Geometry-Based Description for Hydrogen Bond Organization in Small Water Clusters Derived from Spectroscopic and Quantum Chemical Data

Hydrogen-bond organization plays a central role in determining the structure and properties of water from molecular to macroscopic scales. In this study, we propose a geometry-based descriptor for small hydrogen-bonded water clusters, (H2O)n, with n = 2–6. The central element of the proposed geometric framework is the dimensionless geometric index, Sn = d/l, where d is the center-to-molecule distance in a cluster configuration and l is the nearest-neighbor O···O distance associated with hydrogen-bonded water molecules. The geometric descriptor is not intended to replace quantum-chemical calculations or to provide a direct measurement of hydrogen-bond energy, lifetime, or number. Instead, it provides a compact geometric framework for describing the structural organization of small hydrogen-bonded water clusters. The obtained geometric trend is compared with selected Nuclear Magnetic Resonance (NMR), Møller–Plesset perturbation theory (MP2), and radial distribution function data as complementary qualitative and semi-quantitative references. The proposed geometric index Sn = dl was further compared with MP2 quantum-chemical O···O distances for (H2O)n clusters, n = 2–6, using the oxygen atoms as structural nodes of the hydrogen-bonded motifs. This comparison showed that the exponential increase in Sn is consistent with the characteristic O···O donor–acceptor length scale of approximately 2.8 Å, linking the geometric framework with calculated molecular geometries. Over the limited interval n = 2–6, the geometric index Sn increases monotonically and nonlinearly with cluster size. The quantum-chemical reference data previously reported in our study, comprising GIAO-DFT-calculated 1H chemical shifts obtained for MP2-optimized water-cluster geometries, show a rapid nonlinear increase from the dimer to the pentamer, followed by the onset of saturation in the pentamer–hexamer range. The semi-empirical stabilization parameter evaluated in the present study indicates increasing relative stabilization, with a reduced incremental change around n ≈ 4–5. The qualitative consistency of these size-dependent trends supports the use of Sn as a compact geometric descriptor of hydrogen-bond organization in small water clusters, without interpreting it as a direct quantitative measure or mechanistic framework of hydrogen-bond cooperativity. Importantly, liquid water is not treated as a system of closed cyclic clusters; cyclic motifs are used only as frameworked geometric reference configurations for small hydrogen-bonded aggregates. The geometric trend is qualitatively compared with selected quantum-chemical, spectroscopic, and radial distribution function data and should be regarded as an empirical geometric approximation over the limited interval n = 2–6. These findings indicate that geometric, spectroscopic, and quantum-chemical descriptors reflect related, but not identical, aspects of hydrogen-bond organization. The proposed approach links cluster geometry, O···O intermolecular distances, and hydrogen-bond connectivity in a simplified geometric description.