways to connect waters in a cubic ar-
rangement [25]. Among those, the
D
2
d
and
S
4
clusters (structures
1
and
2
of Fig. 3)
are the lowest energy isomers [26], and the only ones observed experimentally [27].
While the molecular coordinates completely specify the unit cells of Fig. 2 or
clusters of Fig. 3, it is natural to distill the H-bond topology from the molecular
geometry and ask to what extent physical properties can be predicted on the basis
of the H-bond topology alone, as first conceived by Radhakrishnan and Herndon
[28]. The abstraction from full atomic coordinates to the H-bond topology can
be described in several ways. The topology can be summarized using directed
graphs [29, 30], either for ice crystals (Fig. 2b) or finite clusters (Fig. 4), in which
each vertex corresponds to a water oxygen and an arrow connecting two ver-
tices indicates the presence of a H-bond and the direction from H-bond donor to
H-bond acceptor. The ice rules require that all vertices in defect-free ice have two
incoming and two outgoing bonds. Another language to describe the mapping of
deep local minima of the potential surface to patterns of H-bond connectivity ab-
straction is to describe a mapping to a spin-lattice model [31, 32]. In fact, there
exist magnetic compounds, known as spin ices, in which the electronic spins obey
ice rules [33-35].
Is there
utility
in abstracting from the atomic coordinates to the H-bond connec-
tivity? Of course, order parameters for order-disorder transitions like ice Ih/XI can
be cast in the language of topological properties. However, can we make physical
predictions based on the correspondence between deep minima on the potential
energy surface of water molecules and directed graphs or spin configurations? In
this chapter, we will hopefully build a convincing case that the answer to these
questions is affirmative. As a preview, consider the fact that, to date, there is no
empirical potential function for water that describes the small energy differences
among H-bond isomers in various phases of ice [36]. While it may be feasible to
perform electronic structure calculations on several H-bond isomers for a small
