Clusters of Coarse-Grained Water Molecules - The Journal of Physical

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Clusters of Coarse-Grained Water Molecules James D. Farrell* and David J. Wales* University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom S Supporting Information *

ABSTRACT: Global optimization for molecular clusters can be significantly more difficult than for atomic clusters because of the coupling between orientational and translational degrees of freedom. A coarse-grained representation of the potential can reduce the complexity of this problem, while retaining the essential features of the intermolecular interactions. In this study, we use a basin-hopping algorithm to locate putative global minima for clusters of coarse-grained water molecules modeled using a monatomic water potential for cluster sizes 3 ≤ N ≤ 55. We characterize these structures and identify structural trends using ideas from graph theory. The agreement with atomistic results and experiment is rather patchy, which we attribute to the tetrahedral bias in the three-body potential that results in too few nearest neighbor contacts and premature emergence of bulk-like structure. In spite of this issue, the results offer further useful insight into the relationship between the structure of clusters and bulk phases, and the mathematical form of a widely used model potential.

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INTRODUCTION In spite of its essential ubiquity, water still conceals from us many of its mysteries. Studies of water clusters provide insight into these questions, as they bridge the gap between the water molecule and the condensed phase. Consequently, a great deal of work in both experiment1−14 and theory15−35 has been devoted to understanding their properties. As the number of minima on a potential energy surface is believed to increase exponentially with the number of degrees of freedom,36,37 existing deterministic global optimization techniques rapidly become intractable for small water clusters, warranting the use of stochastic methods. In addition, this scaling essentially precludes the systematic study of computationally expensive ab initio surfaces. Most high level ab initio studies that appear in the literature are restricted to small cluster sizes and can benefit from the use of a computationally cheap, empirical guiding potential.26,27 Hence, the vast majority of water cluster global optimization studies employ stochastic optimization for a model potential surface.15−25 The most widely used have been the TIPnP family potentials38−41 and their derivatives, which employ long-range electrostatic forces to produce short-ranged tetrahedral ordering of rigid water monomers. The most widely used of the variants, the TIP4P potential,38 agrees reasonably well with both experiment and ab initio results for clusters of up to twelve monomers.18 A wide variety of global optimization techniques have been employed for water clusters in previous work, and we will not attempt to review them all here. Tsai and Jordan15 proposed TIP4P global minima for N = 8, 12, 16, and 20 obtained with Monte Carlo simulated annealing. Wales and Hodges16 located putative lowest energy structures for N ≤ 21 obtained using the basin-hopping (BH) algorithm, results corroborated by Hartke’s17 extended phenotype algorithm. A combination of © XXXX American Chemical Society

temperature dependent classical trajectories, hydrogen bond topology optimization, and diffusion Monte Carlo was used by Kazimirski and Buch19 to find low energy structures for much larger clusters with N = 48, 123, and 293. Kabrede and Hentschke18 used a genetic algorithm to suggest global minima for N ≤ 25, later extended to N ≤ 30 using a BH algorithm with moves derived from a normal-mode analysis.21 Takeuchi investigated the landscape over the same range utilizing hydrogen bond topology altering moves.22 In the latest of a series of contributions,23−25 Kazachenko and Thakkar25 extended this range to N ≤ 55 using a hybrid scheme that exploits genetic topology optimization. Stochastic optimization for empirical water potentials is relatively difficult because of the interplay between orientational and translational degrees of freedom.16,42 Each arrangement of water molecules may be described by the underlying oxygen skeleton topology, onto which a number of hydrogen bond topologies may be superposed. The twenty molecule dodecahedron, for instance, can support 30 026 symmetry distinct hydrogen bond topologies43 that satisfy modified ice rules.44 The energy of a water molecule is strongly dependent on both the skeletal and directional arrangements, producing a hierarchical disconnectivity graph42,45 which impedes a systematic approach to the global minimum. Singer and coworkers43,46 have used the symmetry of the skeletal topology along with ideas from graph theory to efficiently optimize hydrogen bond topologies in bulk ice and small, symmetrical Special Issue: Kenneth D. Jordan Festschrift Received: December 6, 2013 Revised: February 27, 2014

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dx.doi.org/10.1021/jp4119666 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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lowest energy was reached. The BH scheme employed here proceeded as follows: 1. a translational perturbation was applied to the initial coordinates, Xi; 2. the perturbed coordinates, X′i , were quenched to a local minimum, Xn; 3. the new configuration, Xn, was accepted with probability

clusters. However, the low symmetry of most water cluster global minima means that these techniques are not generally applicable. One approach to the complicating effect of these strongly coupled translational and orientational degrees of freedom is to eliminate the latter entirely. Many coarse-grained models of water do just this,47−49 representing a water molecule with a single site without explicit incorporation of the hydrogen atoms. One such model is the monatomic water representation (mW) proposed by Molinero and Moore,49 which exploits the structural, thermodynamic, and dynamic similarities of water and silicon. Both systems form tetrahedral crystals at ambient pressure and have analogously anomalous density maxima50,51 and decreasing viscosity on compression.50,52 The form of the potential is exactly that of Stillinger−Weber silicon (SW),53 V=

p(i→n) = min(1, e−β ΔE)

where ΔE = En − Ei, Ei, and En are the energies of the initial and new configurations and β = 1/kBT. Now we define the step ratio, Srat, to be the probability that a perturbation/minimization step ends in a different basin,

∑ ∑ ϕ2(rij) + ∑ ∑ ∑ ϕ3(rij ,rik ,θjik) i

j>i

i

which is determined using the stochastic distance minimization scheme outlined in ref 70. Two configurations correspond to the same basin if the Euclidean distance between them, minimized with respect to rotation, translation, and permutation of identical particles, is less than 10−3 Å. The temperature ratio, Trat, is defined as the probability that such steps are accepted,

j≠i k>j

a sum of a Lennard-Jones Gaussian form, ⎡ ⎛ ⎞4 ⎤ ⎛ σ ⎞ σ ⎟⎟ ϕ2(rij) = A ϵ⎢B⎜⎜ ⎟⎟ − 1⎥ exp⎜⎜ ⎢ ⎝ rij ⎠ ⎥ ⎝ rij − aσ ⎠ ⎣ ⎦

where rij is the distance between particles i and j, and a threebody tetrahedral potential,

which means the acceptance ratio, R, which includes the probability of remaining in the same minimum, is simply

2 ⎛ γσ ⎞ ⎡ 1⎤ ⎟⎟ ϕ3(rij ,rik ,θjik) = λ ϵ⎢cos θjik + ⎥ × exp⎜⎜ ⎣ 3⎦ ⎝ rij − aσ ⎠

R = 1 − Srat(1 − Trat)

(3)

This decomposition of the usual acceptance ratio employed in the GMIN program69 can provide a parametrization of the BH step size and temperature which, being bounded between zero and one, is independent of the length and energy scales associated with the system, and so more closely reflects the topology of the landscape. In the present work Srat was chosen to be 0.95 and Trat to be 0.05, yielding an overall acceptance ratio of about 10%. The step size and temperature were dynamically adjusted toward these target values at intervals of 100 BH steps. Structural Analysis. Throughout we make use of graph representations of cluster structures in conjunction with graph theoretical tools implemented in the NetworkX package.71 Each particle is represented by a vertex, and an edge is drawn between two vertices if the pair energy u and to node u from the largest w such that w < u, where such a relationship exists. Lower panel: structures of the global minima of (a) mW21, (b) mW28, (c) mW27, (d) mW29, (e) mW40, (f) mW46, (g) mW54, and (h) mW45. Tubes indicate nearest neighbor contacts (pair energy < −0.45ϵ). The extended core is blue, and the remainder of the cluster skeleton is translucent orange.

lowest energy mW21 and mW28 clusters are the same as their TTM2.1-F counterparts. To investigate the similarity of the landscapes for coarsegrained and atomistic models, the TIP4P, TIP4P/2005, TIP4P-Ew, TTM2.1-F, and AMOEBA structures published in ref 25 for cluster sizes 10 ≤ N ≤ 55 were reoptimized at the mW level, taking the centers of mass of the water monomers as the starting coordinates. The average difference in energies between these minima and the mW global minima varies from 0.81 kJ mol−1 per monomer for the AMOEBA structures to 1.05 kJ mol−1 per monomer for the TIP4P/2005 structures. However, the average root-mean-squared deviation (RMSD) between the structures before and after minimization varies between just 0.26 Å for TIP4P and 0.31 Å for AMOEBA, and the topologies are largely retained, which suggests that, although these structures are not favored for the mW potential, they do correspond to local minima. The origin of the different structural preferences may be analyzed by comparing the components of the mW potential for the two sets of minima. Although the two body-term is significantly more favorable for the mW reoptimized atomistic minima, owing to the higher coordination number, the nonideal angles required to obtain these coordination numbers increase the three-body tetrahedral penalty by a greater degree. The

result is that too many three-coordinate water molecules appear on the surfaces of mW clusters. The structural analogy between the atomistic water and mW landscapes is closely similar to that identified by Bromley et al. between water and silica.72 There it was noticed that, although the ground states of water and silica nanoclusters, (H2O)N and (SiO2)N, were mostly different up to N ≤ 25, low energy structures with similar topologies exist on both landscapes. The preference for one cluster type over another was attributed to the different surface termination mechanisms available. In water, the number of dangling hydrogen atoms at the surface can be reduced by distortion from tetrahedral angles, allowing more hydrogen bonds to form. In silica, the number of dangling oxygen atoms may be reduced by silicon atoms sharing more than one oxygen atom. Additionally, silicon−oxygen double bonds are possible, giving rise to stable “pendant units”. In the mW potential, the tetrahedral bias suppresses distortion from tetrahedral angles, and multiple bonds between mW particles are not possible, with the result that none of the mechanisms discussed above are available. The reverse transformation, from coarse-grained to atomistic structure, requires a choice of hydrogen bond topology and is thus much less straightforward. However, we notice that for the size range in question, except in those cases where mW and G



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atomistic clusters have the same structure, the coordination number of the mW minima is always lower. The closest comparison in this regard is with the AMOEBA clusters, which have a mean average coordination number 0.09 larger than mW clusters. This measure is greatest (0.12) for the TIP4P/2005 model. In addition, in atomistic clusters, with directional bonding, three-coordinate species either accept two hydrogen bonds and donate one (AAD), or donate two and accept one (ADD). It is well-known that configurations with adjacent AAD or ADD pairs are energetically unfavorable.73 72% of the mW global have odd-membered rings composed of three-coordinate particles. Because the smallest number of colors needed to color the vertices of an odd cycle so that no two adjacent vertices share the same color is three,74 there exists no hydrogen bond topology for these cluster skeletons that does not necessarily contain such an unfavorable pairing. In the bulk, the tetrahedral bias naturally produces extended four-coordination. However, for isolated clusters, favoring tetrahedral angles does not suffice to reproduce the observed propensity of atomistic models to maximize the number of hydrogen bonds formed and therefore the number of fourcoordinate surface molecules. Emergence of Ice-like Structure. An interesting feature of the mW clusters is the early emergence of ice-like substructure. Spectroscopic observations suggest the onset of crystallization at N = 275 ± 25.13 The lowest energy structure obtained by Kazimirski and Buch19 for TIP4P293 is ice-like. Bandow and Hartke,20 having relaxed and partially optimized spherical fragments taken from a collection of ice phases, suggest that an ice-like core will not be observed in TTM2-F global minima for N < 90. In the latter report, the retention of ice-like order was determined using a geometric definition. The RMSD between the spherical fragments before and after optimization was calculated, and those molecules whose contribution to the RMSD is below some threshold were considered to be in sites consistent with the initial ice lattice. Here, we attempt to identify ice-like structure by choosing some canonical fragment of a crystalline ice phase and searching for subgraphs of a cluster isomorphic with these fragments. The advantage of this method is that no geometric cutoffs are required, and that substructures will be identified even if they are significantly distorted from the ideal geometry. We choose the decameric and icosameric fragments of the cubic and hexagonal ice lattices, shown in Figure 8a,b. The number of molecules belonging to cubic and hexagonal fragments is plotted in Figure 9. In the global minimum structures, cubic substructures are less frequently encountered. Two edge-sharing fragments are found in mW43, and three facesharing fragments in mW46 and mW50. Far more extensive cubic structure can be found, however, in other low energy structures. There exists a minimum for 46 particles 2.8 kJ mol−1 higher in energy than the global minimum, in which 65% of particles are located in cubic fragments. A similar structure exists for 50 particles, 7.9 kJ mol−1 above the global minimum, with 68% of particles in cubic fragments (Figure 8d). Each of the substructures consists of two layers, with the remaining particles arranged as either dimers or five-membered rings, as noted for the hexagonal structures. A single hexagonal subunit is first observed in a global minimum structure of mW39, and a pair of fused subunits occurs in mW40. Three-coordinate particles whose nearest neighbors do not belong to the fragment deviate most from the ideal geometry. With the exception of mW48, all larger even-

Figure 8. Upper panel: structures of (a) the decamer fragment of cubic ice, (b) the icosamer fragment of hexagonal ice, and (c) the pentadecamer fragment of the icosahedral water model. Lower panel: examples of clusters containing these fragments, namely (b) cubic ice substructure in the low energy minimum of mW50 discussed in the text, (e) hexagonal ice substructure in the global minimum of mW46, and (f) icosahedral subunits in mW47. The extended core is blue, and the remainder of the cluster skeleton is translucent orange.

Figure 9. Percentage of monomers belonging to cubic and hexagonal fragments as a function of cluster size.

membered clusters contain at least one hexagonal subunit. The most extensive hexagonal substructure is observed in mW46 and mW50, in which three hexagonal subunits are identified and 57 and 48% of particles are found in lattice sites, respectively (Figure 8e). For these cluster sizes, particles above and below the plane of the hexagonal moiety arrange into five-membered rings, whereas the sides are coordinated by dimers and fourmembered rings. In addition to ice-like structure, we also see the “pentagonal box” fragment that appears in the expanded structure of Chaplin’s icosahedral network model75 (Figure 8c). Two such boxes appear in the global minima of mW47 and mW49, and in a low energy structure of mW45 (Figure 8f). The mW model thus produces bulk-like structure in global minima for cluster sizes much smaller than for atomistic models.19,20 This trend results from the propensity of the mW potential to form tetrahedral angles at the expense of fourcoordinate geometries, as explained in the previous subsection.

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CONCLUSIONS For atomistic water models, cluster global minima tend to be dense, maximizing the number of hydrogen bonds formed. Save for small clusters, there are few simple relationships between H



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clusters of similar sizes. We have demonstrated that, in spite of many successful applications to condensed phase systems,49,60−63 the coarse-grained mW water model predicts cluster morphologies that are often at odds with both those obtained with atomistic potentials and observed experimentally. As a result of the tetrahedral bias in the three-body potential, the average coordination number of an mW global minimum cluster is lowered with respect to its atomistic counterpart, and bulk-like substructure begins to emerge for much smaller cluster sizes. In contrast with atomistic clusters, clear structural trends exist as a function of cluster size. We have identified structural patterns for small clusters (3 ≤ N ≤ 20), which form open geometries with close to tetrahedral angles, and found that the structures of the larger mWN studied here depend strongly on N, with even numbers favoring a morphology based on hexagonal ice, whereas odd-membered clusters favor substructure based on a pentameric ring. It has been demonstrated that an isotropic model of water cannot simultaneously reproduce the structure and thermodynamics of bulk water;48 in the present work we find that the mW coarse-grained water model does not reproduce the global minima of water clusters as well as it appears to represent bulk water. A question that naturally arises is whether or not a single-site representation exists that might perform well in both limits. A coarse-grained water model that attempts to simultaneously reproduce bulk water and the water−vacuum interface has recently been reported76 and suggests that a compromise between local structure and surface geometry must be made. The subtle differences in the structural preferences of water and silica clusters can be explained by the different surface termination mechanisms available. A proper description of the differences between these substances clearly requires inclusion of the orientational degrees of freedom of the monomer units. It is doubtful whether a SW-type potential could be capable of reproducing these effects.

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ASSOCIATED CONTENT

S Supporting Information *

Cartesian coordinates and energies of the global minima in xyz format. This material is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Authors

*J. D. Farrell: e-mail, [email protected]. *D. J. Wales: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to the EPSRC for financial support. REFERENCES

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Clusters of Coarse-Grained Water Molecules - The Journal of Physical

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