Proton transport in water

Ab initio molecular dynamics and ab initio path integrals have been used to investigate the solvation structure and transport dynamics of hydronium and hydoxyl ions in bulk water at 300K.

When placed in bulk water, an excess proton will attach itself to a water molecule, forming a hydronium cation. The primary solvation structure of hydronium in water is one in which it is coordinated by three water molecules, forming the so-called "Eigen" cation, H₉O₄⁺, shown below:

A second solvation complex is also observed to form, however, in which the excess proton cannot be assigned to a particular oxygen but rather appears to be located directly between two water molecules. This is the H₅O₂⁺ cation, which plays an integral role in the proton transfer process. This complex is shown below:

The relative occurrence probability of each of these complexes is approximately 60% and 40%, respectively.

The following sequence of snapshots taken from the dynamical trajectory illustrate the process of proton transport in this system: The first shows the H₉O₄⁺ structure together with the second solvation shell water molecules of the hydronium.

Proton transport occurs first by the breaking of a hydrogen bond between first and second solvation shell, which leaves a water molecule in the first solvation shell with three neighbors. This threefold coordinated water can accept the proton from the hydronium, which occurs by the formation of an intermediate H₅O₂⁺ complex as shown below:

The proton then transfers to the accepting oxygen:

The proton transfer step is complete when the donating water closes its solvation shell by forming a fourth hydrogen bond with another solvent water. Thus, the entire solvation structure of the hydronium has migrated from one site to another through the hydrogen bond network via a proton transfer step. This is known as structural diffusion. It must be remembered, however, that this is only a simplified sketch of a rather complex process that could involve several transfers before the resolvation occurs. The movie below shows a segment of the trajectory depicting this event.

Below is shown the O-O radial distribution functions for the H₅O₂⁺ (dashed line) and the H₉O₄⁺ (solid line) complexes. The distribution functions are with respect to the hydronium oxygen. For the H₅O₂⁺ complex, an average over both oxygen atoms is taken.

The case of hydroxide in water has received considerably less attention in the literature. The reason for this goes back to a century-old notion which views hydroxide as a water molecule with a missing proton (the so called ``proton hole picture''). This idea implies that the mechanism of hydroxide transport can be understood on the basis of the hydronium mechanism by simply reversing all of the hydrogen bond polarities.

This idea does not, however, take into account the fact that the hydronium and hydroxide have a different chemistry in water. The hydroxide case is complicated by the fact that of the two solvation structures formed by the hydroxyl ion, one of them cannot diffuse. The two structures are shown below:

The complex on the left has the hydroxyl oxygen coordinated by four water molecules, with the hydrogen bonds all lying in a plane perpendicular to the OH_- bond axis. The complex on the left has the hydroxyl oxygen coordinated by three water molecules with the hydrogen bonds forming a more open, tetrahedral configuration. the H₉O₅^- structure (on the left) cannot diffuse as a result of the 90 degree angle between the OH^- the coordinating hydrogen bonds. In this configuration, the hydroxyl's accepting a proton from a neighbor would result in the formation of a new water molecule with a 90 degree bend angle, which is energetically highly unfavorable. In the H₇O₄^- structure (on the right), the angle between the OH^- bond axis and the coordinating hydrogen bonds is roughly 107 degrees. Thus the acceptance of a proton from a neighbor in this configuration results in the formation of a new water molecule with an approximately correct bend angle. Thus, this structure can diffuse.

Forming the H₇O₄^- structure from the H₉O₅^- structure requires the breaking of a hydrogen bond in the first solvation shell of the hydroxide ion. Moreover, there is evidence from our calculations that the hydroxide hydrogen can form a weak hydrogen bond to neighboring waters. The role of this hydrogen bond is to coordinate the hydroxide ion like a water molecule in the H₇O₄ ^- state before the proton transfer occurs. In this state, it is ``prepared'' to receive the proton and become a properly solvated water molecule. The full mechanistic process is shown in the figure below:

The blue surfaces designate the so called electron localization function, which indicates spatial regions of high probability to find electrons. The electron localization function shows, strikingly, that the long pairs of the hydroxide ion are not distinct but form a continuous ring, which supports the idea of flexibility in accepting hydrogen bonds and the possibility of hypercoordination (more than three hydrogen bonds). The movie below depicts the process:

Below is shown the O-O radial distribution functions for the H₇O₄^- (dashed line) and the H₉O₅^- (solid line) complexes. The distribution functions are with respect to the hydroxyl oxygen.

In both cases, the proton transfer rate correlates with molecular reorientation times, i.e., the average time required to break a hydrogen bond. This also correlates well with measured proton transfer rates from NMR, approximately 1.5ps.

When nuclear quantum effects are included via ab initio path integrals, no clear distinction between the two solvation complexes, H₅O₂⁺ and H₉O₄⁺, is possible. Rather, the picture that emerges corresponds to a ``fluxional'' complex that evolves continuously between these two ``limiting'' or idealized forms.

This effect can be quantified by examining the two-dimensional distribution of the oxygen-oxygen distance R_OO and the asymmetric stretch coordinate. In order that the analysis be as unbiased as possible, this analysis is carried out in three stages. First, the distribution of all hydrogen bonds is computed. This gives rise to a distribution with very broad wings as shown in the figure below:

However, note that there is a finite probability at zero asymmetric stretch for certain values of the OO separation. This suggests the existence of centrosymmetric complexes in the liquid with the proton shared equally between two oxygens, a fact that rules out a description solely in terms of Eigen's H₉O₄⁺ complex picture.
In the next stage of the analysis, only those hydrogen bonds comprising the first solvation shell of the defect oxygen O* are kept. This oxygen is defined to be the oxygen with three hydrogen neighbors. When this distribution is computed, the result is that shown below:

The distribution still shows two distinct wings while the region around zero asymmetric stretch is enhanced considerably. The two wings are due to two long hydrogen bonds to the first solvation shell members of O*. Thus, a final step of refinement is to focus on the ``most active'' hydrogen bond, i.e., the one with the smallest asymmetric stretch. The distribution corresponding to this single hydrogen bond is shown below:

This distribution has a flat, featureless single peak. This suggests that there is a continuum of possible complexes occuring with approximately equal probability, with the Eigen, H₉O₄⁺, and Zundel, H₅O₂⁺, occuring only as limiting or ideal forms. In other words, there is a ``fluxional'' solvation complex formed by the hydrated proton.
The mechanism of proton transfer, based on second solvation shell dynamics, can be investigated by computing the coordination number of the water molecule across the hydrogen bond with the shortest asymmetric stretch from the O* defect site. The coordination is superimposed on the distribution in color. The color darkens as the coordination number decreases, with the darkest color at R_OO small and zero asymmetric stretch:

This suggests that quantum effects do not affect the essential mechanism of proton transfer in water.
Another manifestation of quantum effects is that the defect can become delocalized over the hydrogen bond network. This effect appears in certain ``non-classical'' configurations, such as the one shown below:

The picture shows the imaginary time path in the region of the defect. In each imaginary time slice, the defect oxygen O* is colored yellow. It is, thus, clear that in different imaginary time slices, there can be a different defect oxygen. In more traditional language, the thermal wave packet corresponding to the defect site delocalizes over the hydrogen bond network, ``probing'' possible directions for the next proton transfer step. Once a proton transfer event is completed, the defect tends to localize for a period on a particular oxygen.

For hydroxide in water, the most significant quantum effect is in the geometry of the threefold coordinated complex. In the classical picture, the proton can only transfer if the angle between the OH^- bond axis and the hydrogen bond through which the proton is transferring is near the perfect tetrahedral value of 107 degrees. When nuclear quantum effects are included, this is no longer the case, and a wide range of angles is allowable. This so called ``corner cutting'' effect could possibly explain why there is a larger observed isotope effect in hydroxide than in hydronium transport. This quantum effect is illustrated below in the distribution functions of the proton transfer coordinate and angle:

where the distributions on the left and right are the classical and quantum distributions, respectively.

References

Ab initio molecular dynamics simulation of the solvation and transport of hydronium and hydroxyl ions in water.
M.E. Tuckerman, K. Laasonen, M. Sprik, and M. Parrinello J. Chem. Phys. 103, 150 (1995).

Ab initio molecular dynamics simulation of the solvation and transport of H₃O⁺ and OH^- ions in water.
M.E. Tuckerman, K. Laasonen, M. Sprik, and M. Parrinello J. Phys. Chem. 99, 5749 (1995).

The nature of the hydrated excess proton in water .
D. Marx, M.E. Tuckerman J. Hutter and M. Parrinello, Nature 397, 601 (1999).