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Water dimer and small clusters

water dimer with parameters from ab initio 6-31G** calculation V Water dimer

V Water dimer inside fullerene C70
V Comparison with trimer, tetramer and pentamer

V Small clusters

Water dimer

Much effort has been expended on the structure of small isolated water clusters. The smallest cluster is the dimer. Typically, in the ambient atmosphere there is over one water dimer for every thousand free water molecules rising to about one in twenty in steam. It has one mirror plane of symmetry (CS ) and dimensions from ab initio 6-31G** calculation as shown right. The charge transfer for this calculation is only 0.03 e from δ+ve acceptor molecule to δ-ve donor molecule. Its molecular orbitals are shown on another page.


The variation of water's fugacity coefficient with pressure


The equilibrium constant for dimer formation

2H2O(g) → (H2O)2(g)

is 0.0501 bar−1 at 298.15 K [2052], with dissociation energy of 13.2 kJ ˣ mol-1 [2210a]; (D2O)2 14.88 kJ ˣ mol-1 [2210b] (but see below). The thermochemical properties of the dimer have been determined [2052] and experimental dimer studies have been reviewed [2314].


The dimer formation causes deviations from ideal behavior in gaseous water (see left) [2356].a

The effect of the hydrogen bond on the stretch vibrations is seen below. The activation energy required to switch from one acceptor position to the other on the acceptor molecule (1.88 kJ ˣ mol−1) is only 40% of that required to switch from one donor hydrogen to the other (4.71 kJ ˣ mol−1) on the donor molecule [2314] via the bifurcated state, both of course far lower than the dissociation energy.

Vibrational bands

symmetric stretch

(v1, cm−1)


(v2, cm−1)

asymmetric stretch

(v3, cm−1)

H2O monomer [607] 3657 1595 3756

HO-H···OH2 dimer donor

3545 [2314] 1669 [2315] 3715 [2314]
HO-H···OH2 dimer acceptor 3600 [2314] 1653 [2315] 3730 [2314]


The high-resolution spectra for the out-of-plane librational vibrations, at around 500 cm−1, of the water dimer has been published [2719] as has the rotational spectra around 4 cm−1 [1977b].


Bond energies calculated using 6-31G** basis set

Calculated bond energies for the water dimer are given right using the 6-31G** basis set. Particularly noteworthy is the steepness of the 'wall' inside the optimum position and the more relaxed structures allowed outside the optimum position that still have significant bond energy. The position of the hydrogen bond stretch vibration energy (~200 cm−1) is shown with vibrational range of about -0.15 Å +0.3 Å. Although stated in the early literature that the behavior is purely electrostatic at larger distances, this is not true. The most energetically favorable water dimer is shown above right using ab initio calculations with the 6-31G** basis set. It is also shown below with a section through the electron density distribution (high densities around the oxygen atoms have been omitted for clarity). This shows the tetrahedrality of the bonding in spite of the lack of clearly seen lone pair electrons; although a small amount of distortion along the hydrogen bond can be seen. This tetrahedrality is primarily caused by electrostatic effects (that is, repulsion between the positively charged non-bonded hydrogen atoms) rather than the presence of tetrahedrally placed lone pair electrons. The hydrogen-bonded proton has reduced electron density relative to the other protons [222]. Note that, even at temperatures as low as a few kelvin, there are considerable oscillations in the hydrogen bond length (~ ±0.2 Å at 50 K with timescales ~0.2 ps) and angles (~ 9 ° at 50 K with timescales ~0.2 ps) [591]. The potential energy surface [1668] and wagging vibration [1743] of the water dimer have been described.

water dimer showing the electron density perturbation along the hydrogen bond



water dimer dimensions

R = 2.976 (+0.000, -0.030) Å, α = 6 ± 20°, β = 57 ± 10° [648]; α is the donor angle and β is the acceptor angle. The dimer (with slightly different geometry) dipole moment is 2.6 D [704]. Although β is close to as expected if the lone pair electrons were tetrahedrally placed (= 109.47°/2), the energy minimum (~21 kJ mol-1) is broad and extends towards β = 0°.


It has been noted that dimers of more distant water molecules (~1 nm) show synchronous behavior due to their interacting electric fields [2086].


Bond energies of dimers with temperature. from [2442]


At lower temperatures the bond energy for (H2O)2 and (D2O)2 dimers reduce with temperature, explained as due to the vibration amplitude growth. However at higher temperatures, the bond energies of both increase with temperature. This phenomenon also occurs in methanol but is difficult to explain [2442].

C70 fullerene containing a water dimer [2517]Water dimer inside fullerene C70

Using clever synthetic methods involving opening up a fullerene, inserting the water molecule(s) under pressure and then re-sealing the fullerene, one molecule of water has been placed in C60 [2516] and C70 fullerenes [2517] and two water molecules placed inside the C70 fullerene [2517] (see right).


The water dimer inside the C70 fullerene is free from any further hydrogen bonding to water molecules and is prevented from dissociating due to the confinement. It has a cis-linear conformation resulting from confinement (3.7 Å x 4.6 Å prolate ellipsoid cavity) and high effective pressure effects inside C70. This has a structure similar to the most energetically favorable water trans dimer (see above) with the important change in that the water molecule on the left (above) is flipped up rather than down (see right).

Comparison with trimer, tetramer and pentamer

The water trimer ring system has been reviewed up to 2003 [2426] and reanalyzed in 2016 [2726]. It appears that quantum delocalization of hydrogen bonded protons between oxygen neighbors occurs in the trimer and pentamer, but not the tetramer and hexamer [2426]. Such aromatic-like delocalization within pentamers and dodecahedra present in supercooled water stabilize their structures and contribute to the stability of ES-clustering, where half the water molecules are within pentamers.


The average dimensions for the trimer, tetramer and pentamer from ab initio 6-31G** calculation are shown below. The charge on the donor hydrogen atoms increase, the hydrogen bond lengths contract and the electron density width within the hydrogen bonds increase as the structure goes from dimer to trimer to tetramer to pentamer.

Structure of water trimer, tetramer and pentamer. The parameters are averaged for similar positions from ab initio 6-31G** calculation


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Small clusters

Many theoretical studies in vacuo and some experimental work in the gas phase have been carried out on small water clusters. Often the underlying theme is that the structures will be of use in understanding the structure of liquid water. This, however, is misleading as the lowest energy structures structures of clusters, with greater than five water molecules, do not occur in liquid water. The reason for this is that the small water clusters described balance the strength of their hydrogen bonding against the number of hydrogen bonds that can be formed, often maximizing the number of hydrogen bonds with each hydrogen bond being of sub-maximal strength (17-19 kJ ˣ mol-1, [2596] ) and with somewhat strained angles (poor directionality). If such clusters were to be immersed in liquid water, many new hydrogen bonds would be established around their periphery with the surrounding water molecules in the liquid; with the result that the original cluster would immediately 'dissolve' and change structure.


The structures of these small water clusters are interesting, however. The most stable structures with up to five water molecules are given as the above planar structures. Examples of larger clusters which have three-dimensional structures are given below.

(H2O)6 prism from [2533]              (H2O)8 cube from [2595]

(H2O)6 prism [2533] with 9 hydrogen bonds                 (H2O)8 cube [2595] with 12 hydrogen bonds


In liquid water the preferred (H2O)6 conformation would be the chair hexamer with just 6 internal hydrogen bonds but with an additional 12 hydrogen bonds to other water molecules in the liquid (about twice as many and stronger hydrogen bonds than in the gas phase) .Also in liquid water the preferred (H2O)8 conformation would be the bicyclo- octamer with just 9 internal hydrogen bonds but with an additional 14 hydrogen bonds to other water molecules in the liquid (also with about twice as many and stronger hydrogen bonds than in the gas phase). Larger structures have been found by ab initio calculation (for example [115] ). As an example, (H2O)16 was shown to be optimal in a tri-stacked cube conformation with 28 hydrogen bonds; a structure highly unlikely to survive in a liquid water milieu.

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a The fugacity (f) of a real gas is the effective pressure which replaces the true mechanical pressure due to the non-ideality; f = φ x P where φ is the fugacity coefficient and P is the pressure. For an ideal gas f = P. [Back]



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