REVIEWS

Water: From Clusters to the Bulk

Ralf Ludwig*
Dedicated to Professor Manfred Zeidler
on the occasion of his 65th birthday

Water is of fundamental importance conceptual models, which in them- ter models for liquid water try to mimic
for human life and plays an important selves reproduce the observed behav- the transition from these clusters to
role in many biological and chemical ior of the liquid. The exploration of bulk water. The important question is:
systems. Although water is the most structural and binding properties of What cluster properties are required to
abundant compound on earth, it is small water complexes provides a key describe liquid-phase behavior?
definitely not a simple liquid. It pos- for understanding bulk water in its
sesses strongly polar hydrogen bonds liquid and solid phase and for under- Keywords: ab initio calculations ´ hy-
which are responsible for a striking set standing solvation phenomena. Mod- drogen bonds ´ molecular clusters ´
of anomalous physical and chemical ern ab initio quantum chemistry meth- vibrational spectroscopy ´ water
properties. For more than a century the ods and high-resolution spectroscopy chemistry
combined importance and peculiarity methods have been extremely success-
of water inspired scientists to construct ful in describing such structures. Clus-

1. Introduction diffraction pattern and proposed models of liquid waterº,[6]

Stillingers ªWater Revisitedº,[7] and Mishima and Stanleys
Water has probably received more scientific and techno- review about ªThe relationship between liquid, supercooled
logical interest than any other substanceÐmainly for two and glassy waterº[8] should be emphasized.
reasons. Firstly, water is a major chemical constituent of our Recent progress in some fields of water science makes it
planets surface and as such it has been indispensable for the worthwhile to summarize important results obtained during
genesis of life. Secondly, it exhibits a fascinating array of the last five years. Topics concern the discovery of new ice
unusual properties in pure form and as a solvent. The phases,[9, 10] new insight into supercooled and glassy wa-
importance of water is clearly described in monographs such ter,[8, 11, 12] and a better understanding of high-mobility trans-
as ªProperties of Ordinary Water Substanceº by Dorsey port of water ions.[13, 14]
(1940),[1] the compendium ªWater. A Comprehensive Treatise, In this review I would like to focus on recent progress in
Volumes 1 ± 7º, edited by Franks (1972 ± 1982),[2] the book calculating and measuring water clusters and their properties.
ªMetastable Liquidsº by Debenedetti (1996),[3] and Balls This interest is mainly a consequence of the fact that
popular survey of the ªhistoryº of water ªH2OÐA Biography investigations on small water clusters are a perfect means
of Waterº (1999).[4] Those standard works go along with with which to characterize structural changes and bonding
thousands of reviews, proceedings, and articles which all show mechanisms in passing from isolated molecules to bulk states.
that water is one of the most appealing of the open puzzles in Thus we can ask whether their is a continuous path from the
science. Among those highlights, Bernal and Fowlers ªA gas to the liquid phase. Is there some evidence from experi-
theory of water and ionic solution, with particular reference to ments or theoretical models that gas-phase structures may
hydrogen and hydoxyl ionsº,[5] Narten and Levys ªObserved also be important constituents of the liquid phase?
An outline of this review is as follows: First we describe
[*] Priv.-Doz. Dr. R. Ludwig some of the anomalies of water. The fascinating array of
Physikalische Chemie unusual properties can be qualitatively understood from
Fachbereich Chemie der Universität Dortmund
Otto-Hahn-Strasse 6, 44221 Dortmund (Germany)
water-bonding characteristics. Thus we discuss hydrogen
Fax: (‡ 49) 231-755-3937 bonds (H-bonds) and their possible arrangements in water.
E-mail: ludwig@pc2a.chemie.uni-dortmund.de The crystal structures of hexagonal ice Ih and clathrate

Angew. Chem. Int. Ed. 2001, 40, 1808 ± 1827  WILEY-VCH Verlag GmbH, D-69451 Weinheim, 2001 1433-7851/01/4010-1809 $ 17.50+.50/0 1809
REVIEWS R. Ludwig

hydrates are presented as typical three-dimensional networks.

Many-body effects are crucial for the size and arrangement of
water complexes. Consequently we discuss recent theoretical
and experimental evidence for cooperativity in H-bonded
systems. This chapter is followed by a brief introduction into
two common water models: the mixture and the continuum
model approaches.
A survey of calculated water clusters ranges from small ring
structures up to icosahedral networks. The main features and
properties of the water structures are discussed in respect to
experimental findings and their possible relevance for water
models.
In the following chapter we describe exciting spectroscopic
methods which recently allowed the detection of small water
clusters: small quasi-planar ring structures (n ˆ 3 ± 5), isomer-
ic hexamers (n ˆ 6) representing the transition from cyclic to
three-dimensional structures, and larger clusters in the ªcageº
regime (n ˆ 7 ± 12). Finally we present recent cluster models
for liquid water. All models are based on calculated water
structures and it is assumed that these may be constituents of
the liquid phase. We discuss whether these models are able to Figure 1. Temperature dependence of the isobar density 1 (a)[16, 18] and the
thermal expansivity ap (b).[19]
explain the properties of liquid water including some of its
anomalies.

water expands by about 11 %. It is that process that allows a

sheet of ice to float on liquid water. Both effects, the density
2. Water Anomalies maximum and the negative volume of melting, cause lakes
and rivers to freeze from the top down.
The behavior of liquid water deviates strongly from that The isothermal compressibility kT[22] passes through a
expected of a simple liquid in almost every respect. The minimum in the normal liquid-water range at 319 K. It then
liquid-phase density maximum is the most prominent and increases further as the temperature decreases and becomes
publicized of the water anomalies. more pronounced in the supercooled region (Figure 2 a). The
The density of liquid water at atmospheric pressure same behavior is found for the heat capacity CP at constant
increases as it is cooled to 277 K, at which temperature the pressure.[23, 24] As shown in Figure 2 b, the minimum value of
density has a maximum value of 0.999972 g cmÿ3.[15, 16] As CP occurs at 308 K, which is right in the middle of the liquid-
shown in Figure 1 a, the density decreases rapidly below water range. Supercooling leads to a strong increase in the
277 K, a trend which continues if the liquid density is followed heat capacity. It takes more heat to raise the temperature of
into the supercooled region below the freezing point at water than to warm up most other substances by the same
273 K.[17±19] Although water is not the only liquid to exhibit a amount. This occurrence results in ocean circulation effects
density maximum, the phenomenon only appears in a few that strongly influence local and global climates.
other liquids, such as SiO2[20] and Ga[21] melts. Water also The dynamic properties of water also show strong devia-
possesses a negative volume of melting (Figure 1 b). The tions from simple liquid behavior. The diffusion constant D
density of most liquids increases as they freeze, however, normally decreases as pressure increases at constant temper-

Ralf Ludwig, born in 1961 in Gladbeck, Germany, studied physics and graduated from the
Rheinisch-Westfälische Technische Hochschule in Aachen with a diploma in 1988. Three years
later he received his PhD in physical chemistry under the guidance of Prof. Manfred Zeidler.
He received a research stipend from the Heinrich-Hertz-Stiftung of the state of North-Rhine-
Westfalia and worked with Prof. Tom C. Farrar as a postdoctoral fellow in the chemistry
department of the University of Wisconsin in Madison, where he improved his skills in liquid
NMR spectroscopy. During his two-year stay in Madison he also conducted theoretical studies
with Prof. Frank Weinhold. In 1995 he returned to Germany and joined the research group of
Prof. Alfons Geiger at the University of Dortmund. There, he extended his methodological
spectrum by molecular dynamics simulations and obtained his habilitation in physical
chemistry in 1999 for studying hydrogen bonding in clusters, liquids, and oligopeptides. His
current research focuses on the structure and dynamics of molecular clusters, pure liquids, and
aqueous solutions of biophysical interest.

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Water Clusters REVIEWS
EA for the molecular motion increases with decreasing
temperature, which indicates there is a change in the
mechanism of molecular mobility.[27] Many other dynamic
properties of the liquid, such as the dielectric relaxation
time[28] and the nuclear spin relaxation times,[29, 30] show the
same kind of accelerating deviation from simple liquid
behavior as the liquid is cooled far from the glass transition
temperature.
Another surprising characteristic of water is its preferential
orientation in the hydration shell of nonpolar solutes and
nonpolar side groups attached to biopolymers. The structure
adopted by liquid water which is in close proximity to
nonpolar solutes is a fundamental characteristic of modern
theories of hydrophobic hydration and hydrophobic effects,
which are of great relevance to our understanding of many
important chemical and biological processes.[31]
Placing a solute molecule in liquid water leads to a
rearrangement of the random H-bond network. Besides
making some space for the guest molecule, water tries to
strengthen its network around the nonpolar solute. This can
Figure 2. Temperature dependence of the isothermal compressibility kT best be done by placing its tetrahedral bonding directions in a
(a)[21] and the constant-pressure specific heat Cp (b).[22, 23]
straddling mode[32] as shown in Figure 4 (left). The water

ature. Again, liquid water displays a reversal of this behavior.

As shown in Figure 3 a, the diffusion coefficient actually
increases with increasing pressure up to about 200 MPa,
above which the usual decrease in D is observed.[25] Cold
water gets more fluid when it is squeezed, whereas most

Figure 4. Water molecules next to a nonpolar solute. Each water molecule

prefers to place its tetrahedral bonding directions in a straddling mode
(left). A possible full arrangement of such a H-bonded network is shown
for a crystal structure of a clathrate hydrate (right).

molecule is aligned with three tetrahedral directions tangen-

tial to the surface of the occupied space in order to preserve
the maximum number of H-bonds. This orientational restraint
leads to a negative entropy contribution for the solution. The
crystal structures of the clathrate hydrates of many nonpolar
substances show that such arrangements are possible in
principle (Figure 4 (right)).

3. Hydrogen Bonds and their Arrangements

3.1. Structure and Properties of the Water Molecule

Figure 3. Pressure dependence of the isothermal diffusion coefficient D at
273 K (a)[24] and the temperature dependence of the isobaric viscosity h at
atmospheric pressure (b).[25, 26] Qualitatively, insight into the origin of these peculiar liquid
properties is available from a consideration of the shape and
bonding characteristics of the water molecule (Figure 5). A
liquids become more viscous under pressure. The pressure simple molecular orbital description provides a useful qual-
and temperature dependence of another transport property, itative picture of the electron distribution in the water
the viscosity h, is also anomalous.[17, 26] Figure 3 b shows an molecule. Four localized regions of excess charge appear in
Arrhenius plot at atmospheric pressure. The deviation of the a tetrahedral arrangement around the central oxygen atom.
data from a straight line indicates that the activation energy Two positive regions are associated with the hydrogen atom

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REVIEWS R. Ludwig

forming four H-bonds with

nearby molecules (Fig-
ure 7). In a so-called ªWal-
rafenº pentamer,[35±40] the
two hydrogen atoms of the
centered molecule act as
acceptors and the two lone
Figure 5. The water molecule (left). The two positive regions at the pairs as donors. It is possible
hydrogen atoms and the negative ªlone pairº orbitals of the oxygen atom to fill three-dimensional
Figure 7. Tetrahedral configura-
(right) obtained by an NBO analysis. space with tetrahedrally co- tion of water molecules spanned
ordinate units, and some by two covalent bonds and two
nuclei, which are significantly stripped of their attendant realization of this is the solid ªlone pairsº of the central mono-
ice Ih crystal, which water mer.
electrons by the highly electronegative oxygen atom.
The excess negative charge that thus appears around the forms on freezing at atmos-
oxygen atom is organized primarily in two lobes, or ªlone- pheric pressure, as well as clathrate hydrates, which are
pairº orbitals, which complete the tetrahedral arrangement of described in detail in the next section. This possibility
the local electron deficit or excess. Overall, the electron indicates that all microscopic and macroscopic properties
distribution in an isolated water molecule can be related to arise from the fact that liquid water is a three-dimensional
the value of the equilibrium bond angle (104.58), the value of tetrahedral H-bonded (HB) network.
the dipole moment (1.85 D), and the tetrahedral coordination
of water molecules in condensed phases.[2]

3.4. Ice Ih and Ice Polymorphs

3.2. Hydrogen Bonding in the Water Dimer
Thirteen crystalline phases are presently known: Ih (h ˆ
The intermolecular attraction between the hydrogen atom
hexagonal), Ic (c ˆ cubic), ice II ± XI[41, 42] and XII.[43, 44] Two
of one water molecule and the lone-pair electrons on another
distinct amorphous water structures, low- and high-density
represents an H-bond. Hydrogen bonds are the dominant
amorphous ice, have also been reported.[8] This uncommonly
interactions between water molecules. Thus much experi-
large number of different solid phases attests to the structural
mental and theoretical effort has been directed toward
versatility of the water molecule. Relative to ice Ih , all the
understanding the nature of the water dimer, which represents
other phases exist at lower temperatures and/or higher
the archetype for hydrogen bonding.
pressures. Those phases differ in the connectivity of the rings
The water dimer (Figure 6) exists in the vapor phase and
and in the position of the hydrogen atom between the oxygen
was measured for the first time by Dyke and co-workers.[33, 34]
atoms, and show bent H-bonds.[45] We will now take a closer
Their molecular beam resonance experiments clearly showed
look at ordinary hexagonal
that the lowest energy arrangement
ice Ih , which is shown in
has a plane of symmetry containing
Figure 8 along with cubic
the hydrogen donor molecule to
ice Ic . The oxygen atoms in
the right and the symmetry axis of
ice Ih possess almost tetrahe-
the molecule to the left. As shown
dral coordination. Each wa-
in following sections, experiments
ter molecule is involved in
Figure 6. Experimental using advanced techniques as well
four H-bonds, with the two
structure of the water dim- as high-level quantum-mechanical
lone pairs as donors and
er as measured from mo- calculations fully support the no-
lecular beam resonance both hydrogen atoms as ac-
tion of linear H-bonds. The meas-
studies by Dyke et al.[33, 34] ceptors. Compared to the
ured bond length in the water
Covalent chemical bonds gas-phase geometry, the dis-
are shown as solid lines dimer of about 2.98 Š is signifi-
tance R(O ´´´ O) is shortened
and H-bonds as dashed cantly longer than the observed
lines.
to 2.74 Š and the bond
distances in both liquid water and
length R(OÿH) is length-
regular ice (about 2.85 and 2.74 Š,
ened to 1.01 Š as a result of
respectively). The shortening of the R(O ´´´ O) distance in
hydrogen bonding. Simulta-
stronger H-bonded networks can be attributed to the
neously, the intramolecular
cooperative nature of hydrogen bonding which will be
bond angle a(H-O-H) is
discussed in detail in following chapters.
widened to the typical tetra-
hedral angle of 109.58. All
3.3. Tetrahedral Structures these structural changes can
be attributed to cooperative Figure 8. The structure of hexago-
The tetrahedral arrangement of the bonding groups in a effects in hydrogen bonding. nal ice Ih (top) and cubic ice Ic
single molecule results in the possibility of the molecule only Ih and Ic are presented here (bottom).

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Water Clusters REVIEWS
because of their structural motifs of chair- and boatlike However, in a computer simulation study of the hydrogen-
hexamers, which play a significant role in the experimental bond network, Geiger et al.[56] found some evidence for the
and theoretical studies discussed later. existence of numerous clathrate-like holes in liquid water. The
same shape of orientational distribution functions have been
observed when studying the orientation of water molecules in
the hydration shell of hydrophobic solutes.[32] Thus the
3.5. Clathrate Hydrates structure was characterized as ªclathrate-likeº, since the
hydration shell molecules are oriented in such a way that one
Clathrate hydrates present the greatest variety and most of the four tetrahedral bond directions points radially outward
intensively studied of H-bond inclusion compounds. The and the remaining three bonding directions straddle the
ability of water molecules to form a wide variety of four- dissolved particle (Figure 4 (left)).
coordinate networks, which results in the polymorphism of The academic interest concerns the water ± water interac-
ice, is also apparent in the hydrate inclusion compounds. The tion in these topologically rather complex systems and the
voids in clathrate hydrates are much larger than in ice, and the guest ± host interaction with guest species that range from
H-bonded networks are unstable unless they are occupied by noble gas atoms to large and polar organic molecules.
guest molecules. After the discovery of crystalline hydrates of Clathrates are also believed to occur in some outer planets
chlorine by Davy[46] and Faraday[47] in 1823 it took more than a at fairly high temperatures. The interest is, however, not only
century before von Stackelberg and Müller[48] and Pauling and academic: The petrol industry suffers from the nuisance of
Marsh[49] could determine the structure of this chlorine hydrocarbon clathrates blocking gas pipelines in arctic
hydrate and its gas hydrates. In the meantime the crystalline regions, and has just started to show interest in the giant
structures of the gas hydrates had been well characterized. natural methane deposits in the deep ocean floor and in
Generally the structures are of the cagelike type-I or type-II permafrost regions.[57, 58]
clathrate structures. The structure-type formed depends upon
the size of the guest gas molecule,[50] which stabilizes the
lattice through nonbonding repulsive interactions.[51, 52] These
3.6. Statistical Networks
structures have a pentagonal dodecahedron (512) of radius
3.9 Š as a principle building block. As this is not a space-filling
So far we have characterized some of the structures which
polyhedron, the crystalline form of those hydrates contain a
occur in the gas phase of water or polymorphs of ice. The
second larger polyhedra to form the lattice; that is a
precise nature of the H-bonded disorder of water in the liquid
tetrakaidecahedral (51262) unit for type-I and a heccaidecahe-
phase is still unknown. Scattering experiments of liquid water
dral (51264) unit for type-II clathrates. All the principle
give a precise description of atomic position disorder, but
building blocks are shown in Figure 9. Ripmeester and
simply do not lead to a uniquely clear picture of the H-bonded
network. Thus computer simulation methods, such as molec-
ular dynamics (MD) or Monte Carlo (MC) simulations, are
helpful for generating a representative set of configurations
for a small region within the solid or liquid of interest. These
calculations predict a totally connected random network of
H-bonds as shown in Figure 10. The melting of ice results in a
latent heat of 1.4 kcal molÿ1 being absorbed. This value is
equivalent to breaking about 10 % of the H-bonds and the
system becoming ªfrustratedº. The water structure at any
instant and on any length scale is amorphous, with many
Figure 9. Clathrate structures of cage type I and II: dodechedron (512),
tetracaidecahedron(51262), and heccaidecahedron (51264). dangling bonds. Its structure becomes ªrandomº and contains
many five- and seven-membered rings, as well as the ice six-
membered ring.[59, 60]
Ratcliffe[53] used 129Xe NMR spectroscopy to identify a new
clathrate hydrate. An X-ray powder pattern diffraction study
has shown this clathrate to have a host lattice isostructural
with a known clathrasil structure, which contains the poly-
hedra 512, 435663, and a larger one, 51268. More recently,
Udachin and Ripmeester reported a complex clathrate hydrate
structure showing bimodal guest hydration.[54] Formally this
structure consists of alternating stacks of structure H and II
hydrates, and might conceivably be found in those settings in
which clathrate hydrates form naturally. The ubiquity of such
motifs in crystalline hydrates led Pauling to formulate a
ªclathrate theoryº of liquid water[55] that was built upon
dodecahedral and tetrakaidecahedral clusters as structure Figure 10. Statistical network of liquid water. Snapshot from molecular
units, but this picture has not received widespread support. dynamics simulations.

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REVIEWS R. Ludwig

4. Many-Body Effects/Cooperativity charge-transfer interaction. In this Lewis-type picture the

donor ± acceptor interaction between a lone pair on an oxygen
The nature of the physical interaction that contributes to atom and an OÿH antibond leads to an energetic stabilization
hydrogen bonding has been the subject of numerous dis- as a result of progressive charge transfer (CT).[72, 73] This CT
cussions in the chemical literature.[61] On one side, H-bonds interaction is responsible for both enthalpic and entropic
are attributed to purely electrostatic interactions, or to factors that stabilize certain H-bond clusters over others. For
electrostatic plus polarization interactions; on the other side example, the cooperative effects in a water pentamer (Fig-
covalent interactions are considered to be extremely impor- ure 12) lead to H-bonding energies that are almost twice as
tant. As discussed, there is increasing experimental evidence
for the partial covalence of the H-bond. The more recent and
prominent ones are the Compton scattering experiments on
ice Ih by Isaac et al.[62, 63, 64] and NMR measurements of 1H-15N
and 15N-15N scalar couplings of several H-bonded biological
systems by Grzesiek et al.[65, 66] It should be noted that the
covalent contributions to H-bonding concluded from Comp-
ton profiles is highly controversial. Parrinello et al.[67] calcu-
lated the same features of the electron distribution in ice Ih ,
but come to a different chemical interpretation.
Cooperativity of H-bonding in water was originally an idea
of Frank and Wen.[68] The formation of a first H-bond
(Figure 11) results in a change in the charge distribution

Figure 12. Overlapping of the oxygen lone pair orbital and the antibonding
OH orbital in an equilibrium structure of the (H2O)5 pentamer. The
average H-bond strength is almost twice that in the dimer as a result of
cooperative effects.

strong as the linear dimer H-bond.[74, 75] Some of the con-

sequences are: Strong CT interactions tend to lengthen the
covalent OÿH bond and to shorten the noncovalent H ´´´ O
hydrogen bond, thus reducing the overall R(O ´´´ O) separa-
Figure 11. Hydrogen-bonded water dimer in the framework of an NBO tion as given in Table 1. On enthalpic grounds, cooperative
analysis.[74] The overlap of an oxygen lone pair orbital and an antibonding
OH orbital is shown.
Table 1. Calculated average distance R(O ´´´ O) and bond length R(OÿH)
[Š], NBO delocalization energies DEn!n‡1 [kcal molÿ1] from oligomer Wn
within the participating monomer in such a way that the to oligomer Wn‡1 , and natural charge transfer qCT [e] in water clusters
hydrogen acceptor molecules becomes potentially an even (RHF/6-31 ‡ G*).[75]
better H-bond donor than before. It is capable of forming a R(O ´´´ O) R(OÿH) DEn!n‡1 qCT
stronger second bond because of the existence of the first W1 ± 0.9476 ± ±
bond. The same is true for the proton donor, which has an W2 2.964 0.9491 9.73 0.009560
enhanced ability to accept a proton as a result of the bond that W3 2.872 0.9514 8.70 0.008917
it has already formed. This idea has been supported for many W4 2.847 0.9531 15.04 0.015397
W5 2.837 0.9536 16.72 0.016404
years by quantum mechanical studies. Kollman and co- W6 2.833 0.9535 16.94 0.016673
workers used a Morokuma analysis[69, 70] of the results from
an ab initio calculation to break down the interaction energy
of the water dimer into four components: electrostatic, bicoordinate ring structures are intrinsically favored over
polarization, charge transfer, and dispersion. The authors open-chain and starlike topologies (Figure 13). On entropic
found out that the contributions of various components varied grounds, two-coordinate connectivities such as chains and
with intermolecular distance. Roughly half the interaction at cycles are favored over three- or four-coordinate networks.
the equilibrium distance of about 2.98 Š could be attributed Compared with two-coordinate closed CT ring forms, higher
to electronic interactions. three- and four-coordinate clusters are disfavored at higher
Weinhold and co-workers performed a natural bond orbital temperature because of strongly hindered intermolecular
(NBO) analysis to eliminate the charge-transfer component bending or stretching modes. These reduce the librational
from the Hamiltonian operator of H-bonded dimers.[71] They entropy contributions and the unfavorable cooperative
reported that this component constituted the major energetic H-bond directionality patterns which lead to significant
contribution to many H-bonds, whereas electrostatic attrac- enthalpy loss. The occurrence of the three-dimensional
tion was largely canceled out by exchange repulsion. Thus, the structures at lower temperatures is caused by the high
interaction energy can be considered to be a consequence of a connectivity and the resulting larger total H-bond energy.

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Water Clusters REVIEWS
energetic criterion. It could be shown that clusters of four-
bonded molecules will be observed as the probability of
bonding to a nearest neighbor increases. The thermodynamic
anomalies of water could be explained by considering those
four-bonded clusters to be associated with regions of larger
molecular volume, lower energy, and higher order. The
authors results held when random configurations of molecular
sites in the continuum, taken from computer simulations of
water, were used.[83, 84]

5.2. Continuum Models

ªContinuum modelsº contrast to the mixture models in that

in the former it is considered that the H-bonds are never
broken in the liquid, but are only more or less distorted from
their optimal form. A more modern example is the ªcontin-
uous random networkº (CRN) model of Sceats and Rice.[85]
Figure 13. H-bond formations of water molecules: ring, star, lasso,
tetrahedron, and chain. This approach is supported by the observation that H-bonds
need not necessarily break for molecules to have the mobility
that is characteristic of the liquid. The model is based on
continuous modifications of the topology of the H-bonded
5. Water Models network rather than disruption of local H-bonded associates.
This view implies there are ªbifurcatedº H-bonds or shared
5.1. Mixtures Models H-bonds between two atoms of one molecule.[86]
The picture of continuous modification of the H-bonded
The combined importance and peculiarity of water has network has been proven useful in explaining the mobility of
inspired scientists for more than a century to construct water molecules in the liquid.[87±89]
conceptual models which reproduce the observed behavior of A distinction between mixtures and continuous models
the liquid. The earliest attempt probably goes back to could be made as follows: in the former there are intact and
Röntgen in 1892. He explained the density maximum as broken H-bonds, while in the latter there is a fully H-bonded
resulting from a shifting equilibrium between small ice network. There is a continuous transition between these
crystallites suspended in a liquid of dissociated individual extremes. For example, the model of Stanley and Teixeira,[82]
molecules.[76] Röntgens concept was based on the idea that which is given above as a modern mixture model, can also be
liquid water can be modeled by a mixture of two locally less- regarded as a continuous model. The introduction of a
and more-dense structures. It thus represents the first of a threshold for H-bonds into a continuous model picture leads
family of so-called ªmixture modelsº for the liquid struc- to a mixture model.
tures.[77] Such models focus on the H-bond structures of the
liquid and distinguish between a population of ªintactº and
ªbrokenº H-bonds. This view was naively confirmed in 6. Survey of Calculated Water Clusters
infrared studies of liquid water.[78] A strong overtone of the
OH stretching mode was found with two distinct components Semi-empirical and ab initio quantum mechanical studies
in the 1.4 to 1.6 mm region. The absorbance of the lower on water clusters are numerous.[90±120] This development went
wavelength component decreases with decreasing temper- along with improved methods and better computational
ature, while the higher wavelength component increases. This capabilities. Our survey concentrates on recent ab initio
behavior has been associated with the expected decrease in Hartree ± Fock (HF) and density functional (DFT) calcula-
the population of ªbrokenº H-bonds as the liquid cools. tions. For clusters larger than n ˆ 30, only semi-empirical
Although other explanations do not require two distinct states calculations are taken into consideration. The presented
of bonding, those experimental results encouraged several water clusters are discussed as a function of size and
scientists to follow the idea of bonded and nonbonded connectivity so that most of them can play a significant role
structures for modeling the properties of liquid water.[79±81] in the experimental and theoretical investigations described
Stanley and Teixeira[82] presented a more modern model in later.
this spirit. Their statistical model does not require that water
molecules are strictly H-bonded or not H-bonded; it simply
describes the degree of connectivity that is observed to occur 6.1. Small Cyclic Water Clusters (n ˆ 3 ± 6)
in a lattice of four-coordinate sites when some random
fraction of the nearest neighbors are considered bonded, and The optimal structures and harmonic vibrational frequen-
the rest are considered unbound. The two groups of interact- cies of small water clusters including ring structures Wn,
ing water molecules are distinguished for example by an n ˆ 3 ± 6, have been determined by Xantheas et al.[121±124] with

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REVIEWS R. Ludwig

Hartree ± Fock (HF) and density functional (DFT) methods, Table 2. Calculated average R(O ´´´ O) distances [Š] in water clusters Wn
as well as Mùller ± Plesset second order perturbation theory (n ˆ 2 ± 6) at various levels of theory with the aug-cc-pVDZ basis set.[121±123]
(MP2) with an augmented correlation-consistent double-zeta Cluster HF B-LYP MP2
basis set (Figure 14). The density functional B-LYP used W2 3.032 2.939 2.911
W3 2.927 2.808 2.799
W4 2.880 2.743 2.743
W5 2.867 2.727 ±
W6 2.855 2.714 ±

The vibrational spectra of small water clusters (n ˆ 4 ± 6)

show a blue shift of about 70 cmÿ1 for the intramolecular
bending mode with respect to the monomer. The correspond-
ing red-shifts in the OH stretches were estimated to be about
50 and 500 cmÿ1 for the free and bridging hydrogen atoms,
respectively, with respect to the monomer. These values are in
excellent agreement with experimental observations both on
small water clusters as well as the bulk, as we will see later.
Xantheas et al. pointed out that geometries, harmonic fre-
quencies, and IR intensities can provide guidance to research
groups studying water clusters; they demonstrated that the
shifts in the OH stretching frequencies correlate well with the
changes in the corresponding equilibrium bond distances and
obey the well-known Badgers rule.[125]

6.2. Isoenergetic Water Hexamer Clusters (n ˆ 6)

An accurate theoretical description of the water hexamer is

Figure 14. Small water clusters (n ˆ 1 ± 6) as calculated by ab initio and
an interesting and fundamental subject. The cyclic hexamer,
DFT methods.[121] as discussed above, is the building block of many ice forms and
it appears to be relevant for liquid water as well.
Prompted by the surprising experimental results for the
combines the Beckes gradient-corrected exchange functional water hexamers in the gas phase, Kim and Kim[126] performed
with the gradient-corrected correlation functional of Lee, extensive ab initio and DFT calculations on the five lowest
Young, and Parr.[123] The authors were particularly interested energy structures of the water hexamers (Figure 15). The
in the correlation of various properties with cluster size. Thus authors demonstrated that the ring, book, bag, cage, and
they performed benchmark studies of the water monomer and
H-bonded dimer to ascertain the minimum level of theory and
basis set that would yield meaningful results for the larger
clusters. Accurate structure, harmonic frequencies, dipole
moments, and polarizability components for the water
molecule, along with several measured properties for the
water dimer, such as the structure and rotational constants,
were chosen as benchmarks. The most important results
concern the structural and spectral trends in small cyclic water
clusters. The study of cyclic clusters (n ˆ 3 ± 6) reveals a
systematic contraction of the nearest R(O ´´´ O) separation
with increasing size, a fact which can be attributed to
nonpairwise additive (cooperative) many-body interactions
as discussed earlier. As given in Table 2, the R(O ´´´ O)
distance decreases nearly exponentially with increasing clus-
ter size. The HF results, although not as accurate as the
correlated data, exhibit the same trends as the DFT and MP2
results. While the R(O ´´´ O) separation decreases, the
length of the bridged OH bonds increase monotonically
with increasing ring size. The R(OÿH) bond length in
the MP2 tetramer is about 2 pm longer than in the water Figure 15. Calculated water hexamer isomers showing quasiplanar and
monomer. cagelike clusters.[126]

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Water Clusters REVIEWS
prism structures have energies that are within 0.7 kcal molÿ1 of 6.4. Water Octamers: Cubic or Cyclic?
each other, that is, they are nearly isoenergetic. Their MP2
calculations showed that the lowest energy conformer is the Some of the forementioned water heptamers can be
cage followed by the book (within 0.1 kcal molÿ1) and the generated from the two most important water octamer
prism structure (within 0.2 kcal molÿ1). The ring and the bag conformers. These cube water structures with D2d and S4
structures are only slightly higher in energy (0.5 and symmetry present isoenergetic global minima.[128] They con-
0.7 kcal molÿ1, respectively) than the cage hexamer. The free tain two kinds of water monomer: donor-donor-acceptor and
energy of the book hexamer above 40 K is slightly lower than acceptor-acceptor-donor molecules. Both cubic octamers are
the cage, which might imply that the book structure would be shown in Figure 17 along with a nearly cyclic and a bicyclic
more populated and thus be detectable. At higher temper-
atures the populations of the five hexamers would be almost
the same. Deuteration did not change the nearly isoenergetic
behavior of the clusters. The cage structure is still the lowest
energy conformer, followed by the two competing structures
of the book and the prism, whose energies are only
0.2 kcal molÿ1 higher at 0 K. Above 55 K the book form
would again be more populated than the cage structure. The
nearly isoenergetic nature of the water hexamers suggests
that the kind of structure that will be detected experimen-
tally strongly depends on the physical and chemical environ-
ment.

6.3. Variety of Water Heptamers (n ˆ 7)

In spite of a spate of studies of various water clusters, only a

few theoretical investigations on the water heptamer are
available. The experimental vibrational spectra of water
heptamers encouraged Kim et al.[127] to perform ab initio Figure 17. Cyclic, bicyclic, and cubic octamers of water obtained from
ab initio calculations.[128, 129]
and DFT calculations on twelve possible water heptamer
structures to explore the conformations as well as the
spectroscopic properties of these water clusters. Two three- octamer calculated by Weinhold.[129] The cyclic topology has
dimensional cagelike structures comprised of a seven-mem- less H-bonds than the polycyclic octamers (8 versus 12) and is
bered ring with three additional H-bonds were found to be the thus energetically disfavored. Its strong thermodynamic
lowest energy heptamer conformers (Figure 16). The global stability is caused by entropic factors. A virtually free torsion
about the H-bond axis in the cyclic octamer leads to lowest
frequency vibrations whereas angular strain factors in the
cubic structures cause higher vibrational temperatures of the
lowest H-bond modes.[129]

6.5. Ice-Like and Clathrate-Like Structures (n ˆ 12 ± 26)

Larger water clusters with up to 26 water molecules were

calculated by Ludwig and Weinhold.[130] The species W12 , W18 ,
Figure 16. Energically low-lying isomers of calculated water heptamers.[127] and W26 (Figure 18) were chosen as representative clusters
with hexagonal units. Those hexagonal facets are well known
structural elements of known crystallographic ice forms.[42]
minimum energy of the most stable species was found to be The W12 cluster has two W6 rings directly coupled face to face
0.5 kcal molÿ1 lower than the other. The Gibbs free energy in a charge-balanced fashion, with each vertex (oxygen atom)
calculations using HF and B3LYP frequencies have shown having trigonal coordination. A more ice-like fragment is the
that both structures are stable up to 100 K. An almost two- W18 cluster, which contains an adamantane-like core but-
dimensional ring conformer lies only 1 kcal molÿ1 above the tressed with three W2 side chains. Although this polycyclic
global minimum at 0 K. Above 150 K, this ring structure is cluster exhibits hexagonal elements of a tetrahedral lattice, it
more stabilized than the three-dimensional heptamers for has no four-coordinate vertices that could truly resemble a
entropic reasons. The vibrational spectra of different hep- typical interior site of bulk ice Ih . The W26 cluster was initially
tamer conformers were discussed and compared with spectra formulated as a tetrahedral diamond-like microcrystal with
of the hexamer and octamer water clusters. central water molecules concentrically surrounded by succes-

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REVIEWS R. Ludwig

6.6. Larger Clusters: Icosahedral Networks (n ˆ 280)

Chaplin[133] formed an icosahedral three-dimensional net-

work of 280 hydrogen-bonded water molecules. In this
structure each water molecule is involved in four H-bonds,
two as donors and two as acceptors. The network is based on
the regular arrangement of 20 slightly flattened tetrahedral
14-molecule units as shown in Figure 19. Within these units

Figure 18. Ice- and clathrate-like water clusters.[130]

sive shells of tetrahedral coordination to give increased

numbers of ªinteriorº four-coordinate monomers at each
shell. However, even the W26 cluster contains only six four-
coordinate (plus eight three-coordinate and twelve two-
Figure 19. An expanded icosahedral water cluster consisting of 280 water
coordinate) monomers, which leads to an average coordina-
molecules with a central dodecahedron (top) and the same structure
tion number of 2.77. This value is less than in the trigonal collapsed into a puckered central dodecahedron (bottom). The figure is
coordination of the fullerene-like topologies W20 , W24 , and reproduced with the kind permission of Elsevier Science and M. F.
W28 (bucky water). While pentagonal and hexagonal facets Chaplin. [133]
are well-known structural elements of crystallographic ice
forms,[42] the intact bucky-water polyhedra are principally four water molecules form the corners of the tetrahedron and
recognized as crystallographic elements in certain clathrate- are involved in both boat-form hexamers and pentamers. The
type hydrates, such as the pentagonal dodecahedral (512), remaining ten molecules form an adamantane-type ring
tetrakaidecahedral (51262), and the hekkaidecahedral (51264) structure, identical to a 10-molecule unit recently found in a
units. The large number of calculated clusters are topologi- crystalline supramolecular complex,[134] and also as found
cally similar; they differ only in the proton ordering around within the 18-molecule cubic ice cell. Four of these molecules
each vertex. The most stable species of the cage types are are involved in hexamers in both the boat and chair
shown in Figure 18. The comparable tetrakaidecahedral conformation, with the remaining six molecules form pen-
cluster structures W24 , W25 , and W26 were also calculated by tamers and hexamers in the chair conformation.
Khan.[131] He used semi-empirical methods for his calculations Pentamers of water have bond angles of 1088, which are
and described the structural features and stability of these 1.478 closer to the supposedly most stable a(H-O-H) angle in
clusters. water vapor (104.528) than are the tetrahedral angles
Most recently, Khan calculated multiple-cage clusters to (109.478) in ice. The clusters can grow in three dimensions;
examine whether these fused structure formations became each cluster has twelve potential sites at its icosahedral
more favorable as the cluster-size increases.[132] Indeed, the 35- vertices for use as centers for neighboring, overlapping
mers having two fused dodecahedral cages consistently show a clusters. The structure becomes more distorted as the network
greater stability than their single-cage isomers. grows.

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Water Clusters REVIEWS
7. Survey of Measured Water Clusters 7.2. Water Hexamers: Cage, Quasiplanar Ring, or Chair?

7.1. Small Cyclic Water Clusters (n ˆ 3 ± 6) 7.2.1. The Cagelike Hexamer in the Gas Phase
As discussed in Section 6, ab initio studies on the hexamer
Enormous progress in laser spectroscopy has facilitated
reveal the existence of several low-lying structures, but predict
new, highly detailed studies of water clusters. Studies of the
that the minimum-energy form is the three-dimensional cage
structure and dynamics of isolated small clusters of water
shown in Figure 21. This cagelike hexamer agrees best with
molecules provide a means of quantifying the intermolecular
forces and hydrogen-bonded rearrangements that occur in
condensed phases. Far-infrared (FIR) vibration-rotation-tun-
neling (VRT) spectroscopy of clusters has recently been
developed by Saykally and co-workers[135±145] to address such
questions. Low-frequency van der Waals vibrations in clusters
can be measured with tunable FIR lasers to resolve rotational
and tunneling motions. The resulting VRT spectra can be
analyzed in terms of permutation-inversion (PI) group theory
and scattering theory to yield pair potentials of unprecedent-
ed accuracy and detail for weakly bonded systems. FIR-VRT
spectroscopy is also a powerful probe of the tunneling Figure 21. Experimentally found water hexamers: the cage structure
dynamics that occur in hydrogen-bonded clusters. This detected in the gas phase,[146] the quasi-planar cyclic structure trapped in
liquid helium,[147] and the chairlike cyclic structure measured in organic
method should allow for the investigation of the cooperative
hosts.[148]
(nonpairwise) effects of hydrogen bonding in water clusters.
In a series of beautiful experiments Saykally and co-workers
characterized the water dimer,[135] the cyclic water trimer,[136] the measured rotational constants from VRT measurements
the tetramer,[137] and the pentamer.[138] The results unambig- performed by the Saykally group.[146] The fact that both high-
uously established that the structures of the water clusters level ab initio calculations and diffusion quantum Monte
responsible for the observed spectra were indeed the quasi- Carlo (DQMC) results[147±149] predict the cage as the lowest
planar rings predicted by theory.[139] These spectra permitted energy form, and that no structures other than the most stable
estimates of the R(O ´´´ O) distances to be determined for have ever been detected by VRT spectroscopy in super-sonic
each of the clusters and yielded a quantitative experimental argon jets, is fairly compelling evidence that the cage form is
measure of the hydrogen-bond cooperativity. The R(O ´´´ O) indeed the most stable water hexamer. Clearly the water
distance obtained from VRT spectroscopy and theoretical molecules have enough time to find the absolute minimum
studies of water clusters are plotted in Figure 20. Consistent structure in these gas-phase experiments at 5 K. Ab initio
studies also predict that zero-point vibrational effects are
crucial for the stability and that they can alter the energy
ordering of the low-lying hexamer structures. Thus, there were
some hints that other hexamer isomers could be detected by
changing the chemical environment and temperature.

7.2.2. Quasiplanar Hexamers in Liquid Helium

Nauta and Miller recently reported the experimental
observation of the cyclic water hexamer, which was a higher
energy isomer than Saykallys cage structure previously
Figure 20. The R(O ´´´ O) distance versus cluster size obtained from VRT characterized in the gas phase.[150] The ring hexamer shown
spectroscopy[136±139] and three different levels of theory: Hartree ± Fock
in Figure 21 was formed in liquid helium droplets and studied
(HF), Mùller ± Plesset second-order perturbation theory (MP2), and
density functional theory (B-LYP).[121±123] The experimental R(O ´´´ O) by infrared spectroscopy. Three main results of this beautiful
distances in liquid water at 298 K[155] and hexagonal ice Ih at 183 K[41, 42] are study are remarkable. Firstly, this isomer is formed selectively
given for comparison and are indicated by dotted lines. as a result of unique cluster-growth processes in liquid helium.
The experimental results indicate that the cyclic hexamer is
with the occurrence of cooperative effects, all the methods formed by insertion of water molecules into smaller, pre-
produced an exponential contraction of the R(O ´´´ O) formed cyclic complexes such as trimers,[151] tetramers, and
distance with increasing cluster size which converged to the pentamers. Buck and co-workers[152] obtained different results
bulk (ordered ice) value of about 2.74 Š. Experiment and for growing methanol clusters in helium. Once the cyclic
theory strongly suggest that the water trimer, tetramer, and trimer was formed and cooled, insertion of the fourth
pentamer have cyclic, quasi-planar minimum energy struc- methanol molecule into the ring was inhibited by the lack of
tures. Larger water clusters were expected to have three- energy needed to open the ring. The result was a tetramer
dimensional geometries, with the hexamer representing the structure corresponding to a cyclic trimer with the fourth
transition from a cyclic to a three-dimensional structure. molecule hydrogen bonded to the outside. As a possible

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REVIEWS R. Ludwig

explanation for the difference between the water and and the three-dimensional cage structure will involve a great
methanol systems Nauta and Miller suggested that insertion deal of H-bond rearrangement, which is expected to be
into the water ring is facilitated by tunneling of the hydrogen difficult in liquid helium.
atoms through associated barriers. In contrast, ring insertion
may be inhibited in methanol because it involves more motion
of heavy atoms. Secondly, the infrared spectrum for the OH 7.2.3. The Chairlike Hexamer in Organic Matrices
stretches of water clusters formed in liquid helium were
compared with those of the corresponding complexes formed Recently, the supramolecular association of cyclic water
in a free jet expansion experiment. The vibrational frequency hexamers with an ice-like chair conformer into one-dimen-
shifts resulting from the interaction with the helium were sional chains inside an inclusion organic host was reported.[154]
negligible and the bonds for the dimer, trimer, tetramer, and The geometric parameters of the hexamers are summarized in
pentamer essentially coincidence with the gas-phase bands.[153] Table 3. The average R(O ´´´ O) distance of 2.776 Š is about
An additional peak red-shifted relative to the pentamer was the same as the analogous value of 2.759 Š in ice Ih at 183 K.
assigned to the cyclic isomer of the water hexamer. The most
compelling support for this assignment comes from compar- Table 3. Experimentally determined distances [Š] and angles [8] for water
hexamers detected in an organic host.[154]
ing the frequency shifts for all of these cyclic complexes with
the corresponding ab initio and DFT calculations on the R(O ÿ H) R(O ´´´ O) R(H ´´´ O) aO-H ´´´ O aO ´´´ O ´´´ O
clusters (Figure 22). As already shown for intra- and inter- 0.960 2.711 1.758 171.3 113.2
molecular geometries, the frequency shifts vary smoothly with 0.956 2.785 1.844 167.6 96.0
1.264 2.833 1.703 145.0 140.2

However, there is a wide variation of the angles a(O-O-O);

the average value is 116.58, which is considerably deviated
from the corresponding value of 109.38 which occurs in
hexagonal ice. The hexamers are self-assembled by OÿH ´´´ O
H-bonds into extended chains along the channels, which
consist of fused four- and six-membered water rings. The
observed interhexamer oxygen ± oxygen distance of 2.854 Š is
very similar to the separation of 2.85 Š found in liquid
water.[155] This supramolecular association of water molecules
in chains is presumably enforced by the shape of the hosts
Figure 22. Experimental[150, 153] and calculated[121±123] red-shifts of the OH
vibrational frequency for cyclic water clusters from the trimer to the
channels, whose relatively narrow openings preclude the
hexamer. The shifts are taken in both cases relative to the average of the formation of the more stable three-dimensional clusters found
symmetric and asymmetric OH stretches of the monomer. The frequency in the gas phase. There are no OÿH ´´´ N H-bonds between the
shifts are essentially the same for the gas-phase and liquid helium data up to water chains and the organic host. Thus the water clusters can
the pentamer. The band for the cage hexamer detected in the gas phase is
be removed by heating without changing the structure of the
more red-shifted than that of the hexamer in liquid helium relative to a
quasi-planar ring structure. The solid curves present guidelines for the eye. host. In contrast to previously described inclusion complexes
They are obtained by scaling the fitted experimental curve to the of water clusters that have strong interactions with the host,
theoretical data. the water clusters more closely resemble structures found in
liquid water or ice. The IR spectra suggest that the water
cluster size. The cyclic hexamer peak is precisely where it is chains have more similarities with liquid water than with
expected theoretically, whereas the cage band is shifted hexagonal ice.
further to the red region. Better agreement with experimental Ice-like clusters with chair and boat conformations were
and theoretical values can not be expected given that the also observed by Barbour et al.[134, 156] in the solid state. The
theoretical values are based on harmonic frequency calcu- intervening voids of a cobalt cage complex are filled with
lations. From a theoretical viewpoint it is important that the clusters of ten water molecules that adopt an ice-like
calculated shifts can be scaled to the experimental data by one conformation. The water cluster is sufficiently flexible to
scaling factor for all cyclic clusters. Increasing cluster sizes respond to small changes in its environment, but the overall
clearly do not require different scaling factors. Thirdly, Nauta conformation is robust.
and Miller used superfluid liquid helium as a growth medium However, free chair- or boatlike hexamers can not be
to access a different portion of the energy surface as in the detected. It clearly requires neighbors in a periodic system
gas-phase experiment, and could observe the cyclic water such as are present in hexagonal and cubic ice or organic
hexamer. This species is one of the prominent morphologies matrices.
found in computer simulations of liquid water[60] and is the
structural motif of ice Ih .[41, 42] The cage isomer characterized
previously by Saykally and co-workers[146] has a most intense 7.3. From Heptamers to Decamers
OH vibrational band which has a much greater red-shift.
Several calculated local minima[126] lie lower in energy than The understanding of the evolution of clusters larger than
the cyclic hexamer. Clearly the path between this hexamer hexamers in the ªcageº regime was the challenge of Buck and

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Water Clusters REVIEWS
co-workers.[157±159] Prior to their measurements, experimental The authors reported that the presented experimental
results focused on the OH stretch of large clusters without size technique will allow the measurements to be extended down
selection[160, 161] and n ˆ 7,8 clusters attached to benzene.[162±164] to cluster sizes of n ˆ 6 and up to n ˆ 20 by changing the
Recently, Buck et al. presented the first infrared measure- carrier gas and the collision partner.
ments of the OH stretch mode of pure water clusters with size
n ˆ 7 ± 10. Their experimental method is a combination of size
selection (by momentum transfer in collisions with rare gas 8. Cluster Models for Liquid Water
atoms) with an infrared depletion technique.[165] In the first
step the different clusters are dispersed into different angles 8.1. Quantum Cluster Equilibrium Theory of Liquid
according to their masses and detected by a mass spectrom- Water
eter. Then the OH stretch vibrational mode of the water
molecules is excited by IR laser radiation. The detector Weinhold[129] developed a method for calculating equilib-
records the depletion in the cluster signal caused by dissoci- rium properties of liquids by extending the standard statistical
ation of the clusters by the absorbed radiation. Additional thermodynamic treatment of chemical equilibria to the
calculations on the heptamers resulted in numerous energy analogous equilibria between molecular clusters, as charac-
minima, which can be divided into two structural families. The terized by modern ab initio techniques. The quantum cluster
experimental spectra are reproduced quite well using the equilibrium (QCE) theory of liquids is based on the role of
lowest energy structures similar to those found by Kim H-bonded molecular clusters as fundamental constituent
et al.[127] and shown in Figure 16. Inclusion of the zero-point units. Standard quantum statistical thermodynamic methods
motion effects was crucial for reproducing the spectra. The are used to treat the equilibria between clusters in the
spectra of the two lowest isomers can be derived from the canonical ensemble, which leads to predictions of macro-
cubic S4 octamer by removal of either one double donor or scopic thermodynamic and spectroscopic properties. Mean-
one double acceptor water molecule. The two cubic octamer while QCE has been shown to provide practical, quantitative,
isomers with D2d and S4 symmetry were characterized, along or semiquantitative descriptions of H-bonded fluids such as
with nonamers and decamers, earlier by Buck et al.[157] using amides[166±169] and alcohols.[170±173] The quantum cluster equi-
the same experimental method. The proposed lowest energy librium formalism was first applied to the most important
nonamer and decamer structures (Figure 23) are derived from H-bonded fluid, namely, liquid water. In these first papers,
emphasis was not placed on achieving high quantitative
accuracy, but rather to illustrate qualitatively how the QCE
model ªworksº, and included the interplay between micro-
scopic water clusters and macroscopic phase behavior, the
stability of QCE predictions with respect to changes in the
theoretical model or inclusion of other clusters, and the
importance of nonpairwise additive cooperative effects in
aqueous condensation phenomena. The simple seven-cluster
QCE(7)/3-21G model for liquid water comprised clusters
from the water monomers W1 up to the cyclic n-mers W5 , W6 ,
and W8 .
It could be shown that structures found in the gas-phase
experiments, such as the cubic octamer with D2h symmetry are
completely negligible in the QCE population, even though
they are lower in energy. The reason for these dramatic
differences in thermodynamic stability can be traced to
energetic and entropic factors. The H-bonds in the cubic
clusters are much more highly strained by the severe non-
Figure 23. Structures of water clusters (n ˆ 8 ± 10): one cubic octamer (top linear O-H-O angles required along the cube edges, since the
left), one nonamer (top right), and two decamers (bottom) as measured by
IR depletion techniques.[157±159]
ideal 908 angle at each corner is much smaller than the
equilibrium H-O-H bond angle. Severe angular strain in
polycyclic octamers lead to higher vibrational temperatures of
the octamers by insertion of one and two two-coordinate the lowest H-bond modes and result in much larger unfavor-
molecules, respectively, into the cube edges. The two lowest able entropic contributions. Such a process renders this
energy decamers can be viewed as two fused pentamers with species essentially irrelevant for describing equilibrium prop-
the same and the opposite orientation of H-bonds in the two erties of water.
cycles, respectively. However, the spectrum calculated for The QCE liquid-phase population as shown in Figure 24
these two minima did not match the experiment very well. A consists mainly of cyclic octamers W8 . The next highest
better agreement was obtained for another ªbutterflyº mini- concentration of clusters are W5 and W6 . Thus the equilibrium
mum structure, which can be viewed as a D2d octamer with liquid phase is pictured as a mobile distribution of ring
two extra two-coordinate donor ± acceptor molecules inserted isomers that are packed roughly with van der Waals contact
at opposite edges. separation. This packing leads to characteristic near neighbor

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REVIEWS R. Ludwig

ated[174] and recently to tritiated[175] water clusters. They

characterized the microstructural composition and macro-
scopic properties relative to those obtained for protonated
species. The main goal was to compare the phase diagrams
and thermodynamic properties of isotopically substituted
water. It was found that the QCE triple point is shifted to
higher temperatures by about 2.0 K upon deuteration and by
about 3.0 K upon tritiation compared to the measured 3.8 K in
heavy and 4.5 K in super-heavy water. The shifts of the
melting point to higher temperature is also modeled correctly.
The QCE theory allows for a change in the properties of light,
Figure 24. Cluster population P [mass %] from QCE(18)/3-21G model
water at one atmosphere of pressure. The contributions of leading clusters heavy, and super-heavy water clusters (masses, momentum of
are shown in each phase: W24 , W8 , and W6 in the solid phase, W8 , W6 , and inertia, zero-point energies, and vibrational frequencies)
W5 in the liquid phase, and W1 and W5 in the gas phase.[130] individually and/or in any combination. In this way the
influence of different properties on cluster populations can be
investigated. An interesting result is that the shift of the triple
R(O ´´´ O) distances of about 2.8 Š within the rings and a
point to higher temperatures is a net effect: Lower zero-point
next-nearest O ´´´ O separation of about 4.5 Š. These values
energies for deuterated and tritiated water alone strongly
are crudely consistent with known features of the radial
raise the melting points, whereas a reduction in the vibrational
distribution function of liquid water. What is certainly not in
frequencies for the isotopically substituted species alone
agreement is the low number of nearest neighbors compared
lowers the melting point. The combination of the opposite
to the experimentally found number of 4.4. Also most of the
effects result in a small net shift of the melting point to higher
thermodynamic properties, such as the liquid ± vapor co-
temperatures as shown in Figure 26. The larger masses and
existence curve and heats of vaporization, could be repro-
rotational temperatures in heavy and super-heavy water
duced in reasonable agreement. It was more difficult to
clusters have practically no effect on the melting point.
reproduce a liquid ± solid(ice) phase transition; for that it was
necessary to calculate three-dimensional four-coordinate
water structures similar to crystalline ice. Thus Ludwig and
Weinhold[130] extended the QCE model of liquid water to
include larger ice-like clusters, such as the tetrahedral and
fullerene-like clusters with up to 26 water molecules. A low-
energy tetrakaidecahedral W24 cluster (Figure 18) leads to a
new low-temperature phase that borders on both liquid and
vapor regions in first-order transition lines and gives rise to a
true QCE triple point (Figure 25). The authors characterized

Figure 26. Calculated QCE triple points (*) for light, heavy, and super-
heavy model water. The partial triple points are obtained by individually
replacing zero-point energies (zpe), vibrational frequencies (freq), trans-
lational masses (mass), and rotational temperatures (rot).[175]

It is also interesting to note that isotopic substitution leads

to different cluster populations in the liquid range. The
dominant cyclic octamers W8 are slightly replaced by W5 and
W6 ring structures. It is usually assumed that shifts in
thermodynamic and dynamic properties in going from H2O
to T2O can be ascribed to zero-point-energy-induced thermal
Figure 25. Phase diagram of the light and heavy QCE(18)/3-21G model offset and keep the structural properties nearly identical.
water. The available experimental triple points for light (~) and heavy (~)
water are given for comparison.[130]

8.2. Icosahedral NetworksÐA Structure Proposal for

the microstructural composition and macroscopic properties Water
of this ªbucky iceº phase. Although it differs significantly
from physical ice Ih (for example, the melting point is 20 K too Chaplin[133] proposed a fluctuating network of water
high and the molar volume is 5 % too low), it manifests molecules with localized icosahedral symmetry to explain
qualitatively correct thermodynamic features of true ice many of the anomalous properties of water. This structure is
polymorphs, which suggests an important role of voluminous built up from a mixture of hexamer and pentamer rings and
clusters in the liquid/solid transition region. In further studies contains cavities capable of enclosing small solutes. The
Ludwig and Weinhold extended the QCE theory to deuter- model was developed by arranging alternating sheets of boat

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Water Clusters REVIEWS
and chair conformations of water hexamers from the lattices number of nearest neighbors of about 4.34 is also in good
of hexagonal and cubic ice. This structure was folded to form agreement with the reported 4.4 nearest neighbors calculated
an icosahedral three-dimensional network with capacious from diffraction data.[42] The radial distribution functions for
pores that is capable of partial collapse as a result of the low-density structure can be compared with experimental
competition between the formation and destruction of data obtained for solutions, supercooled water, and the low-
H-bonded interactions. The stability of the network is bal- density amorphous ice. The distance between the cavities of
anced; it is able to fluctuate between an expanded low-density 5.4 Š is close to the value of about 5.5 and 6 Š for supercooled
one of about 0.94 g cmÿ3 (Figure 19 (top)) and a denser water[56] and the Ne ´´´ Ne distance in water,[56] respectively.
collapsed one of about 1.00 g cmÿ3 (Figure 19 (bottom)) with- The radial distribution function of the low-density structure
out breaking any H-bonds during the small changes that occur also shows similarities to those of the low-density amorphous
in the H-bond strength relative to the nonbonded interactions. structure. Both include features similar to cubic and hexag-
The expanded structure is formed when structuring solutes or onal ices.[181] Neutron-diffraction data has been used to
surface interactions are present which result in stronger suggest the presence of pentamers, boat and chair conformers
H-bonds. Weak H-bonds yield the partially collapsed of hexamers, and partial dodecahedra.[182] By using Gaussian
structure as a consequence of the formation of puckered broadening for the high-density structure, the resulting four
dodecahedra. The resulting densities of both structures as peaks at 2.8, 4.6, 6.7, and 9.0 Š show close agreement with the
given above can be related to well-known measured densities. values of 2.79, 4.56, 6.95 and 8.60 Š measured by X-ray
The lower density structure may be compared with that of experiments.[183] The agreement with neutron scattering
low-density water found around macromolecules[176] (0.96 data[184] was even better.
gcmÿ3), of supercooled water (ÿ 458; 0.94 g cmÿ3), and of low-
density amorphous ice (0.94 g cmÿ3).[177, 178] The high-density
structure compares with the density of water at 273 K
8.3. A Simple Two-Structure Model for Liquid Water
(1.00 g cmÿ3). The collapse of all dodecahedral structures
gives a density of 1.18 g cmÿ3, which is similar to that of high-
A simple two-structure model for liquid water was pro-
density amorphous ice at 1.17 g cmÿ3.[178]
posed by Benson and Siebert.[185] The authors were able to
The structures allow the explanation of many of the
construct a model involving isomerization between clusters
anomalous properties of water, such as its density maximum
which reproduces, within two per cent, the anomalous heat
as a function of temperature, and the viscosity minimum as a
capacity of liquid water from the melting up to the boiling
function of pressure. Additionally, the radial distribution
point. The clusters are polycyclic, cubic-shaped octamers
pattern, the presence of both pentamers and hexamers, the
which can dissociate into two cyclic tetramers. These clusters
change in properties and the ªtwo-stateº model of super-
(Figure 28) are held together by H-bonds and although very
cooling, as well as the solvation properties of ions, hydro-
phobic molecules, carbohydrates, and macromolecules can be
reproduced.
The strongest direct evidence for this model is the agree-
ment with the radial distribution functions. The high-density
structure showed comparable radial distribution functions
with that from X-ray[179] and neutron scattering[156, 180] data. As
shown in Figure 27 there is a peak maximum at 2.8 Š. The

Figure 28. Heat capacity Cp of water as a function of temperature as

calculated by Benson and Siebert[185] in comparison to measured data. The
two-structure model includes tetramers and cubic octamers only.

labile, they appear to be the principal species present in liquid

water according to this model. The fact that this two-structure
model is capable of giving reasonable heat capacities is not
too surprising. Structural theories of liquid water depend on
the assumption that H-bonds can be considered as ªintactº or
ªbrokenº. Benson and Siebert fitted their model directly to
Figure 27. Comparison of the calculated radial distribution function goo(R)
of the high-density structure (ÐÐ) with the X-ray diffraction data of bulk data, such as the sublimation energy of ice, the enthalpy
water[179] at 277 K (ÐÐ). The X-ray data yield a weighted sum of all atom of melting, and estimates of broken H-bonds. A satisfactory
pair distribution functions, which is mainly determined by the O ´´´ O description of one or more of the properties of water over a
contribution. The model peaks have been broadened by using a normal particular range of temperature and density is usually
distribution with a standard deviation of 0.1 Š. The radial distribution
function of oxygen atoms, as determined by neutron scattering,[156] is also
obtained when an equilibrium of two species differing
shown (- - - -). The figure is reproduced with the kind permission of Elsevier significantly in structural order and the number of H-bonds
Science and M. F. Chaplin.[133] is used. As Benson and Siebert pointed out, the heat capacity

Angew. Chem. Int. Ed. 2001, 40, 1808 ± 1827 1823

REVIEWS R. Ludwig

of water could also be obtained by assuming equilibria describing its dynamics. These configurations are never
between polycyclic decamers and monocyclic pentamers. present in low-lying minimum structures in ab initio calcu-
However, results from scattering experiments are not in lations at 0 K.
agreement with the proposed structures.[186] If an R(O ´´´ O) The combination of calculated clusters and their properties
distance of about 2.89 Š is assumed, then maxima in the pair with new laser spectroscopy experiments are likely to
correlation function should result at about 4.09 and 5.01 Š.[2] generate a wealth of information regarding the intricate
A maximum at 4.5 Š is found experimentally, which supports details of how water molecules interact. Recent experiments
a tetrahedral structure. have yielded detailed information about structural and
Recently, Rodriguez et al. considered the water hexamer dynamic aspects of small clusters. The clusters occur as nearly
and octamer in a theoretical study of isomerization, melting, planar ring structures up to pentamers, whereas three-dimen-
and polarity of model water clusters.[106] The model includes sional clusters are energetically favored with clusters larger
five hexamers of similar energy with different geometries and than the pentamers. Large intermolecular zero-point energies
dipole moments, as well as two nonpolar octamers with D2d of the H-bonds become crucial, and can alter the energy
and S4 symmetry. The melting transitions were studied by ordering of the low-lying hexamer structures. This is the
using ab initio methods and empirical force field models. The reason why the cage hexamer is found in the gas phase and the
melting transitions for the hexamer and the octamer were quasi planar ring hexamer in liquid helium. The physical and
found at 50 and 160 K, respectively. The authors conclude that chemical environment strongly determine the occurrence of
their results provide a comprehensive picture of the relation- different isomers. The geometries and vibrational frequencies
ship that exists between the spatial and polarization fluctua- have both been experimentally characterized up to the
tions that occur in small water aggregates. hexamer structures. The measured bond distances and vibra-
tional frequencies provide strong support for the cooperative
nature of hydrogen bonding. Growing cluster size, and thus
9. Summary and Outlook increasing H-bond strength, lead to a shortage of intermo-
lecular distances and a red-shift of the OH stretching modes
Small water clusters and their properties can be calculated as predicted by ab initio calculations. These findings certainly
by high quality ab initio and DFT methods using extended support theoretical predictions and experimental results that
basis sets. Most of the theoretical predictions for geometries show the covalent character of hydrogen bonding.
and vibrational frequencies, as well as their behavior with Three water models based on the assumption that calcu-
increasing cluster size, are supported by experimental infor- lated molecular clusters and structures may be constituents of
mation. To the best of our knowledge these state of the art the liquid phase were recent presented. The quantum cluster
calculations are applied to clusters comprised of up to eight equilibrium (QCE) model assumes that water species calcu-
water molecules. For larger clusters Hartree ± Fock calcula- lated from current ab initio methods are adequate represen-
tions are performed on smaller basis sets. DFT is the method tatives of the ªflickering clustersº of the liquid-phase struc-
of choice for including electron correlation where post- ture. This treatment of liquids is thus in the tradition of
Hartee ± Fock methods are too expensive or impossible. mixture models. The QCE model exhibits many characteristic
These calculations are known for clusters with up to 30 water features of a true gas/liquid phase transition (macroclustering,
molecules. For most larger clusters, such as Chaplins icosa- volume collapse, specific heat increase, Clausius ± Clapeyron
hedral water structures with 280 water molecules, only semi- pressure dependence). The inclusion of larger ice- and
empirical methods can be applied. The problem with these fullerene-like clusters leads to a new low-temperature phase
methods is that H-bonds and their cooperative behavior are that bounds both liquid and vapor regions in first-order
certainly not calculated in an appropriate way. Although transition lines, and gives rise to a true triple point. Although
calculations yield accurate structures and properties for small the bucky-ice phase differs in significant respects from
clusters, there remain shortcomings. Growing cluster sizes physical ice Ih , it manifests qualitatively correct thermody-
may lead to a dramatic increase in the number of isomers for namic features of true ice polymorphs, which suggests an
each species which is no longer manageable. It requires a lot important role of voluminous clusters in the liquid ± solid
of experience and knowledge to pick out the plausible low- transition. Nevertheless quantitative differences between
energy structures. The harmonic approximation is a further theory and experiment currently remain. For example, the
limit for calculating reasonable vibrational frequencies. The absence of a density maximum cannot be reproduced. The
calculated frequencies are usually overestimated and correct- failure to accurately describe higher order temperature
ed by a factor typical for the chosen method and basis set. It is derivatives suggests the importance of incorporating vibra-
some comfort that the measured trend of red-shifting of the tional anharmonic frequencies. Improved ab initio treatments
OH stretch in water clusters can be smoothly reproduced by of cluster ± cluster interactions and molecular excluded-vol-
using only one correction factor. However, this is no longer ume effects are also desirable. Although the current QCE
true for low-lying frequency vibrations, which are particularly model permits description of many interesting liquid proper-
important for the use of calculated frequencies in the vibra- ties at a useful level of chemical accuracy, rather high levels of
tional partition function. Another deficiency is the calculation theory as well as larger clusters may be necessary to achieve
of true minimum structures. Molecular dynamics and Monte convergence to a quantitatively realistic water phase diagram.
Carlo simulations have shown that bifurcated structures may Chaplin calculated large icosahedral structures, including
play a significant role in liquid water, particularly for 280 fully H-bonded molecules, to investigate long-range

1824 Angew. Chem. Int. Ed. 2001, 40, 1808 ± 1827

Water Clusters REVIEWS
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