**3. Sequential Aufbau of 2-, 3-, 4-Coordinate Windowpane Water Clusters**

The properties of each water cluster *W*<sup>k</sup> of an envisioned class are dictated by its specific H-bond coordination pattern. As primary descriptors of this pattern, we expect that each water molecule may generally be involved in two-, three-, or four-coordinate H-bonding to other molecules of the cluster (with singly coordinated "dangling" molecules excluded in leading clusters of the equilibrium thermodynamic distribution). For labeling purposes, the coordination pattern of each cluster may be usefully described by the number of quadruply (*q*), triply (*t*), or doubly (*d*) coordinated sites, appended as presuperscripts (viz., q,t,dWn) to the cluster symbol. In this notation, the cyclic water tetramer of Figure 1 is labeled 0,0,4W4, with each monomer doubly coordinated in chain-like linkages to the substrate.

The structural logic for sequential *Aufbau* construction of windowpane clusters is straightforward. Starting from an existing cluster of this class, such as the cyclic water tetramer of Figure 1, one can choose any edge-type coordination (such as that between O(1) and O(10) in Figure 1) as a "base" for a new windowpane by attaching a water dimer in parallel fashion with two new H-bonds, as shown in the left panel of Figure 2. For maximum stabilization in forming this new H-bond attachment (e.g., from emanating H(12) at O(10)), the Grotthuss-type proton ordering should be continued around the edges of the newly formed windowpane that joins to O(1). The net result of this particular attachment is that sites O(10) and O(1) become tri-coordinate (*t* → *t* + 2), while other sites remain di-coordinate, leading to an overall 0,0,4W4 → 0,2,4W6 change in labeling. Some of these clusters, such as 0,0,4W4 itself or the cubane-like 0,8,0W8 described below, are featured in many previous cluster investigations, but the emphasis here is on hierarchical families of clusters that can be associated with a well-defined mechanistic *Aufbau* sequence of dimer additions, particularly leading to higher four-coordinate (*q*-type) motifs.

By alternating the sign of folding angles between panes, such additions can be continued indefinitely in "ladder-like" procession, as shown in successive panels of Figure 2. Each panel of Figure 2 includes (in parentheses) the per-monomer energy and standard-state Gibbs free energy change with respect to free water molecules, which serve to exhibit the important cooperative (nonadditive) effects of Grotthuss-ordered coordination patterns. The first four panels (0,0,4W4, 0,2,4W6, 0,4,4W8, 0,6,4W10) show the addition of successive rungs to the ladder pattern, up to the four-pane member. The ensuing 1,4,4W9 (row 3, left) is the alternative "2 × 2" four-pane cluster, which adopts a buckled saddle-shape deformation from planarity with a central four-coordinate monomer. From the starting two-pane ladder ( 0,2,4W6) at the upper right, one can also attempt to add another rung that curls backward (*E*-like) rather than forward (*Z*-like), but this optimizes to the cubane-like 0,8,0*W*<sup>8</sup> cluster (row 3, right). The cubane motif becomes an evident building block for extensions to two-cube (4,8,0*W*12), three-cube (8,8,0*W*16), or longer rod-like clusters, as illustrated in the final row of the figure.

**Figure 2.** *Cont*.

**Figure 2.** Calculated *Aufbau* sequence of windowpane clusters q,t,dWk from starting cyclic tetramer 0,0,4W4 (**upper left**), showing parenthesized per-monomer changes (kcal/mol) in energy (Δ*E*) and Gibbs free energy (Δ*G*(0)) from free water molecules in each panel.

An alternative *Aufbau* starting point is provided by the twisted two-pane (1,0,6W7) cluster shown in the upper-left panel of Figure 3. This cluster features "Möbius-like" coordination with a continuous Grotthuss-ordered chain passing twice through the unique four-coordinate central monomer to form a closed loop. Remaining panels of Figure 3 show selected clusters that are obtained by successive Grotthuss-ordered dimer additions to 1,0,6W7, aimed at increasing *q* numbers of saturated four-coordinate sites. The resulting structures all incorporate the higher density coordination angles of the windowpane motif, but they exhibit irregular overall shapes that appear suitable as possible contributions to bulk liquid or amorphous solid phases. As seen in Figures 2 and 3, the 8,8,0W16 cluster (Figure 2, lower right) achieves the largest number of three- and four-coordinate sites (*q* = *t* = 8) and the deepest per-monomer energy (−10.62 kcal/mol) in the depicted sequences. However, whether some or all of these clusters contribute significantly to known roots of the QCE equations, or whether (like the buckyball-type clathrate clusters previously studied [60]) they can serve as leading contributors to entirely new roots (phases) of the QCE phase diagram remains to be investigated.

**Figure 3.** *Cont*.

**Figure 3.** Similar to Figure 2, for successive q,t,dWk windowpane clusters built from the Möbius-like 1,0,6W7 cluster (**upper left**).

It is evident that each *Aufbau* cluster shown in Figures 2 and 3 may have alternative isomeric rearrangements of the proton network without altering the *q/t/d* descriptors of <sup>O</sup>···(H)···O coordination linkages. Such alternative q,t,dWn (alt) isomers may have higher point group symmetry, different proton orderings (e.g., Grotthuss cycles around individual panes rather than overall periphery), and higher or lower energy than the *Aufbau*-derived clusters described above. Figure 4 displays two such alternative high-symmetry forms of the 0,2k,4W2k+4(sym) sequence (*k* = 1, 2), with respective *C*<sup>s</sup> (*k* = 1), *C*<sup>i</sup> (*k* = 2) symmetry. The *Cs*-symmetric 0,2,4W6 (Cs) structure (Figure 4, left) is slightly higher in energy than 0,2,4W6 of Figure 2, but *C*i-symmetric 0,4,4W8 (Ci) (Figure 4, right) is slightly lower in energy than its low-symmetry counterpart in Figure 2. The inherent chirality of the coordination pattern about *each* O atom of higher-coordinated water clusters of Figures 2 and 3 indicates that reduced symmetry (net chirality) is a high-probability feature of equilibrium water cluster distributions in any phase involving their participation.

**Figure 4.** Alternative higher-symmetry 0,2k,4W2k+4(sym) clusters (*k* = 1,2), one (*C*s) of higher energy, the other (*C*i) of lower energy than the corresponding low-symmetry structure of Figure 2.

Note that although H-bonds are considered weak noncovalent attractions, the cumulative energy release from larger cluster formation (viz., Δ*E* ≈ 170 kcal/mol for the 8,8,0W16 cluster) can readily exceed that necessary to dissociate a strong covalent bond, as in the ion pair clusters involved in self-dissociation (pH) of liquid water [54,55]. The per-monomer free energies of formation shown in Figures 2 and 3 remain slightly positive under standard-state conditions, but the windowpane clusters are expected to gain increased stability relative to the ice-like clusters of the near-ambient regime as pressure increases. Full thermochemical and vibrational spectroscopic values for each cluster are included with the optimized coordinates in SI.
