Structural Correspondences between the Low-Energy Nanoclusters of

J. Phys. Chem. C 2008, 112, 18417–18425

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Structural Correspondences between the Low-Energy Nanoclusters of Silica and Water Stefan T. Bromley,*,†,‡ Bernhard Bandow,§ and Bernd Hartke| Departament de Quı´mica Fı´sica & Institut de Quı´mica Teo`rica i Computacional (IQTCUB), UniVersitat de Barcelona, C/ Martı´ i Franque`s 1, E-08028 Barcelona, Spain, Institucio´ Catalana de Recerca i Estudis AVanc¸ats (ICREA), E-08010 Barcelona, Spain, Regionales Rechenzentrum fu¨r Niedersachsen, Leibniz-UniVersita¨t HannoVer, Schloβwender Straβe 5, D-30159 HannoVer, Germany, and Institut fu¨r Physikalische Chemie, Christian-Albrechts-UniVersita¨t zu Kiel, D-24098, Kiel, Germany ReceiVed: July 30, 2008; ReVised Manuscript ReceiVed: September 11, 2008

Nanoclusters of two well-known tetrahedrally coordinated systems, (SiO2)N silica and (H2O)N water, are compared in the size range N ) 8-26 with respect to energetic stability and T-atom topology (where T ) Si and O, respectively). Because of the similar cage-forming tendencies of both systems in a number of bulk phases (e.g., clathrates), we focus on clusters with an even number of T-atoms for which cluster structural decomposition into a union of convex polyhedra (i.e., face-sharing cages with T-atom-vertices) is always possible. For (H2O)N, in the range N ) 8-24, the even-N ground-state clusters are almost exclusively formed from face-sharing cubes and pentaprisms of T-atoms. We show that (SiO2)N clusters having the same T-atom topology as these prismatic water clusters are also low in energy (per SiO2 unit with respect to the corresponding lowest energy (SiO2)N cluster), and have the same groundstate structure for N ) 12. Along with similarities, an important difference is discussed, namely that, rather than a converging tendency for the two systems to have similar prismatic topologies with increasing size, for (H2O)N at N ) 24, self-solvating amorphous clusters compete strongly with the prismatic ground states, becoming ground states themselves for N > 24. Contrary to this shift in water cluster structural preference, (SiO2)24 clusters with the same T-atom topology as the amorphous (H2O)24 clusters are found to be high in energy with respect to both the (SiO2)24 ground-state and (SiO2)24 prismatic isomers. We discuss possible reasons for the existence and eventual breakdown of the topological analogy between (H2O)N and (SiO2)N nanoclusters with relation to the different bonding characteristics in each system. 1. Introduction Silica and water are known to display numerous analogous bulk phases in which the semicovalently bonded SiO2 network topologically corresponds to an isomorphic framework of hydrogen-bonded H2O molecules through a {Si, O}silicaT{O,H}water oneto-one mapping. The success of this structural crossover is mainly due to the fact that both systems can formally be regarded as networks being built up from corner-sharing tetrahedral TL4 units, with atoms T in the centers and atoms L at the corners (where T ) Si, O and L ) O, H for silica and water, respectively). This general perspective naturally highlights the common structural heritage of the two systems, and it also helps to rationalize the links between the two systems in terms of their particular physical properties. The existence of tetrahedral ordering has, for example, been invoked to help explain various peculiar properties exhibited by both systems in the liquid phase (e.g., density maxima,1,2 two distinct amorphous phases,3,4 phases where viscosity decreases upon compression,5,6 and “fragile-to-strong” transition7,8). In the solid state this shared tetrahedral tendency is also displayed in a range of analogous crystalline polymorphs. For example, the dense water ice types: Ih, Ic, and III are isostructural with the dense silica polymorphs cristobalite, tridymite, and keatite, respectively. Similarly, a number of open crystalline clathrate hydrate water ices (e.g., * Corresponding author. † Universitat de Barcelona. ‡ Institucio ´ Catalana de Recerca i Estudis Avanc¸ats. § Leibniz-Universita ¨ t Hannover. | Christian-Albrechts-Universita ¨ t zu Kiel.

sI, sII, and sH) have topologically corresponding structures as all-silica clathrates (e.g., MEP, MTN, DOH),9,10 both materials families exhibiting structures based upon (TL2)N cages with even-N. Although corner-sharing tetrahedral units, having opportunities for extension in four distinct directions, are ideal for spanning three-dimensional space in bulk phases, when faced with a breaking of periodic symmetry (e.g., at an extended surface), it is far from clear that a structural interpretation based upon corner-sharing tetrahedra will remain valid. In the most extreme case, we may imagine reducing the number of participating tetrahedra in a sample until most of them have an incomplete corner-sharing environment (i.e., less than four nearest neighbors). This situation is exemplified in the case of silica and water nanoclusters whereby nearly all atoms can be thought of as being at the surface. At this length scale one should remember that the constituent “tetrahedra” are in fact higherlevel abstract building units, which in reality are composed of relatively independent molecular/atomic components. This extra degree of freedom allows for greater structural diversity than that imposed by the corner-sharing paradigm, and, in principle, allows molecular/atomic-scale energy-lowering mechanisms to override any bulk propensity for tetrahedral ordering. In this study we investigate the low-energy structural tendencies exhibited by both water and silica nanoclusters. In order to identify any remnants of common structural motives in water and silica at the nanoscale, we have specifically sought nanoclusters in both systems with similar structures that have a low energy with respect to their corresponding nanocluster

10.1021/jp806780w CCC: $40.75  2008 American Chemical Society Published on Web 10/31/2008

18418 J. Phys. Chem. C, Vol. 112, No. 47, 2008 ground-state energy. In particular, we concentrate on those nanoclusters that share an analogous T-atom topology but, unlike in the bulk, we do not insist that the L-atom linkages form a near-perfect tetrahedrally symmetric environment around each T-atom. In this way we retain the essential topological isomorphism at the level of the “tetrahedral” T-atom centers while allowing for the L-atom linkages to adopt a minimum energy configuration in order to best adapt to their broken symmetry environment. Although the L-atoms in silica and water nanoclusters are found to cope with their under-coordinated surroundings in different ways, in line with their chemically disparate characters, we nevertheless find a remarkable correspondence between the T-atom topologies of a set of lowenergy prismatic water and silica nanoclusters possessing an even number of T-atoms in the range (TN, N ) 8-26). At a size of N ) 24, however, this nanoscale similarity appears to start to breakdown with the emergence of low-energy nonprismatic amorphous clusters for water, which are not energetically favorable for silica. Comparing a range of lowenergy (TL2)24 isomers for silica and water, we rationalize why such an evolution in cluster structure is not followed by silica at this size. These findings extend the known bulk similarities between water and silica to the extremes of the nanoscale and further emphasize the limitations of strictly applying principles derived from bulk systems to nanosystems. The remainder of this article is organized as follows: In section 2, we briefly list the most important technical aspects of this study. In section 3, we present our results on similarities and differences between water and silica nanocluster structures and discuss possible explanations for the trends we find. The article concludes with a brief summary in section 4. 2. Methodology In part 2.1 of this section, we describe the databases of water and silica clusters used in this study, how they were constructed, and how our method of exploration provides a robust coverage of the low-energy end of the isomer spectra. In part 2.2, we give details on the mapping procedure we used to establish structural isomorphism between water and silica nanoclusters, and we explain different ways we have used to order these two different cluster systems on common energy scales. 2.1. Derivation of Low-Energy Cluster Isomers. The first step of our study was to reliably establish the low-energy isomer spectrum of each system (i.e., (SiO2)N and (H2O)N) and for each cluster size N ) 8-26. Because of the complexity of the potential energy surface (PES) in both systems, attempts to provide trustworthy near-exhaustive collections of low-energy isomers by manual design quickly becomes untenable for all but the very smallest of clusters. In this study, where we consider clusters beyond this limited size range, global optimization methods are employed as powerful but relatively unbiased means to obtain low-energy clusters in each system. Although global optimization methods primarily aim to efficiently locate only the lowest energy cluster of a particular size (i.e., the global minimum structure), typically they additionally give rise to large a number of low-energy minima. Using a wide range of initial conditions (e.g., random and selected initial cluster geometries, variable algorithm parameters), one can vary the regions of the low-energy potential energy landscape that a global optimization algorithm probes, thus increasing the probability for finding a more diverse range of low-energy isomers. Collating the results of a series of such exploratory searches can provide a rich set of local minima close in energy to the global minimum. Such an approach, although often extremely effective, is neither

Bromley et al. deterministic nor complete, hence it is possible that we may have missed important low-energy local (or even global) minima in one or both of the two systems. However, we believe that the probability for this is quite low, with the resulting lowenergy isomer spectrum in each system being very densely populated. All studied water clusters were generated by an evolutionary algorithm hybridized with local searches.11,12 This algorithm has been shown to reliably reproduce known global minimum energy structures of TIP4P (H2O)N clusters up to N ) 21.13 Recently, two of the present authors established a new parallel implementation of this evolutionary algorithm that completely avoids serial bottlenecks by replacing the traditional generational order by a pool concept.14 This algorithm was proven to yield the same results as the earlier generational version, but it saves up to 50% real time and can be used with optimal scaling on arbitrary numbers of processors. Running on up to 64 processors in parallel, it has been used to establish global minimum energy structures of TTM2-F water clusters up to N ) 3414 and to explore putative low-energy structures up to N ) 108.14 The TTM2-F water potential15 was used in all calculations. This potential has been shown to yield structures and energetics of various water cluster isomers up to N ) 21, in quantitative agreement with ab initio MP2/aug-cc-pVDZ calculations.15-18 For selected structures, we have performed quantum-chemical test calculations at various levels of theory (including RI-DFT (BP86/aug-TZVP) using Turbomole v5.719 and LMP2/aug-ccpVDZ using Molpro2006.2,20 with (local) geometry relaxation of all degrees of freedom and without imposing symmetry), never finding any qualitative discrepancies, neither in structural parameters nor in relative isomer energies. Using our massively parallel evolutionary global optimization, we have produced a huge number (approximately 8 × 105) of local and putative global minima for the TTM2-F water system. While this decreases the likelihood for having missed important minima to practically zero, it also precludes a scan of all structures and their sorting into structural classes, since this ultimately has to be done by hand to ensure the absence of biases based on predefinition of structural classes. However, having a comprehensive, energy-sorted list of structural classes is a prerequisite for the present comparison of structural preferences of different cluster systems. Therefore, we had to resort to the following ad-hoc expedient of reducing the number of clusters for visual inspection: For each cluster size, all minima in our database are sorted by energy. From the lowest-energy structure upward, the next structure is selected that differs in energy by at least 0.01 kJ/mol. It should be pointed out that this energy criterion is potentially dangerous. Structural classes span rather wide energy regions that can partially overlap with each other leading to possibly very different clusters with energies only differing by less than 0.5 kJ/mol. Furthermore, energy differences of orientational isomers of the same structure can be as large as a few kJ/mol. Therefore, the energy difference criterion used for our water cluster classification merely is our best compromise between minimization of work by hand and completeness of the classification. This automatic energy-based selection step is iterated until a collection of about 30 clusters per size is accumulated. These are then grouped into classes by inspection. The resulting groups are the basis for comparison with the SiO2 system in this work. All silica clusters were obtained through a two-step process consisting first of global optimization searches, and second, a structural and energetic refinement stage. The global optimizations employed the Basin hopping algorithm,21 which uses a

Structural Analogy between (H2O)N and (SiO2)N combination of Metropolis Monte Carlo sampling and energy minimization to sample the phase space of cluster configurations. Although the BH algorithm is one of the least hindered global optimization methods with respect to the specific topology of the PES,22 its success with respect to clusters of a real material, as with all global optimization methods, relies further on a sufficiently accurate yet efficient representation the PES of that material at the nanoscale. For the smaller (SiO2)N clusters (approximately N < 16), we have used an interatomic potential set that has been specifically parametrized to accurately predict the energies and structures of silica nanoclusters.23-26 The potential has a two-body Buckingham form complemented with electrostatics, and, for small clusters, has been found to be superior to similar bulk silica potentials and semiempirical methods with respect to ab initio nanocluster energies and structures.23 For the larger clusters (approximately N g 16), we have additionally used the TTAM27 silica potential, which, although parametrized for bulk silica, has also been shown to be of some use in global optimization studies of silica clusters.26,28 From the resulting large number of isomers from the global optimizations (typically a few hundred for each (SiO2)N cluster size), the 20-30 lowest energy structures and selected higher energy isomers with high symmetry were taken for energetic and structural refinement with density functional theory employing no symmetry constraints using the GAMESSUK29 code. The postoptimizations employed the B3LYP30 functional and a 6-31G(d,p) basis set, which has been shown in numerous previous studies to be a suitable level of theory for calculating accurate structures and energetics of silica nanoclusters.31 2.2. Comparison of (SiO2)N and (H2O)N Clusters. The mapping and comparison between water and silica cluster structures is done by first extracting the coordinates of the N T-atoms from one structure and using them to make a suitably scaled TN skeleton using T-atoms of the other type. To form a new (TL2)N cluster isomorphic to the original, 2N L atoms of the appropriate type are then added to the scaled TN cluster in an unspecific, yet chemically sensible manner. Retaining the TN toplogy, a global optimization of the L-connectivity pattern for the (TL2)N cluster is performed (locally optimizing all atom positions for each L-connectivity) to obtain the lowest energy cluster of this type. Possible changes in orientations, positions, and bonding patterns after this procedure, which may break the intended isomorphism, are detected by visual inspection. Although our main insistence is on maintaining T-atom topology, we further make sure that only when a L-link between T-atoms exists in one cluster should it also exist in its topological isomer. This rule forbids the creation of new L-linkages between T-atoms that didn′t already have at least one T-L-T link in the original cluster, but allows for multiple L-links between T-atoms that may have only one L-link in the corresponding isomorphic cluster. All the comparisons reported below are only between the lowest energy T-isomorphic clusters found after this mapping process. The energies involved in the hydrogen-bonded water system versus that in strongly covalently bonded silica clusters are massively different. For a meaningful comparison of the energetics in each system some normalization of the energy scale is thus necessary. We have attempted such a normalization is two ways. First, taking advantage of the analogous TL2 stoichiometry in each system and the fact that the bonding of a TL2 unit is often topologically similar in each case (i.e., each TL2 unit tends to bond with the remaining TL2 units to form a tetrahedral network), we can employ the binding energy per

J. Phys. Chem. C, Vol. 112, No. 47, 2008 18419 TL2 unit to make natural common energy scale. To account for the numerical differences in the energy per TL2 unit in each system, we quote it as a percentage above the binding energy per TL2 of the respective ground-state cluster. To give some idea of what such percentages mean in each system, we provide an example: for water, 1% equates to 0.38-0.46 kJ/mol/H2O above the (H2O)N N ) 8-24 ground states, while for silica 1% is between 0.04-0.05 eV/SiO2 above the correspondingly sized (SiO2)N ground states. As a relatively arbitrary measure, we consider low-energy clusters in each case to be within 2.5% of the ground state (i.e., approximately 1 kJ/mol/H2O and 0.1 eV/ SiO2, respectively). In all cases where this energy scaling is employed, a percentage sign will be shown. A second way to the compare the energies of the two distinct systems is again to employ the binding energy per TL2 unit but, instead of a percentage with respect to a size dependent ground-state energy, to use a standard fixed point of reference in each system to normalize the energy scale to be the same for both. Although there are a number of possible choices for such fixed points, we regard a natural one to be the energy of the (TL2)12 ground state, which uniquely has the same T-atom topology in both silica and water. Taking the binding energy per TL2 unit for the ground-state of (H2O)12 and (SiO2)12 to be the same, we arrive at a general normalized energy scale for both systems which can be further scaled fit within any convenient interval. This energy scaling is arguably better suited to comparing the overall changes in cluster energetic stability with increasing size than the former percentage binding energy method, which can be affected by ground-state energy fluctuations. In all cases where this energy scale is employed, it will be referred to as the normalized energy scale. 3. Results and Discussion Part 3.1 of this section starts out with an introduction into our comparative results, by first presenting a brief overview of water cluster structures. We then draw the parallel to silica clusters, but, in doing this, it turns out to be necessary to introduce a few characteristic structural differences between these two cluster systems, namely, different “terminations” of clusters. This allows us to let this part culminate with a comparative overview of the full size range N ) 8-24 under study. Part 3.2 is devoted to a closer examination of the various water and silica cluster structures at the size N ) 24, in order to discuss how and why the similarities at lower sizes disappear at larger sizes. We split this section into two subsections in order to treat the specific cases of the stability of interior unit clusters for (SiO2)24 (section 3.2.1) and (H2O)24 (section 3.2.2). Finally, in section 3.3 we discuss how the observed structural differences for nanoclusters for N > 24 may be reconciled with the reappearance of similarities in the bulk. 3.1. Comparison of Low-Energy Prismatic (SiO2)N and (H2O)N Clusters (N ) 8-24). In Figure 1 we show the lowest energy (SiO2)N and (H2O)N N-even clusters for N ) 8-24. For N ) 8-20,24, the lowest energy water clusters (labeled GS in Figure 1 to denote ground state) have compact disk or columnar forms based on unions of cubes and/or pentaprisms sharing one or more of their square faces. The (H2O)22 ground-state water cluster, which, although it may be viewed as being based on shared cubes and pentaprisms, has a open cage-like topology due to a shared O-atom edge having no H-linkage.14 The lowest energy (H2O)22 square-face-sharing (SFS) compact prismatic water cluster (see Figure 1), is, however, essentially degenerate with the (H2O)22 ground state (only 0.1% higher in binding energy per H2O), and thus the SFS prism rule is approximately

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Figure 2. Two characteristic terminations observed in silica clusters: (a) an NBO- defect with a folded-back three-coordinated oxygen center, (b) two Si2O2 rings formed from two folded-back oxygen atoms. Note that these terminations are not found in water clusters.

Figure 1. Comparison of the low-energy SFS prismatic cluster isomers of (H2O)N and (SiO2)N for sizes N ) 8-24. Percentages are binding energies per TL2 unit with respect to the corresponding ground state. Key: red - oxygen atoms; small white - hydrogen atoms; large white silicon atoms.

held even for N ) 22. Although, generally for even-N (TL2)N clusters of size N ) 8-14, there is only one unique way to make SFS prismatic clusters from only pentaprisms and/or cubes, for sizes N g 16 there is more than one possible isomer that maintains this energetically preferred prismatic SFS topology. For N ) 16, for example, we may have three SFS cubes or two SFS pentaprisms, while for N ) 18, an SFS column of two cubes and a pentaprism and a disk made from the SFS of two pentaprisms and a cube are two possibilities. In both cases (N ) 16, 18), the columnar isomers with most cubes are the ground states (see Figure 1) for water. For N ) 20-24 a similar structural competition exists (see Figure 1), but, for this larger size range, the disk isomers with the most pentaprisms are found to be the most energetically favored SFS prismatic water isomers. A speculative rationale for this switch in structural preference upon cluster growth may be that packing an increasing number of cubes leads to energetic disadvantages for these systems. In other words, the growth of one-dimensional (1D) stacks of cubes leads to longer and thinner stacks with a worse ratio of surface to volume (with associated surface strain energy) than for stacks of pentaprisms. In two- (2D) and threedimensional (3D) growth, using cubes leads more quickly to over coordinated vertices (as, in the case of the missing

H-linkage, the (H2O)22 cluster mentioned above) than using pentaprisms. The cube-based column to pentaprism-based disk structural transition in water clusters is strongly reminiscent of that reported for ground state (SiO2)N silica clusters occurring at N ) 24.26 The cube-based column and pentaprism-based disk (SiO2)N ground states are, however, not fully structurally isomorphic with the ground-state columns and disks of water, largely because of the different energetically preferred cluster terminations in each system. In water clusters, each T-L-T linkage is asymmetric with one-half covalent and the other a much weaker hydrogen bond (i.e., O-H · · · O). In the water clusters, every edge (formed by such a linkage) that does not form part of a shared square-face gives rise to two terminal oxygen atom vertices: (i) an oxygen atom corner coordinated to three hydrogen atoms (a “dangling lone pair” of electrons), and (ii) an oxygen atom corner coordinated to four hydrogen atoms, where one hydrogen atom is “dangling”. In water, this type of termination in unavoidable and necessarily means that a few less hydrogen bonds are formed, which is only a small energetic penalty. In silica, the corresponding topology would entail dangling bonds with unpaired electrons on both the terminal four-coordinated oxygen atoms on the terminal three coordinated silicon atoms, which would cause energetically costly perturbations to the electronic structure of the cluster. Although no termination is as stable as fully coordinated relaxed T-L-T linkages, silica has a number of structural resources to atomically reconstruct itself at terminations and avoid dangling bonds. First, in one such way oxygen atoms on terminating edges can “fold-back” and form bonds with the three coordinated silicon corner sites. On a terminating square face of a cluster, either one or two dangling oxygen atoms can foldback in this way. In the first case, the oxygen atom folds back to bond with two three-coordinated silicon corner sites, leaving one remaining dangling oxygen atom (see Figure 2a), whereas in the second case both dangling oxygen atoms each bond with one of the three-coordinated silicon corner sites, leaving a fully coordinated face with two Si2O2 rings (see Figure 2b). In the first case, the dangling oxygen atom takes an electron from the folded-back three-coordinated oxygen center to form a closedshell nonbridging oxygen anion (NBO-) and a reasonably tetrahedral oxygen coordination environment around each silicon atom.32,33 In the latter case, we recover the 4-fold coordination of each silicon center, albeit in a strained nonperfectly tetrahedral way, also with no unpaired electrons. The energetic preference for either type of fold-back termination depends on cluster type and size (compare terminations in D20-24 and C20-24 for (SiO2)20-24 in Figure 1 for example). Although via oxygen foldback terminations silica clusters can maintain a T-atom isomorphism with the SFS prismatic water clusters, it is noted that water itself does not use the analogous mechanism of folding back H-bonds, as this is energetically very unfavorable.

Structural Analogy between (H2O)N and (SiO2)N Another manner in which silica clusters terminate is via nontetrahedral pendant oxygen double bonds (i.e., SidO); a specific chemical bonding mechanism impossible to reproduce in water. In the N-even (SiO2)N cluster ground states for N ) 8-10, and N ) 14-24, SidO terminations are found to be dominant over fold-back terminations.24-26 This means that, for these sizes, the most stable silica clusters are not directly T-isomorphic with the corresponding water ground states. For (SiO)12, however, the structure of the ground-state is two SFS cubes with tetrahedral fold back terminations, exactly following the structure of the (H2O)12 ground-state cluster (see Figure 1). Considering the high energetic stability of the T-isomorphic prismatic (TL2)12 ground-state structure in both water and silica, we further checked the energies of the T-isomorphic silica versions of all low-energy prismatic SFS (H2O)N even-N clusters from N ) 8-24. In addition, we searched the low-energy isomer spectrum of even-N silica clusters for any other SFS prismatic silica clusters which may also have low-energy water isomorphic analogues. The lowest energy clusters, for both silica and water, found from this procedure are shown in Figure 1. The lowest energy (SiO)N cluster isomers only based on SFS T-atom cubes and/or pentaprisms are generally found to be quite low in energy with respect to the respectively sized silica ground-state clusters (ranging between 0-5.8% higher with respect to binding energy per SiO2). The two smallest clusters: (SiO2)8 and (SiO2)10, having the form of a single cube and a single pentaprism respectively, are the least stable prismatic isomers, which is probably due to the high percentage of T-atoms involved in the necessary fold-back terminations with respect to the total number of T-atoms. The even-N (SiO)N SFS prismatic clusters in the range N ) 12-24, however, all have binding energies per SiO2 unit well within 2.5% of that of the respective ground state. Moreover, the lowest energy SFS prismatic clusters found for (SiO2)N N ) 8-18 take the same topological form as the correspondingly sized prismatic SFS water cluster ground states. Specifically, for the most stable three SFS prismatic clusters in this range (i.e., (SiO2)N for N ) 12, 16, 18, all having binding energies within 0.9% of the respective silica ground states), the clusters are either the ground-state for silica (i.e., for N ) 12) or are only bettered energetically by cluster isomers that possess SidO terminations. For clusters in the size range N ) 20-24, the (SiO2)N SFS prismatic cluster analogues of the respectively sized SFS prismatic water groundstate clusters are still low in energy (between 1-1.5% higher than the corresponding ground-state with respect to binding energy per SiO2) but are not the most energetically favored (SiO2)N SFS prismatic clusters. Instead of the SFS pentaprismbased disks exhibited by the (H2O)N N ) 20-24 ground states, the correspondingly sized most stable SFS prismatic clusters in silica are columnar, and are mainly based on SFS cubes. Testing the stability of these SFS columns as water clusters reveals that they are also low in energy in water (between 1-2.1% higher than the corresponding ground-state with respect to binding energy per H2O). This shows a somewhat symmetrical divergence in the tendency with respect to the most energetically favored type of SFS prismatic cluster as we increase the cluster size from N ) 8 to N ) 24. Nevertheless, in both water and silica, this type of cluster has high energetic stability energy. In Figure 3 we summarize the results of this section via a graph of the normalized binding energy of the SFS prismatic clusters of both water and silica reported in Figure 1. From visually comparing the fluctuations in the bold lines in the graph in Figure 3, we can clearly see that, for both water and silica,

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Figure 3. Normalized binding energies (per TL2) of low-energy cluster isomers of (H2O)N and (SiO2)N for sizes N ) 8-26.

the lowest energy SFS prismatic clusters for even-N in the range N ) 8-24 follow a similar increasing binding energy dependence with respect to size. The graph shows that the lowest energy SFS prismatic silica clusters are especially low in energy for N ) 14-24 (with respect to the corresponding ground states). The divergence with respect to the column versus disk structural preference of silica and water at N ) 18 is marked in Figure 3 by the dashed lines separating above and below the lines representing the energies of the SFS prismatic silica clusters. Further, in Figure 3 we extend the size range to N ) 26 to specifically show the change in structural preference in the ground states of water. For N ) 26, the SFS prismatic clusters are superseded in energetic stability by “self-solvating” water clusters, which contain an interior H2O unit (open blue symbols in Figure 3). Even at this structural crossover, however, it is noted that (i) SFS prismatic clusters in both silica and water continue to be low in energy (0.8% and 0.7% above the groundstate with respect to binding energy per TL2 unit for (H2O)26 and for (SiO2)26 correspondingly), and (ii) that the similarities in the trends in fluctuations in the energetics of both water and silica SFS prismatic clusters (bold lines in Figure 3) also continue at least to a size of N ) 26. Thus, although the most energetically favorable clusters for even-N for both silica and water for N > 24 are not SFS prismatic isomers, in both systems, this type of cluster is still likely to be found in the low-energy isomers of larger cluster sizes and will only gradually be pushed higher up the energy scale. 3.2. Comparison of (SiO2)24 and (H2O)24 Clusters. In order to investigate the crossover from SFS prismatic cluster to other cluster types, we have specifically focused on the cluster size N ) 24. At N ) 24, although an SFS prismatic isomer is the ground-state for water, the low-energy isomer spectrum for (H2O)24 contains many non-SFS prismatic clusters that are very close in energy with the SFS prismatic ground state. These nonSFS prismatic clusters are generally more spherical and less structurally ordered than the SFS prismatic isomers, and all have the distinguishing property of containing an interior H2O unit (i.e., no atom of the unit is at the surface of the cluster). In Figure 4 we show an example of such a cluster, and we indicate the normalized binding energy of the lowest energy cluster of this type for (H2O)24 in Figure 3. Although for (H2O)N N > 24 such clusters seem to dominate up to a size of least N ) 34,14 for silica clusters in this size range, our investigations indicate

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Figure 5. Comparison of the energetics of various cluster isomers of (H2O)24 and (SiO2)24. Energies are percentage binding energies per TL2 unit with respect to the corresponding ground state.

Figure 4. Representative structures of (TL2)24 cluster types as employed in the energetic comparison in Figure 5. The blue atoms denote the position of the interior T-atom in the “interior unit” clusters. The blue circles denote the position of the pendant TL2 unit in the “pendant unit” clusters.

that such “interior-unit” clusters are always higher in energy than SFS prismatic isomers, which tend to remain quite close in energy to the respectively sized silica cluster ground states. In order to understand this change in structural preference in water clusters, and why it is not also observed in the low-energy isomers for silica, we have analyzed the stability and structure of a range of (SiO2)24 and (H2O)24 cluster isomers. Specifically we have looked at five classes of clusters which we name (1) interior unit, (2) disk, (3) column, (4) cage, and (5) pendant unit. Representative cluster isomers from each class for both water and silica are shown in Figure 4. The percentage binding energy difference with respect to the corresponding (TL2)24 ground-state for four examples of each of the representative cluster classes is shown in Figure 5. As expected, the SFS-based disks and columns are relatively stable for both (SiO2)24 and (H2O)24 and form an energetically overlapping set of cluster isomers. For (SiO2)24, the cage cluster isomers and cluster isomers with pendant units, however, are much more stable than the corresponding (H2O)24 isomers. This can be explained by the presence of SidO terminations (in the pendant unit clusters) and folded-back closed Si2O2 rings (in the cages), which silica is able to utilize with relatively little energetic cost. As discussed above, these termination mechanisms are impossible in water which instead must use a higher proportion of energetically unfavorable terminal hydrogen atoms and less hydrogen bonding in order to best mirror the corresponding silica cluster structures. 3.2.1. Energetic Stability of Interior Unit Clusters for (SiO2)24. The final comparison between isomers of water and silica that contain an interior unit is more difficult to rationalize. Taking first the silica clusters that contain an interior unit, we find that they all lie >5% above the ground state with respect to binding energy per SiO2 unit. In silica, the interior SiO2 unit appears to be very confined and causes the bonded atomic

network to be highly distorted away from perfect tetrahedrality. Although the Si-O-Si bridge is flexible and may be distorted with relatively little energetic expense, the O-Si-O angle is much more rigid and very rarely deviates significantly from perfect tetrahedrality throughout the range of known silica bulk polymorphs.34 Although, in all (SiO2)24 clusters, there exist Si centers with oxygen environments distorted away from perfect tetrahedrality (e.g., in Si2O2 two-rings), we would expect that, in low-energy clusters, the dominant O-Si-O angle would be close to 109.5°. In Figure 6, we show the normalized collated distributions of the O-Si-O angles from a set of three (SiO2)24 columns, a set of three (SiO2)24 disks, and a set of three cages (as used in Figure 5). We note that we omit the pendant unit clusters from this comparison because they possess strictly nontetrahedral dSidO terminations. The column-disk-cage collated distribution is compared with the normalized collated distributions of the O-Si-O angles for a set of three (SiO2)24 clusters that contain an interior SiO2 unit (again corresponding to points in Figure 5). We see that silica interior-unit clusters give rise to a relatively broad O-Si-O angle distribution, whereas the collated column-disk-cage distribution is strongly peaked in a (2.5° range about 109.5°. This difference provides strong evidence to explain the observed high relative energy of (SiO2)24 clusters containing an interior SiO2 unit, and which can also probably be extrapolated to larger sizes. It is noted that both O-Si-O distributions shown in Figure 6 also have a small peak around 90°, which can be ascribed to the two foldedback Si2O2 rings present in all (SiO2)24 clusters. 3.2.2. Energetic stability of interior unit clusters for (H2O)24. The remaining question to be answered is why water energetically prefers clusters with an interior unit over SFS prismatic clusters and all other cluster types for N > 24. For N ) 24, water cluster isomers with an interior unit strongly compete energetically with the SFS prismatic ground state. As the number of water molecules, and thus the number of covalent O-H bonds, is constant in any fixed sized water cluster, we may be tempted to compare the variable number of O · · · H bonds and of nontetrahedral under-coordinated surface sites in various cluster isomers of the same size as a possible measure of energetic stability. For (H2O)24, the ground-state SFS prismatic cluster has 42 hydrogen bonds and 12 surface water molecules

Structural Analogy between (H2O)N and (SiO2)N

J. Phys. Chem. C, Vol. 112, No. 47, 2008 18423

Figure 6. Distribution of O-Si-O angles taking into account non-“interior unit” (SiO2)24 clusters (left) versus the distribution taking into account only “interior unit” (SiO2)24 clusters (right). Note: the y-axis in both graphs corresponds to a similarly normalized abritrary cummulative measure.

having only three hydrogen-bonded nearest neighbors. The lowest competing interior unit cluster, however, has 40 hydrogen bonds and 17 under-coordinated surface water molecules. The higher number of hydrogen bonds and lower number of three coordinated water molecules in the SFS prismatic ground-state in this case may indeed help it to have a lower energy. If, however, we compare the (H2O)26 interior unit ground-state with the correspondingly sized lowest energy SFS prismatic cluster, we find that the former has less hydrogen bonds and more undercoordinated surface water molecules than the latter (41 versus 47 and 13 versus 10 respectively). Evidently the energetics due to local hydrogen bond counting on its own does not give rise to a preference for interior unit water clusters. As an alternative measure, tetrahedral angular order parameters have previously proven to be of use in characterizing and comparing the bulk liquid phases of both silica and water.35,36 Although for silica the degree of tetrahedrality of the O-Si-O angle distribution seems to be a good indicator as to the relative stability of some types of clusters (see above and Figure 6), the energy scale of relative cluster isomer stability does not seem to be associated with the analogous H-O-H angle distribution in water. We note that a minority contribution to the H-O-H angle distribution comes from the strong internal intramolecular OH bonds of the water molecules. The H-O-H angle within each water molecule is negligibly affected in any water cluster, and essentially all H-O-H-dependent energy differences comes from the much weaker H-O · · · H and H · · · O · · · H contributions. Although these three types of H-O-H angles do tend to form tetrahedral centers, as only one of them is effectively fixed, and there is not a severe energy penalty for the latter two to deviate away from the ideal 109.5°, the relation between energetic stability and tetrahedrality appears to be weak. Our analyses of the H-O-H angular distributions (not reported) have not revealed any convincing correlation with cluster energetic stability, tending to support the greater structural flexibility of the water system with respect to tetrahedral TL4 centers than in silica. Unlike for the H-O-H angles, the O-H-O angles in water arise from only one type of interation (i.e., the O-H · · · O hydrogen bonding of an OH group of a water molecule with an

Figure 7. The average O-H-O angle for a the lowest energy prismatic SFS isomer and the lowest energy interior unit isomer for (H2O)N clusters for sizes N ) 20, 22, 24, 26. Note: no low-energy interior unit isomer could be found for N ) 20.

oxygen center of another water molecule). In water systems in general, the O-H-O angles tend to be close to 180° which is strongly suggestive of the energetic preference for directional hydrogen bonding. Other theoretical calculations have indeed pointed to the degree of deviation of the O-H-O angles away from 180° being correlated to the stability of clathrate hydrate clusters.37 In Figure 7, we show the average O-H-O angle of the lowest energy SFS prismatic (H2O)N isomer and interior unit (H2O)N isomer for N ) 20, 22, 24, 26. The data show a trend for low-energy interior unit clusters to have an increasing average O-H-O angle (toward 180°) with increasing size, whereas the low-energy SFS prismatic clusters appear have an average O-H-O angle that generally gets smaller (away from 180°) with increasing size. At the size N ) 26, the difference

18424 J. Phys. Chem. C, Vol. 112, No. 47, 2008 in average O-H-O angle between the lowest energy SFS primatic cluster and the interior unit ground-state is quite significant (approximately 4.5°). Considering, as mentioned above, that the interior unit ground-state cluster has six less hydrogen bonds and three more under-coordinated surface water molecules than the higher energy SFS prismatic cluster (+10 kJ/mol), the O-H-O angles may play an important role in water cluster energetics. Because of the relatively weak flexible nature of the O · · · H bond, it is easier for a cluster to adapt to topologies that are much more energetically costly in the more tetrahedrally constrained strongly bound silica system, thus explaining the absence of low-energy (SiO2)N clusters with an interior unit for N e 26. For water it appears that, although the energetic constraints on tetrahedrality are much looser, an important criterion for relative energetic stability appears to be the average O-H-O angle. The preference for interior unit (H2O)N clusters for N > 24 seems to be related, at least in part, to the capacity of this type of cluster to accommodate O-H · · · O hydrogen bond linkages that are closer to 180° than in other cluster topologies. 3.3. Structural Tendencies of (H2O)N and (SiO2)N Approaching the Bulk. Although these system-specific arguments provide a basis for insight into the comparative nanoscale cluster behavior for water and silica, it is interesting to speculate generally on how TL2 systems are likely to evolve with increasing size. As (TL2)N low-energy clusters get larger, it is expected that they will attempt to mimic more and more the features of the thermodynamically most favored crystalline bulk phases. At first, however, for relatively small scales, the cluster energetics will be dominated by local bonding and terminating defects, and the absolute number of viable cluster isomers is relatively limited. Once beyond a very small scale where the clusters are essentially molecular and/or terminations dominate the form of the clusters, the low-energy isomers will start to show tetrahedral ordering. Because of the relatively few choices of compact cluster forms available, these first tetrahedrally ordered clusters will likely display small prismatic motifs (e.g., cubes and pentaprisms) regardless of the specific interaction strength and directionality.38 With increasing size, the number of isomer possibilities increases rapidly and nonprismatic clusters become a viable alternative. At this point, small differences in interaction strength/directionality in each system do start to matter, hence the onset of structural divergence between water and silica. Beyond this stage, we suggest that, in general, nonprismatic more isotropic interior unit clusters should be better able to mirror the long-range electrostatics (i.e., the Madelung field) of the bulk phase in a more uniform manner throughout the cluster while also reducing the overall electrostatic dipole of anisotropic isomers (e.g., SFS disks and columns). This is probably a weak effect and will only come into play when local bonding energetics are equally well satisfied by a number of cluster isomer types. Following this exploratory line of argument, with increasing size, we similarly may expect that the low-energy clusters of silica will eventually display relatively disordered isotropic interior unit isomers, as observed for water. Preliminary interatomic potential based global optimization calculations on the detailed low-energy isomer spectrum of (SiO)N N ) 28-32 do indeed increasingly start to show low-energy disordered clusters, some of which contain interior units, but whose stability has yet to be verified by refinement using density functional calculations. In both systems, any such

Bromley et al. preference for interior unit isomers will only be an energetically preferred structural type for intermediate sized clusters, which, with further growth, will eventually again start to display the ordered motifs characteristic of the respective low-energy crystalline bulk phases. We tentatively suggest that the transition between ordered small clusters, and ordered bulk phases, via that of disordered interior unit clusters, with increasing size may be a characteristic signature of tetrahedrally ordered systems and perhaps of other systems with nonisostropic local interactions. 4. Conclusions We have compared topologies and energetics of low-energy (SiO2)N and (H2O)N clusters for even-N in the size range N ) 8-26. Although the constituent H2O and SiO2 monomers are quite different from a chemical perspective, surprisingly, viewing finite collections of these basic units being made up of tetrahedral building blocks is so strong that many closely similar structures of low energy occur even among the smallest cluster sizes. This nanoscale link between the two systems is all the more impressive if we consider that almost all units of the compared clusters are at the surface, and each system has quite distinct ways to cope with surface terminations. On the basis of our results on two very different TL4 materials, we tentatively suggest that the emergence SFS prismatic clusters may be relatively chemistry-independent, and will be present in the low-energy range of many (TL4)N nanocluster systems. Curiously, the analogy between silica and water at this small scale seems to be closest only in a limited finite size range, and beyond N ) 24 divergences with respect to the most energetically favored structures set in. By comparing the low-energy cluster isomers in each system for N ) 24 with respect to the system specific differences, we attempt to rationalize this trend. References and Notes (1) Vedamuthu, M.; Singh, S.; Robinson, G. W. J. J. Phys. Chem. 1994, 98, 2222–2230. (2) Angell, C. A.; Kanno, H. Science 1976, 193, 1121. (3) Mishima, O.; Calvert, L. D.; Whalley, E. Nature 1985, 314, 76. (4) Lacks, D. J. Phys. ReV. Lett. 2001, 86, 3207. (5) Angell, C. A.; Cheeseman, P. A.; Tamaddon, S. Science 1982, 218, 885. (6) Tsuneyuki, S.; Matsui, Y. Phys. ReV. Lett. 1995, 74, 3197. (7) Saika-Voivod, I.; Poole, P. H.; Sciortino, F. Nature 2001, 412, 513. (8) Ito, K.; Moynihan, C. T.; Angell, C. A. Nature 1998, 396, 329. (9) Koh, C. A. Chem. Soc. ReV 2002, 31, 157–167. (10) Baerlocher, Ch. Lynne, B. McCusker, D. Olson, D. H. Atlas of Zeolite Framework Types, 6th ed.; Elsevier: The Netherlands, 2007. (11) Hartke, B. J. Comput. Chem. 1999, 20, 1752. (12) Hartke, B. Z. Phys. Chem 2000, 214, 1251. (13) Hartke, B. Phys. Chem. Chem. Phys. 2003, 5, 275. (14) Bandow, B.; Hartke, B. J. Phys. Chem. A 2006, 110, 5809. (15) Burnham, C. J.; Xantheas, S. S. J. Chem. Phys. 2002, 116, 5115. (16) Fanourgakis, G. S.; Apra, E.; Xantheas, S. S. J. Chem. Phys. 2004, 121, 2655. (17) Hartke, B. Phys. Chem. Chem. Phys. 2003, 5, 275. (18) Lagutschenkov, A.; Fanourgakis, G. S.; Niedner-Schatteburg, G.; Xantheas, S. S. J. Chem. Phys. 2005, 122, 194310. (19) (a) Ahlrichs, R.; Ba¨r, M.; Ha¨ser, M.; Horn, H.; Ko¨lmel, C. Chem. Phys. Lett. 1989, 162, 165. (b) Deglmann, P.; May, K.; Furche, F.; Ahlrichs, R. Chem. Phys. Lett. 2004, 384, 103. (20) MOLPRO, version 2006.1, a package of ab initio programs. Werner, H.-J. Knowles, R. Lindh, M. Schutz, P. Celani, T. Korona, F. R. Manby, G. Rauhut, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel, C. and Hetzer, A. W. Lloyd, S. J. McNicholas, W. Meyer, W. and Mura, A. Nicklass, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, R.; Thorsteinsson, T. See http://www.molpro.net. (21) Wales, D. J.; Doye, J. P. K. J. Phys. Chem. 1997, 101, 5111. (22) Doye, J. P. K.; Wales, D. J. Phys. ReV. Lett. 1998, 80, 1357. (23) Flikkema, E.; Bromley, S. T. Chem. Phys. Lett. 2003, 378, 622.

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