**1. Introduction**

The earliest applications of ab initio natural bond orbital (NBO) analysis [1–4] consistently revealed a "donor–acceptor" (resonance–covalency-type "charge transfer") picture of hydrogen bonding that was sharply at odds with then-prevalent "electrostatic" conceptions of intermolecular interactions [5,6]. Although the IUPAC *Gold Book* definition of H-bonding was subsequently revised to acknowledge the importance of covalency in Hbonding [7], superficial "dipole–dipole" rationalizations of H bonding continue to survive in many freshman-level expositions [8]. Arguments against the charge-transfer picture or in support of classical-type long-range, multipole, or "electrostatically driven" conceptions of H-bonding continue to appear [9,10] (vs. replies in [11–13]) in the research literature, and similar simplifying approximations persist in the empirical force fields of popular molecular dynamics (MD) simulation methods [14] that are commonly adopted to describe H-bonding in condensed phases.

The daunting task of describing macroscopic phases of liquid water or other H-bonded fluids may seem to demand the drastic long-range approximations of intermolecular ("noncovalent") interactions as compared to the exchange-type ("covalent") interactions of the short-range molecular regime. However, a more practical and accurate approach to describing intermolecular H-bonding is achieved by adopting *supramolecular clusters* [15] {*C*n} as the conceptual "building blocks" of the macroscopic liquid-phase description, based on the known *continuity* of high-density liquid and low-density gaseous phases around the

**Citation:** Weinhold, F. High-Density "Windowpane" Coordination Patterns of Water Clusters and Their NBO/NRT Characterization. *Molecules* **2022**, *27*, 4218. https:// doi.org/10.3390/molecules27134218

Academic Editor: Miroslaw Jablonski

Received: 3 May 2022 Accepted: 27 June 2022 Published: 30 June 2022

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**Copyright:** © 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

fluid critical point [16]. More specifically, quantum cluster equilibrium (QCE) theory [17–19] provides a practical numerical implementation of such "cluster mixture" [20–25] modeling of macroscopic phase properties, based on accurate values of electronic and vibrational properties of H-bonded {*C*n} clusters that can be obtained at any chosen ab initio or density functional theory (DFT) level. The key input for QCE-based thermodynamic modeling of an aqueous phase is the data set of supramolecular clusters whose self-consistent (*T*,*P*) dependent equilibrium populations are determined from the computed partition functions for each cluster by the standard methods of quantum statistical thermodynamics [26].

Among the many H-bonded fluids of practical interest, water itself presents the most studied yet still most perplexing phase behavior of the terrestrial regime [27]. Even the microscopic structure and properties of "ordinary" liquid water under near-ambient conditions remain matters of controversy [28]. Further mysteries surround the phase behavior of water at higher temperatures and pressures, where both theory [29–32] and experiments [33–36] have suggested the existence of an alternative high-density phase of liquid water that could lead to a liquid–liquid critical point and an exotic new domain of thermodynamic behavior near 220 K and 1–2 kbar.

The primary goal of present work was to computationally search for a new class of water clusters {*W*n} based on the quadrilateral ("windowpane") coordination motif of the cyclic tetramer (Figure 1) that might contribute to equilibrium QCE populations in the neighborhood of the proposed high-density phase. In each case, we restricted attention to clusters that maintain maximal Grotthuss-type proton ordering for the powerful effects of cooperative stabilization [37,38], as exemplified by the clockwise ordering of in-ring OH bonds in the view of Figure 1. The near −90◦ coordination angles of the windowpane class correspond to reduced next-neighbor distances and increased mass/volume ratios compared to the characteristic tetrahedral angles and chair–hexagon coordination motifs of ice-I-like clusters. The search for cooperatively stabilized windowpane clusters is organized in *Aufbau* fashion toward increasing numbers of fully four-coordinate sites that more adequately sample the intermolecular interactions expected to dominate in the phase behavior of the low-temperature and high-pressure regime. The resulting windowpane clusters can serve as computational input for subsequent QCE studies to examine their possible role in the equilibrium cluster distributions of the water-phase diagram.

**Figure 1.** Equilibrium structural properties of cyclic (H2O)4 "windowpane" cluster (B3LYP/6- 311++G\*\* level).

A secondary goal of this study was to characterize each computed cluster in deeper conceptual terms that can clarify distinctive features of the underlying H-bond interactions. Such characterization should include aspects of overall cluster stability, strengths of individual coordinative linkages, shifts in atomic charge distribution, and other orbital-level features of free vs. coordinated water molecules. For these purposes, we employed NBO analysis [39,40] to obtain localized descriptors of molecular and intermolecular bonding features. Of particular interest are natural resonance theory (NRT) bond orders [41], which are expected to exhibit useful correlations with bond lengths [42,43], bond energies [44,45], bond stretching frequencies [46–48], NMR 1J and 1hJ spin-coupling constants [49], and other experimentally measurable properties.

Although the present study of novel water clusters was primarily directed toward equilibrium thermodynamic properties, it is important to note that such studies can also yield information on the kinetics and mechanisms of water cluster reactions. This is particularly true when, as in the present case, each cluster of the class is created in a sequential *Aufbau* manner from a previous member, e.g., by successive dimer additions of the form

$$\rm{W\_k} + \rm{W\_2} \rightleftharpoons \rm{W\_{k+2}} \tag{1}$$

where *W*<sup>k</sup> = (H2O)k is a *k*-mer of a chosen coordination pattern. Analogous to elementary A+B - C chemical reactions, one can compute the transition state (*W*k···*W*2) ‡ and other features of the intrinsic reaction coordinate [50] (IRC) for each such cluster reaction. Similarly, for other cluster species satisfying the simultaneous QCE equilibrium conditions,

$$\rm{W\_{\dot{j}} + W\_k \leftrightharpoons W\_{\dot{j}'} + W\_{k'} \ (\dot{j} + k = \dot{j}' + k')} \tag{2}$$

standard quantum chemical methods can be employed to determine transition-state features and associated absolute rate constants along the associated IRC [51]. However, such deeper mechanistic aspects of cluster formation were not addressed in the present work.
