ARTICLES
DOI: 10.1002/cphc.200((will be filled in by the editorial staff)) 

Hydrogen-Bond Networks in Water Clusters (H₂O)₂₀: An Exhaustive Quantum-Chemical 

Analysis 

Andrei M. Tokmachev, [a] Andrei L. Tchougréeff,[a,b] and Richard Dronskowski[a]

Water aggregates allow for numerous configurations due to different
distributions of hydrogen bonds. The total number of possible
hydrogen-bond networks is very large even for medium-sized
systems. We demonstrate that the targeted ultra-fast methods of
quantum chemistry make an exhaustive analysis of all configurations
possible. The cage of (H2O)20 in the form of the pentagonal
dodecahedron is a common motif in water structures. We calculated
the spatial and electronic structure of all hydrogen-bond
configurations for three systems: idealized cage (H2O)20 and defect
cages with one or two hydrogen bonds broken. More than 3 million
configurations studied provide unique data on the structure and
properties of water clusters. We performed a thorough analysis of the
results with the emphasis on the cooperativity in water systems and
the structure-property relations.

Водородные связи в водяных кластерах. Квантово-химический анализ

Introduction 

Most of anomalous properties of water are attributed to the
cooperative behaviour of strong hydrogen bonds (H-bonds)
between water molecules. Extended H-bond networks first
appear in water clusters. Many of water clusters are important
components of the atmospheric chemistry,[1] cloud and ice
formation, thereby linked to the earth's radiation balance and
precipitation patterns. Sometimes even liquid water is thought of
in terms of flickering water clusters[2] although this hypothesis is
debatable. 

Not all water clusters are equally stable and important,
however. Protonated clusters H+(H2O)n exhibit exceptional
stabilities for some "magic numbers" n. The smallest of such
numbers is n=21, and the enhanced stability of this cluster was
confirmed by numerous experiments based on different
experimental conditions and techniques.[3] It was suggested[4] that
H+(H2O)21 is a pentagonal dodecahedron with the H3O+ ion
trapped inside the cage. 

Titration of dangling hydrogen atoms with trimethylamine
(TMA) confirms this hypothesis: the cluster H+(H2O)20 forms a
complex with 11 molecules of TMA, while the cluster H+(H2O)21
can coordinate only 10 molecules of TMA.[5] It is also consistent
with the XPS spectrum of O 1s core level not exhibiting any
internal structure[6] and spectroscopic (IR) results,[7] pointing to a
highly symmetric structure formed by three-coordinated water
oxygen atoms. The pentagonal dodecahedra are probably highly
stable, being major structural elements for all three common
types of gas clathrate structures: sI, sII, and sH hydrates.[8]
The hypothetical character of the above structural
predictions calls for theoretical studies. If one considers clusters
of a fixed size n, different forms (morphologies) of the oxygen-
atom framework are possible. In the case of (H2O)20, four major
structural classes were proposed[9] (see Figure 1). Their relative
stabilities are determined by a fine balance between hydrogen
bonding and strains in the rings, and each of the classes was
predicted as an energy minimum.[10] The dodecahedral structure
can be stabilized due to a larger number of dangling O-H bonds
interacting with other molecules.

Figure 1. Major classes of water clusters (H2O)20: a) dodecahedron; b) edge-
sharing pentagonal prisms; c) fused cubes; d) face-sharing pentagonal prisms.
When the morphology of the cluster is defined, there is still
a lot of freedom for placing H atoms. Normally, the “ice rules”[11]
(basically requiring that water molecules are not ionized) are
imposed on the positions of dangling O-H bonds and directions of
H-bonds. The number of isomers is usually large even for
medium-sized water systems and each of them corresponds to
some local extremum on the potential energy surface: for
example, there are 30,026 symmetry-independent H-bond
arrangements in the case of the dodecahedral cluster (H2O)20. A
variety of methods (force fields,[12] DFT,[13] semiempirical[14] and
ab initio methods[15]) has been used to find and characterize the
H-bond networks with the lowest energy (or a few of them) for the
dodecahedral cluster. At the same time the energy difference
between the H-bond networks is relatively small and many of
them can be thermally populated, thus affecting the physical
properties of the cluster. Therefore, it is desirable to study a large
number of H-bond configurations, preferably all of them. The only
reported study of the whole set of H-bond isomers for this
cluster[16] is made by the OSS2 empirical force field. 

A full quantum-chemical analysis of all possible configurations
aimed to extract statistical data would be a great step forward in
understanding the H-bond networks. The development of highly
efficient linear-scaling methods[17] brings new possibilities to
large-scale calculations. Here we report the first exhaustive
quantum-chemical study of all symmetry-distinct H-bond
configurations of the dodecahedral cluster (H2O)20 as well as
more complex systems with the same morphology. Of course, the
interaction of the cage with the chemical environment, which is
normally the case, may significantly affect the stability of the H-
bond configurations or even bring a partial order to the positions
of H atoms but we believe that the regularities found and the
insights gained from the present analysis of the unperturbed
cluster are quite general. 

Results and Discussion 

Before starting to present the results of the calculations it is
necessary to discuss their potential accuracy. Although the
specialized ultra-fast method used in the present work well
reproduces the properties of small water systems (see the
“Computational Methods” section) there is an obvious question
about the reliability of the results. Cluster of 20 water molecules is
a very complex system. Reputable methods of computational
chemistry predict different most favourable morphologies,
different most stable H-bond networks, and different binding
energies.