Robustness, cross-term corrections, and causal thermodynamic significance of the ring-topological origin of the split first peak in liquid water's structure factor

Machine-learning simulations have shown that the split first peak of liquid water's static structure factor can be decomposed into contributions from hydrogen-bonded rings of different sizes, with 5–8-membered rings explaining the experimentally observed bimodality and pentagonal rings remaining distinct above room temperature. This established result leaves the mechanism unestablished in three linked respects: its robustness to interaction-model choice and machine-learning training-data bias, the quantitative error introduced by neglecting inter-ring cross-terms in the ring-wise decomposition, and whether the pentagon–hexagon topological competition causally underlies water's thermodynamic anomalies and the hypothesised liquid–liquid critical point. The problem is to settle these three questions for liquid water, each by a definite quantitative finding — a robustness assessment, an estimate of the neglected cross-terms, and a causal determination, respectively — whether that finding is affirmative or negative, thereby converting a suggestive topological correlation into a validated, causally grounded microscopic mechanism.

Background

When X-rays or neutrons scatter from a liquid, the angular pattern is summarized by the static structure factor $S(q)$, a function of momentum transfer $q$ whose peaks encode the preferred intermolecular separations and any medium-range (beyond nearest-neighbor) correlations. In liquid water the principal peak of $S(q)$ at low $q$ (around 2 Å⁻¹) is observed to split into two maxima upon supercooling, i.e., cooling below the freezing point while remaining liquid; neutron and X-ray diffraction showed that the separation of the two components grows as temperature decreases (Skinner et al. 2014; Esmaeildoost et al. 2021). The conventional interpretation assigns the two components to two coexisting local environments: more open, tetrahedrally coordinated regions and more compact, disordered regions, an idea connected to water's thermodynamic anomalies (density maximum, compressibility minimum) and to the two-liquid scenario, in which deeply supercooled water is hypothesised to end at a liquid–liquid critical point (LLCP) separating high- and low-density liquid states (Poole, Sciortino, Essmann, and Stanley 1992). Shi and Tanaka (2019, 2020) computationally traced this splitting to local tetrahedral ordering and the coexistence of two types of local structures in the scattering function.

The hydrogen-bond network (HBN) of water — the graph whose vertices are water molecules and whose edges are hydrogen bonds — can also be described topologically through rings, closed loops of hydrogen-bonded molecules of size $n$ (triangles, squares, pentagons, hexagons, ...). Ring statistics are known to be sensitive to the water model used: Martelli (2021) showed that classical force fields differ markedly in HBN ring topology, and ring-level motifs have been linked to water's anomalies (Martelli 2019). In non-aqueous network glasses, ring-size distributions have been connected quantitatively to diffraction features (Tavanti et al. 2020 on amorphous Ge$_x$Se$_{1-x}$; Zhou et al. 2021 on silicate glasses, including an experimental method to extract ring statistics from diffraction).

Faure Beaulieu, Deringer, and Martelli (2026) bridged the two viewpoints for water. A machine-learning interatomic potential (MLIP — an interaction model trained on quantum-mechanical reference data; here the equivariant message-passing MACE model, trained on the dataset of the atomic-cluster-expansion ACE water model of Ibrahim, Lysogorskiy, and Drautz 2024) reproduces the experimentally observed splitting. For each ring size $n$, the authors retain only the atoms belonging to $n$-membered rings and compute a ring-resolved structure factor $S_n(q)$. They find that the low-$q$ and higher-$q$ components of the split peak are carried by different ring populations — 5–8-membered rings dominate, and pentagonal rings keep a distinct structural signature up to at least 350 K — supporting a picture of competition between hexagonal (ice-like, ordered) and pentagonal (disordered, geometrically frustrated) motifs. The authors state, however, that (a) the MACE model shows low-$q$ discrepancies and tends to over-structure the network, plausibly due to ice-like configurations dominating the training set; (b) the ring-wise decomposition removes all cross-terms between atoms in different rings, so the absolute intensities of $S_n(q)$ are not physically meaningful and only peak positions are used; and (c) the analysis establishes a diffraction-level link only — no causal connection to thermodynamic response functions or to the LLCP is demonstrated. They additionally propose, as untested future work, extending the ring-resolved scattering analysis to other network-forming liquids, glasses, and biologically relevant aqueous environments.

Problem Statement

Establish the ring-topological explanation of the split first peak of liquid water's structure factor as a quantitatively valid, model-robust, and causally consequential microscopic mechanism. Concretely, for liquid water simulated with machine-learning (and, for reference, classical) interaction potentials, determine:

1. Robustness to model and training data. How sensitive are the hydrogen-bond-network ring populations, ring lifetimes, and the ring-resolved structure factors $S_n(q)$ — including the qualitative claim that 5–8-membered rings carry the split first peak and that pentagonal rings retain distinct signatures up to roughly 350 K at ambient pressure — to the choice of interaction potential and to the composition of its training dataset (in particular the predominance of ice-like configurations)?

2. Cross-term correction. The ring-wise decomposition of $S(q)$ removes all cross-terms between atoms in different rings, making the absolute intensities of $S_n(q)$ non-additive. Determine quantitatively how large the neglected inter-ring cross-term contributions are, and whether a cross-term-inclusive (or otherwise exact) ring-resolved decomposition changes the conclusion that different ring sizes carry the two components of the split peak — i.e., whether the topological interpretation survives a decomposition in which intensities are physically meaningful.

3. Causal link to thermodynamic anomalies and the LLCP scenario. Determine whether the identified topological competition between pentagonal (disordered) and hexagonal (ordered) rings causally underlies specific thermodynamic response functions of supercooled water — for example, whether growth or suppression of specific ring populations tracks, predicts, or perturbs extrema in the isothermal compressibility or other response functions along isobars and isochores — and what this implies for the hypothesised liquid–liquid critical point. A correlational observation alone does not suffice: the answer must either demonstrate and characterize a causal mechanism or establish that no such causal link exists — a definite negative finding also resolves this sub-question.

The source work additionally proposes, as an explicitly untested generalization direction rather than part of this completion boundary, extending ring-resolved scattering analysis to other network-forming liquids, glasses (including pressure series), and biologically relevant aqueous environments; progress on that axis is welcome context but is not required to answer this problem.

References

1. Zoé Faure Beaulieu, Volker L. Deringer, and Fausto Martelli, 'Topological contribution to the split first peak of the liquid water structure factor using the MACE machine-learning potential', The Journal of Chemical Physics 165, 071101 (2026), DOI: 10.1063/5.0337533, https://doi.org/10.1063/5.0337533 (preprint: arXiv:2601.05891, 'Topological origin of peak splitting in the structure factor of liquid water', https://arxiv.org/abs/2601.05891)
2. L. B. Skinner, C. J. Benmore, J. C. Neuefeind, and J. B. Parise, 'The structure of water around the compressibility minimum', The Journal of Chemical Physics 141, 214507 (2014), DOI: 10.1063/1.4902412, https://doi.org/10.1063/1.4902412
3. N. Esmaeildoost, H. Pathak, A. Späh, T. J. Lane, K. H. Kim, C. Yang, K. Amann-Winkel, M. Ladd-Parada, F. Perakis, J. Koliyadu, A. R. Oggenfuss, P. J. M. Johnson, Y. Deng, S. Zerdane, R. Mankowsky, P. Beaud, H. T. Lemke, A. Nilsson, and J. A. Sellberg, 'Anomalous temperature dependence of the experimental x-ray structure factor of supercooled water', The Journal of Chemical Physics 155, 214501 (2021), DOI: 10.1063/5.0075499, https://doi.org/10.1063/5.0075499
4. Rui Shi and Hajime Tanaka, 'Distinct signature of local tetrahedral ordering in the scattering function of covalent liquids and glasses', Science Advances 5, eaav3194 (2019), DOI: 10.1126/sciadv.aav3194, https://doi.org/10.1126/sciadv.aav3194
5. Rui Shi and Hajime Tanaka, 'Direct Evidence in the Scattering Function for the Coexistence of Two Types of Local Structures in Liquid Water', Journal of the American Chemical Society 142, 2868–2875 (2020), DOI: 10.1021/jacs.9b11211, https://doi.org/10.1021/jacs.9b11211
6. Fausto Martelli, 'Unravelling the contribution of local structures to the anomalies of water: The synergistic action of several factors', The Journal of Chemical Physics 150, 094506 (2019), DOI: 10.1063/1.5087471, https://doi.org/10.1063/1.5087471
7. Fausto Martelli, 'Topology and complexity of the hydrogen bond network in classical models of water', Journal of Molecular Liquids 329, 115530 (2021), DOI: 10.1016/j.molliq.2021.115530, https://doi.org/10.1016/j.molliq.2021.115530 (preprint: arXiv:2101.09160)
8. Fausto Martelli, 'Steady-like topology of the dynamical hydrogen bond network in supercooled water', PNAS Nexus 1, pgac090 (2022), DOI: 10.1093/pnasnexus/pgac090, https://doi.org/10.1093/pnasnexus/pgac090
9. Eslam Ibrahim, Yury Lysogorskiy, and Ralf Drautz, 'Efficient Parametrization of Transferable Atomic Cluster Expansion for Water', Journal of Chemical Theory and Computation 20, 11049–11057 (2024), DOI: 10.1021/acs.jctc.4c00802, https://doi.org/10.1021/acs.jctc.4c00802
10. Peter H. Poole, Francesco Sciortino, Ulrich Essmann, and H. Eugene Stanley, 'Phase behaviour of metastable water', Nature 360, 324–328 (1992), DOI: 10.1038/360324a0, https://doi.org/10.1038/360324a0
11. Qi Zhou, Ying Shi, Binghui Deng, Jörg Neuefeind, and Mathieu Bauchy, 'Experimental method to quantify the ring size distribution in silicate glasses and simulation validation thereof', Science Advances 7, eabh1761 (2021), DOI: 10.1126/sciadv.abh1761, https://doi.org/10.1126/sciadv.abh1761
12. Francesco Tavanti, Behnood Dianat, Alessandra Catellani, and Arrigo Calzolari, 'Hierarchical Short- and Medium-Range Order Structures in Amorphous Ge$_x$Se$_{1-x}$ for Selectors Applications', ACS Applied Electronic Materials 2, 2961–2969 (2020), DOI: 10.1021/acsaelm.0c00581, https://doi.org/10.1021/acsaelm.0c00581