JOURNAL OF CHEMICAL PHYSICS
VOLUME 118, NUMBER 8
22 FEBRUARY 2003
Short H-bonds and spontaneous self-dissociation in „H2 O… 20 :
Effects of H-bond topology
Jer-Lai Kuo
Department of Chemistry, Ohio State University, Columbus, Ohio 43210
Cristian V. Ciobanu
Department of Physics, Ohio State University, Columbus, Ohio 43210
Lars Ojamäe
Department of Chemistry, IFM, Linköping University, SE-581 83 Linköping, Sweden
Isaiah Shavitt
Department of Chemistry, University of Illinois, Urbana, Illinois 61801
Sherwin J. Singer
Department of Chemistry, Ohio State University, Columbus, Ohio 43210
共Received 21 October 2002; accepted 25 November 2002兲
There are 30026 symmetry-distinct ways to arrange 20 water molecules in a dodecahedral cage with
nearly optimum hydrogen bond lengths and angles, analogous to the arrangements that give rise to
the zero-point entropy in ice-Ih. The energy of hydrogen bond isomers in (H2 O) 20 , assumed to be
similar in the past, differs by up to 70 kcal/mol. The isomers differ widely in their hydrogen bond
lengths, some exhibiting bond lengths as short as ⬃2.4 Å. The differences among the isomers
extends to their chemical properties: In some arrangements one or more water molecules
spontaneously self-dissociate, giving rise to spatially separated excess proton and hydroxyl ion units
in the cluster. Isomers that exhibit these unusual properties can be identified by features of their
hydrogen bond topology. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1538240兴
I. INTRODUCTION
hydronium ion in its interior has been proposed by Kassner
and Hagen7 to explain the exceptional abundance of
H⫹ (H2 O) 21 in molecular beam experiments.8,9 Several key
pieces of experimental evidence support this hypothesis.
Most notably, Castleman and co-workers found that 10 trimethylamine molecules bind preferentially to H⫹ (H2 O) 21
clusters, in accord with the 10 dangling hydrogens characteristic of the dodecahedral cage.9 However, no theoretical calculation has found that either dodecahedral (H2 O) 20 or
H⫹ (H2 O) 21 is exceptionally stable.2–5,10–15 All calculations
performed to date for H⫹ (H2 O) 21 find that placing a hydronium inside a dodecahedral cage leads to highly distorted
structures.3–5 Therefore, the structure of (H2 O) 20 and
H⫹ (H2 O) 21 remain issues of great interest in the study of
water clusters.
We have previously determined that 30026 symmetrydistinct H-bond arrangements are possible in the (H2 O) 20
dodecahedron.16 Several of the many possible isomers are
shown in Fig. 1. Each of the water molecules in the dodecahedron is either a double-donor water 共2DW兲, in which both
of its hydrogens are involved in an H-bond, or a doubleacceptor water 共2AW兲, in which one hydrogen participates in
an H-bond and the other hydrogen dangles from the cage. It
would seem that the 30026 distinct isomers of the (H2 O) 20
dodecahedron, all with 20 three-coordinate waters in a similar geometry and 30 H-bonds, should have similar characteristics, but we will show that this is hardly the case. Long
before water clusters were studied, multiple H-bond arrangements for a given lattice of oxygen atoms were recognized to
Commonly held wisdom asserts that each hydrogen
bond between water molecules stabilizes a structure by ⬃5
kcal/mol.1 Therefore aqueous clusters that differ only by the
direction of hydrogen bonds, but otherwise have the same
number of H-bonds and placement of oxygen atoms should
have approximately the same energy. This belief implicitly
lies behind several calculations performed to date2– 6 for the
(H2 O) 20 dodecahedron and the associated H⫹ (H2 O) 21
formed by adding a hydronium ion. In each of those studies,
only one or a handful of arbitrarily chosen hydrogen bond
arrangements were considered. The H-bond arrangement in
the dodecahedral cage was presumed to have a minor effect
on the properties of this cluster.
In this work we show that H-bond topology strongly
affects the structure and energy of (H2 O) 20 . Furthermore,
we find that the H-bond arrangement strongly affects the
chemistry of (H2 O) 20 . In fact, we report that certain seemingly unremarkable arrangements of the H-bonds in
(H2 O) 20—no unusually strained bond lengths or angles—
leads to spontaneous self-dissociation of a water molecule,
producing spatially separated excess proton and hydroxide
ion units in the cluster. 共The terms autoionization and autoprotolysis are also used in the literature to designate the selfdissociation of water into ionic fragments.兲 This is the first
report of such water cluster configurations that can proceed
to a lower energy state through self-dissociation.
The (H2 O) 20 dodecahedron is a well known building
block of type I ice clathrates. The (H2 O) 20 cage containing a
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© 2003 American Institute of Physics
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3584
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
Kuo et al.
be a key feature of ice-Ih, ordinary ice. In 1935 Pauling
estimated,17 later confirmed with remarkable accuracy,18,19
that ( 23 ) N H-bond arrangements are possible for N water molecules in ordinary hexagonal ice, which is consistent with the
experimental residual entropy of ice at 0K. 20 The question of
whether H-bond arrangements are completely random and
the nature of a phase transition to an ordered structure are
matters of current interest.21–30 Therefore, understanding
how H-bond topology affects water clusters is also relevant
to our understanding of ice and liquid water. Finite cluster
studies, like ours, are most relevant to the surface of ice,
where less than 4-coordinate water molecules exist.
II. CLUSTER STABILITY AND H-BOND TOPOLOGY
FIG. 1. Isomers of the (H2 O) 20 dodecahedron: 共a兲 most stable isomer; 共b兲
least stable isomer yet observed; 共c兲,共d兲 two isomers with five nearest neighbor 2AW’s 共discussed below in relation to Fig. 2兲; 共e兲,共f兲 isomers which
have formed zwitterions.
Studies of Cs⫹ (H2 O) 20 by Smith and Dang,14 as well as
our studies16 of the 30026 isomers of the (H2 O) 20 dodecahedron, have previously suggested that nearest neighbor
2AW’s, that is, nearest neighbor dangling hydrogens, are
strongly disfavored. In these previous studies, either empirical potential models or semiempirical electronic structure
methods were employed. Here we confirm this effect with
electronic density functional theory 共DFT兲 calculations.31 In
the left panel of Fig. 2 both the OSS2 empirical potential32,33
and B3LYP density functional theory calculations34,35 exhibit
a generally increasing trend in energy when plotted against
the number of nearest neighbor 2AW’s. Even when the number of neighboring 2AW’s is fixed, there is considerable
variation of the energy, indicating there are other significant
factors determining the stability of these clusters, factors that
FIG. 2. Energy of (H2 O) 20 isomers plotted on the left against the number of nearest neighbor dangling hydrogen pairs or 2AW’s. The energies are calculated
using the OSS2 empirical model 共Refs. 32,33兲 共light gray points兲, or B3LYP electronic density functional theory 共Refs. 34,35兲 with the cc-pVDZ basis set
共Ref. 42兲 共black dots兲. The B3LYP energy points marked 共c兲 and 共d兲 correspond to isomers with those labels in Fig. 1. From the 30026 symmetry-distinct
structures, we selected for electronic density functional theory calculations predominantly those structures that are predicted by the OSS2 model to be either
near the top or the bottom of the energy range, with fewer cases in between. On the right, the OSS2 energy is compared with the B3LYP energy 共dots兲. The
gray line indicates perfect agreement. When the number of nearest neighbor 2AW’s is large, we encountered isomers in which the H-bond topology changed
significantly upon B3LYP optimization, or for which a dangling hydrogen rotated to point toward the interior of the cage. These isomers were excluded from
the plot.
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J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
we identify below. In other work we have calculated the
energy of all 30026 isomers using the OSS2 model. We demonstrated that graph invariants, symmetry-invariant functions
of bond variables, provide a more sophisticated and successful way to link energy to H-bond topology in the isomers of
the (H2 O) 20 dodecahedron.36 While it is not feasible to perform electronic structure calculations for all 30026 isomers
of the dodecahedron, the right panel of Fig. 2 does verify the
general agreement between quantum calculations and the
OSS2 model, although it appears that the OSS2 model underestimates the energy gap between the least and most
stable structures.
The topology of the dodecahedron allows a minimum of
three nearest neighbor 2AW’s 关Fig. 1共a兲兴, which comprise the
most stable class of structures, and a maximum of 10 关Fig.
1共b兲兴, generally the least stable group.16 Both the OSS2
model and quantum chemical studies indicate that the range
in energy between the most and least stable H-bond isomers
is very large, about 50 kcal/mol in the OSS2 model and 70
kcal/mol in B3LYP calculations. This is a surprisingly large
difference, considering that many of the previous theoretical
treatments of the (H2 O) 20 dodecahedron only considered either one or a few arbitrary structures, assuming they were
roughly equivalent in their properties.2– 6 For example, the
dodecahedron pictured in work by Laasonen and Klein5 is a
member of the least stable class of structures, those with 10
nearest neighbor dangling hydrogen pairs. The structure pictured by Lee et al.6 contains four nearest neighbor dangling
hydrogen pairs, only one pair away from the optimum value.
Wales and Hodges,37 using the TIP4P empirical potential
model38 and the powerful basin-hopping search method,39
found a lowest energy (H2 O) 20 dodecahedron with just three
nearest neighbor dangling hydrogen pairs. They did investigate other H-bond isomers of the dodecahedron located by
the basin-hopping search. Wales and Hodges estimated the
energy range of dodecahedral H-bond isomers to be 15.5
kcal/mol for the TIP4P potential and 24.4 kcal/mol for the
ASP-W4 model,40 considerably less than we find in the
present work.
Besides the (H2 O) 20 dodecahedron, the range of energies among the H-bond isomers of smaller water clusters has
previously been calculated. Tsai and Jordan41 found a spread
of 2.7 kcal/mol among eight isomers of the (H2 O) 8 cube, but
larger differences 共5.8 kcal/mol among a subset of six isomers兲 from MP2 calculations at the TIP4P-optimized geometry. McDonald et al.16 enumerated all 14 H-bond isomers of
the (H2 O) 8 cube, and calculated their energy using the OSS2
model,32,33 which was found to agree with previous MP2
calculations better than the TIP4P potential. They found a
spread of energies among the H-bond isomers of the (H2 O) 8
cube of 10.7 kcal/mol. Even on a per-molecule basis, these
energy ranges are small compared to our findings for the
(H2 O) 20 dodecahedron.
III. SELF-DISSOCIATION AND ZWITTERIONIC
STRUCTURES
Not only are some of the isomers of dodecahedral
(H2 O) 20 surprisingly high in energy, they are chemically reactive. An example of such behavior is shown in Fig. 3. This
Short H-bonds in (H2O) 20
3585
FIG. 3. Spontaneous self-dissociation in (H2 O) 20 . Structures shown were
calculated using B3LYP density functional theory, as described in Fig. 2.
The structure on the left is from a very flat portion of the potential surface
where there is a short H-bond 共2.425 Å兲, outlined in the figure. H2 O selfdissociates in this cluster with no barrier and, following charge migration
through the cluster, yields the locally stable zwitterionic structure shown on
the right, 14.3 kcal/mol below the starting energy, containing a hydroxide
ion and an excess proton held in an H5 O⫹
2 -like unit. At other levels of
theory, there is a small barrier to self-dissociation 共shown schematically by
the dashed curve兲.
isomer can lower its energy upon dissociation of a water
molecule to H⫹ and OH⫺ . According to both the OSS2 empirical potential32,33 and Hartree–Fock level ab initio calculations carried out by us, the neutral structure in Fig. 3 is a
local minimum with one unusually short H-bond, representing a partially ionized structure. Displacing the hydrogen of
that bond slightly in the direction of ionization leads, after
overcoming a very low barrier, to a lower minimum exhibiting charge separation. At other levels of theory, such as
B3LYP density functional theory34,35 with a double-zeta, correlation consistent 共cc-pVDZ兲 basis set,42 optimization from
the OSS2 neutral structure initially leads to a neutral structure. However, at the B3LYP level the gradients in the vicinity of the neutral structure become very small but not zero,
and the cluster eventually optimizes to a zwitterionic structure. Hence at this level the neutral structure corresponds to a
very flat region of the potential surface but not a true local
minimum. In the example shown in Fig. 3, self-dissociation
of a water molecule results in an energy drop of 14.3 kcal/
mol according to B3LYP-DFT with the cc-pVDZ basis set.
Since this result was surprising, we checked the energy difference between the same two structures using second order
Møller–Plesset43– 46 共MP2兲 ab initio electronic structure
theory with the cc-pVDZ basis set. We still found a strong
energy drop of 11.4 kcal/mol upon self-dissociation. The energy change upon self-dissociation is quite large. While the
precise value will depend upon level of theory and basis set,
we do not expect the qualitative conclusion to depend upon
the electronic structure method. After the dissociation, the
excess charge does not remain fixed but migrates until the
excess proton, symmetrically solvated as an H5 O⫹
2 unit in the
example of Fig. 3, and the hydroxide are in favorable
positions,47 which we observed to be 2DW’s adjacent to the
excess proton and 2AW’s adjacent to OH⫺ .
This observation then suggests that the most stable
(H2 O) 20 zwitterions contain the H3 O⫹ and OH⫺ linked by
H-bonds pointing 共according to the usual sign convention,
from donor to acceptor兲 as much as possible from the H3 O⫹
to the OH⫺ . This trend has previously been noted by Anick
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3586
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
for zwitterions of the (H2 O) 8 cube.48 One highly symmetric
class of such structures contains the H3 O⫹ and OH⫺ at opposite ends of the dodecahedron. This considerably restricts
the candidates for the most stable zwitterion, which we can
easily list using the enumeration methods we have described
elsewhere.16,36 The most stable zwitterion of this class 关Fig.
1共e兲兴 found using B3LYP density functional theory34,35 and
the cc-pVDZ basis set is only 8.87 kcal/mol above the most
stable neutral dodecahedral structure. The energy difference
for single point calculations at the same structures changes to
9.95 kcal/mol and 11.59 kcal/mol, respectively, for B3LYPDFT and MP2 calculations with the aug-cc-pVDZ basis
set.49 We also find other, less symmetric, zwiterionic minima,
such as the one shown in Fig. 1共f兲. In this example, only 7.23
kcal/mol above the most stable neutral structure according to
B3LYP-DFT with a cc-pVDZ basis, the excess proton and
hydroxide are contained in the configuration characteristic of
⫺
the isolated H5 O⫹
2 and H3 O2 ions. The energy difference
becomes 10.40 kcal/mol when the calculation is repeated for
the same structures with an aug-cc-pVDZ basis.
The enthalpy of dissociation of water to H⫹ and OH⫺ in
bulk water is 13.5 kcal/mol.1 In small aqueous clusters, local
zwitterionic minima containing spatially separated H3 O⫹
and OH⫺ units have been discovered, all lying roughly 20
kcal/mol above the neutral ground state.48,50–54 Therefore, it
is surprising that self-dissociation in a small cluster can either be spontaneous 共Fig. 3兲, or, relative to the most stable
neutral isomer, requires less energy than in bulk water 关Fig.
1共e兲兴. The surprising feature may alternatively be considered
to be the wide energy range of the H-bond isomers, allowing
neutral local minima to be interspersed with zwitterionic
structures. Again, while basis set and level of theory does
have significant quantitative impact on our results, the major
conclusions are not sensitive to these factors.
Kuo et al.
FIG. 4. Hydrogen bonds between three-coordinate waters fall into three
major classes 共top panel兲 in which 共a兲 a 2AW donates to a 2DW, 共b兲 a 2DW
donates to a 2AW, or either (c1 ) a 2AW donates to a 2AW or (c2 ) a 2DW
donates to a 2DW. Each major class is further broken into minor classes
according to the topological index ␰ 关Eq. 共1兲兴. Examples a1 , a2 , a3 , a4 , and
a5 共middle panel兲 illustrate H-bonds of type 共a兲 for which ␰⫽0, 1, 2, 3, and
4, respectively. The normalized bond length distribution accumulated for
H-bonds of type 共a兲 is shown in the bottom panel. The bond lengths were
obtained from the B3LYP/DFT optimized structures whose energies are
given in Fig. 2.
␰ ⫽ 共 number of 2DW neighbors of the H-bond donor兲
⫹共number of 2AW neighbors of the H-bond acceptor兲 .
IV. SHORT H-BONDS
Analysis of the H-bond topology of various (H2 O) 20 isomers reveals a pattern governing the occurrence of exceptionally short H-bonds, and a connection between this pattern
and the stability of protons or hydroxides embedded in the
dodecahedral cage. The H-bonds of the dodecahedron fall
into three major classes according to the two water molecules involved in the H-bond, and each major class is further broken into five minor classes according to the four
water molecules adjoining the H-bond. In the major class
with the shortest H-bonds 共2.43–2.61 Å兲, a 2AW donates to
a 2DW 关Fig. 4共a兲兴. The class with the longest H-bonds
共2.73–2.82 Å兲 has the opposite arrangement, a 2DW donates
to a 2AW 关Fig. 4共b兲兴. Finally, the class with intermediate
bond lengths 共2.58 –2.70 Å兲 are bonds between two 2DW’s
or two 2AW’s 关Figs. 4(c1 ) and 4(c2 )]. Bonds of the type
shown in Fig. 4(c1 ), nearest neighbor 2AW’s, have already
been associated with decreased cluster stability.
Each of the three classes of H-bond described in the
previous paragraph can be further refined into minor classes
according to the neighbors of the two water molecules participating in the H-bond in question. Let ␰ be a topological
index defined as follows:
共1兲
Figures 4(a1 ), 4(a2 ), . . . , 4(a5 ) give examples of a 2AW
donating to a 2DW with ␰ equal to 0, 1, 2, 3, and 4, respectively. Only bonds with ␰ ⫽0 and ␰ ⫽4 are unique. The others admit different placements of the neighbors for which ␰
has the same value. As shown in Fig. 4, there is a strong
correlation between the topological index ␰ and the length of
the H-bond for bonds in which 2AW donates to 2DW, the
types shown in Figs. 4(a1 ) – 4(a5 ). The same trend is observed for bonds of all the major classes 关Figs. 4共a兲– 4共c兲兴,
and it was previously pointed out for smaller water
clusters.55
According to our criterion for stability of the zwitterion,
H-bonds of the type shown in Fig. 4(a1 ) are the best candidates for proton transfer. Before proton transfer, the dangling
hydrogen of the donor is surrounded by two other dangling
hydrogens of neighboring 2AW’s, shown in Fig. 2 to be a
high energy arrangement. Transfer of a proton from donor to
acceptor creates an H3 O⫹ surrounded by two 2DW’s and
OH⫺ surrounded by two 2AW’s, the most stable configuration for a zwitterion. Indeed the reactivity of the bond pattern
of Fig. 4(a1 ) is born out in numerous calculations, in which
the pattern of Fig. 4(a1 ) either produces outright proton
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Short H-bonds in (H2O) 20
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
transfer and the spontaneous creation of a zwitterion, or an
exceptionally short H-bond which can now be understood as
incipient proton transfer.
Understanding the role of proton transfer in relieving the
instability of neighboring 2AW’s explains some of the large
energy differences observed in Fig. 2. The two configurations marked c兲 and d兲 in Fig. 2 are shown in Figs. 1共c兲 and
1共d兲. In both structures there are five nearest neighbor dangling hydrogen pairs. In Fig. 1共c兲, those neighboring 2AW’s
lie in a ring, shown at the top of the figure. All the dangling
hydrogens within that ring are part of H-bonds which are
unfavorable for proton transfer. There is no pathway for
structure 共c兲 to partially or fully transfer a proton to relieve
the strain of neighboring 2AW’s. By contrast, in structure 共d兲
there is an H-bond 关middle foreground of Fig. 1共d兲兴 with ␰
⫽0 of the type shown in Fig. 4(a1 ), the most favorable for
partial or full proton transfer. In structure 共d兲, a proton is
partially transferred: the original covalent OH bond has
stretched to 1.14 Å, the length of the H-bond OH is 1.29 Å,
and the OO distance is 2.43 Å. Many of the structures whose
DFT energy is shown in Fig. 2 are either zwitterions or exhibit partial proton transfer. Some are even double zwitterions, containing two hydroxide ions and two excess protons.
The OSS2 potential is able to capture these effects because it
is a dissociating water potential.
The relationship of energy and bond lengths to H-bond
topology has been considered in two recent papers by
Anick.56,55 He proposes an interesting H-bond enumeration
scheme,56 different from the one we have employed.16,36 The
topological features of the H-bond topology he chooses for
correlation with physical properties are chosen in an apparently ad hoc fashion, and in this respect we prefer the graph
invariants we have introduced which can be arranged in a
hierarchy of increasing complexity.36 Anick analyzed the dependence of bond length on H-bond topology in terms of the
arrangement of 2AW’s and 2DW’s. Although he approaches
the classification of H-bond types in a different manner, the
end result is a very similar trend to those we have found. In
particular, the pattern of fitting coefficients Anick gives for
the (H2 O) 8 cube in Table IV of Ref. 55 suggest that 2AW’s
共2DW’s兲 adjacent to the H-bond donor 共acceptor兲 modify the
length of the H-bond to approximately the same degree.57
Hence, the overall trend is well summarized by the topological parameter ␰ . The trend first discovered by Anick for the
(H2 O) 8 cube is more dramatic in the (H2 O) 20 dodecahedron.
Anick reports H-bond lengths as short as 2.53 Å, while for
(H2 O) 20 the bonds are as short as 2.42 Å and selfdissociation occurs.
V. DISCUSSION
The central finding of this work, that the H-bond topology has a major effect on the energy, and even on the chemical reactivity, of a small water cluster, has implications in
other situations where broken H-bonds also are present in
significant numbers. The surface of ice is characterized by
dangling hydrogens and variable H-bond topology. If local
variations in the H-bond topology at the surface of ice also
imply local variations in the chemical reactivity even fractionally as powerful as on the surface of the (H2 O) 20 dodeca-
3587
hedron, then the H-bond topology cannot be ignored. Since
the local H-bond topology is seen to control the propensity
for self-dissociation in aqueous clusters, this naturally suggests that the H-bond topology be monitored as a possible
reaction coordinate for self-dissociation in liquid water, although we recognize that chemical processes in water clusters may be quite different from liquid water at 300 K. Recent simulations have not revealed a link between selfdissociation and H-bond topology.58 Perhaps this work will
provide new ways to formulate the connection between
H-bond topology and the behavior of the ice surface and
liquid water. In both cases, the key issue is what role the
topology of the surrounding H-bond network plays in the
activation of aqueous chemical reactions.
ACKNOWLEDGMENTS
The authors gratefully acknowledge support from the
U.S. National Science Foundation 共CHE-0109243兲 and the
Swedish Research Council 共VR兲. The calculations were
made possible by resource grants from the Ohio Supercomputer Center, Swedish National Supercomputer Center
共SNAC - NSC兲 and the EMSL Molecular Science Computing Facility at Pacific Northwest National Laboratory.
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