Global minima of protonated water clusters

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Abstract

Candidate global minima are obtained for H3O+⋯(H2O)n clusters with n⩽20 using a basin-hopping algorithm and an empirical, polarizable model potential. We have reoptimized the lowest minima for each system using a more accurate model and find extensive reordering of the potential energy surfaces, especially for larger n. For both model potentials a distorted dodecahedron surrounding an H2O molecule is the global minimum for n=20, in good agreement with experiment. The gap between the latter structure and the lowest minimum with H3O+ in the centre is about 10–20 kJ mol−1.

Introduction

There have been many previous studies of protonated water clusters at various levels of theory. The most accurate calculations have, of course, been conducted for the smallest systems; overviews of the literature for n=1 and n=2 can be found in our recent studies of rearrangement mechanisms in these clusters 1, 2. Benchmark calculations on H5O2+ by Valeev and Schaefer are noteworthy for revealing the sensitivity of the potential energy surface (PES) to basis set and correlation effects [3]. Other studies have employed a variety of techniques including MP2, density functional theory (DFT) and semi-empirical methods 4, 5, 6, 7, 8, 9, 10.

For larger systems, model potentials parametrized using experimental or ab initio data provide a computationally efficient alternative to more accurate techniques. One early study by David [11]used a polarization model [12]to optimize a dodecahedral structure for n=20. Kozack and Jordan developed empirical interaction potentials for H+⋯H2O and H3O+⋯H2O, which we refer to as KJ(H+) and KJ(H3O+) respectively, and used them to study n⩽6 [13]and later n=19–21 clusters [14]. Corongui et al. [15]complemented DFT calculations with results based on a model potential composed of the MCY form [16]for the H2O⋯H2O interactions and a potential developed by Fornili et al. [17]for the H3O+⋯H2O interactions. One of us has developed a sophisticated anisotropic site potential (ASP) [18]and used this to study the structure and energetics of clusters with n=1–7, and more recently Shevkunov and Vegiri [19]developed another potential and applied it to clusters up to about n=50. The KJ(H+) potential has also been used by Svanberg and Pettersson [20]in Monte Carlo simulations for n=8, 20 and 39. These are just a few examples of previous theoretical studies, and a useful review of the subject has been presented by Kochanski et al. which includes a historical account of the study of these systems, discussing the use of the various methods mentioned above [21].

The most basic structural question one can ask is the geometry of the global potential energy minimum of the cluster. If it is kinetically accessible on the experimental timescale then this minimum will dominate the low temperature behaviour of the system. The problem of global minimization has been studied quite extensively for atomic systems and proteins (for a recent overview see Ref. [22]) and we recently published a study for neutral water clusters containing up to 21 molecules [23]using the empirical TIP4P potential [24], supplemented with more accurate energy evaluations using the polarizable ASP-W4 model [25]. In Ref. [23]putative global minima were located using the basin-hopping technique [26]and the conclusions of our study have since found support in further calculations by Day et al. [27]. Here we present an analogous study for the case of protonated water clusters. The only previous basin-hopping study that we know of for any of these systems was performed by Singer et al. [28]who used it to complement jump-walking Monte Carlo (MC) simulations. The MC calculations were not successful in locating the global minimum for n=7 and numerous lower-energy minima were found using the basin-hopping technique. For n=15, only MC calculations were performed, and a minimum was constructed by hand that was lower in energy than those previously found, even though this structure was approximately planar with considerably fewer hydrogen bonds than would be expected for 3-dimensional morphologies.

Our aim in this contribution is to search the PESs of protonated water clusters to locate low-energy minima and to identify favourable morphologies. The coordinates of our lowest energy structures will be made available for downloading from the Cambridge Cluster Database [29]. One concern with some of the previous studies is that rather limited sets of configurations have been considered, and we intend to generate a large database of minima that will be made available as a reference for more accurate calculations. To illustrate the scale of the problem of searching the potential energy surfaces for these clusters, we note the work of McDonald et al. [30]who used graph theoretical techniques to enumerate the number of viable minima for cubic morphologies with n=7 and dodecahedral morphologies with n=19. This method provides a systematic way of investigating the possible hydrogen-bonding patterns for a given set of locations of the oxygen atoms. For n=7, 11 initial configurations were generated from which six minima were obtained using a model potential from the OSS family [31]. For n=19, the number of hydrogen-bonding topologies was found to be nearly 90 000 for structures including a H3O+ unit acting as triple proton donor, of which about 200 were selected for further analysis using an empirically determined fitness criterion based on topological properties. The above study illustrates that a comprehensive sampling of low-energy structures is likely to become computationally unfeasible even for quite small clusters, and that locating the global minimum for the larger systems will be a very challenging problem. As we noted for neutral water clusters [23], the interplay of centre-of-mass and angular degrees of freedom produces a more complex PES than for atomic clusters with a similar number of degrees of freedom.

Section snippets

Methods

The details of the global optimization calculations are much the same as in our previous study [23], though here we use the KJ(H3O+) model potential of Kozack and Jordan [13]. All calculations were performed with a modified version of the Orient program [32]. We performed preliminary runs of 30 000 quenches for n=9 starting from randomly generated geometries, with various temperatures and acceptance ratios, and a common lowest-energy minimum was found in all runs. The minimum number of quenches

Basin-hopping with the KJ(H3O+) model

One or three basin-hopping runs, consisting of 30 000 quenches each, were performed for n⩽9 and n=10–20, respectively. The lowest three energies found and the quench at which they were first located are collected in Table 1. The lowest 10 energies are given in Table 2.For n<10, the lowest-energy minimum was located in under 1000 quenches in all cases and there is little doubt these are the global minima for this particular potential. For n=10–20 the likelihood that we have found the true global

Conclusions

We have performed basin-hopping simulations with the empirical KJ(H3O+) model potential for a number of protonated water clusters and have presented candidate global minima up to n=20. There does not usually appear to be any connection between the H3O+⋯(H2O)n and (H2O)n+1 global minima. The protonated water clusters are generally more disordered than their neutral counterparts and it is likely that the additional proton has a large influence on the local structure. Clathrate structures are

Acknowledgements

M.P.H. thanks the Leverhulme Trust for financial support.

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