Machine-learned one-bead coarse-grained (CG) water models — ML-BOP and ML-BOP_dih (Tersoff-type bond-order potentials, the latter with a four-body dihedral term) and ML-mW (Stillinger–Weber form) — deliver a near-experimental melting point, density anomaly, and roughly two-orders-of-magnitude speedups for mesoscale ice/water simulation, while leaving four documented accuracy limits unresolved: over-structured liquid water (too-deep oxygen–oxygen radial distribution function minimum near 3.4 Å and underestimated first-to-second solvation-shell exchange); a Stillinger–Weber parameterization that cannot simultaneously reproduce the melting point and the experimental temperature-of-maximum-density-to-melting-point ordering; a four-body ice-energetics term trained on DFT-PBE data of contested accuracy; and unquantified temperature- and size-dependent kinetics of the stacking-disordered to hexagonal ice transformation. The models are in addition short-ranged, one-bead and non-polarizable, so they cannot represent explicit electrostatics, polarizability, ions or charged species — a property of the model class that is not posed as a research target. The problem asks to resolve the four documented limits — by constructing and validating improved ML-parameterized CG water models that remove the defects at retained efficiency, or by rigorously establishing which defects are intrinsic to short-range one-bead CG functional forms.

Background

Molecular dynamics (MD) simulations of water underpin studies of ice nucleation, cloud and climate microphysics, hydrates, and biomolecular solvation, but all-atom water models that resolve every oxygen and hydrogen (for example the rigid four-site TIP4P/2005 model, or the accurate but expensive polarizable MB-pol) are too costly for the multi-million-molecule, microsecond-scale simulations these phenomena require. Coarse-grained (CG) models regain that scale by mapping each whole H2O molecule onto a single interaction center ('bead'), gaining roughly two orders of magnitude in speed at the price of accuracy. The monatomic water (mW) model of Molinero and Moore represents each water molecule as one bead interacting through a Stillinger–Weber (SW) potential — a pair term plus a three-body angular term that favors tetrahedral coordination; its interactions are deliberately short-ranged, which is what makes it fast. mW freezes at nearly the right temperature but places the temperature of maximum density (TMD, the anomaly by which liquid water is densest at 277 K at 1 bar) at about 251 K and overpredicts the density of ice.

Chan et al. (Nature Communications 10, 379 (2019)) trained CG water potentials with a multilevel evolutionary strategy (a 'hierarchical objective genetic algorithm', HOGA) that fits not only first-principles energetics and experimental data but also temperature-dependent properties inferred on the fly from MD trajectories. Three models resulted: ML-BOP, a Tersoff-type bond-order potential (in a bond-order potential the strength of the interaction between two beads is modulated by their local coordination environment, with an explicit smooth cutoff function that truncates the interaction at finite range); ML-BOP_dih, which adds an on-the-fly four-body dihedral term — an energy depending on the torsion angle formed by four neighboring bonded units — so that the model can distinguish cubic ice (Ic) from hexagonal ice (Ih) energetically; and ML-mW, a re-optimization of the mW Stillinger–Weber form. ML-BOP reproduces the melting point Tm=273±1T_m = 273 \pm 1 K and the TMD (276–279 K) — reported as the first water model to capture both, and with them the experimental ordering TMD−Tm≈+4T_{\mathrm{MD}} - T_m \approx +4 K — along with densities within 1.4% of experiment and near-experimental diffusivity. The four-body term of ML-BOP_dih was trained on the hexagonal–cubic ice energy difference computed with the PBE density functional (a common generalized-gradient-approximation exchange–correlation functional), about 1.4 meV per H2O (Engel, Monserrat and Needs, 2015), and the authors note that PBE may not be definitive for these sub-meV energetics.

Four accuracy limits were documented and left unresolved, alongside a further property of the model class that is not posed as an open question. (i) ML-BOP-type models over-structure liquid water: the oxygen–oxygen radial distribution function (RDF, the probability density of finding a molecular pair at a given separation, relative to an ideal gas) has too deep a minimum near 3.4 Å, and the exchange of molecules between the first and second solvation (coordination) shells is underestimated — a defect known for short-range CG water models generally, and one that feeds directly into nucleation and growth predictions. (ii) ML-mW, though improved over mW in the density anomaly and ice density, over-predicts the melting point (289 K) and gets the TMD–TmT_m ordering wrong (−10-10 K instead of +4+4 K); the authors trace this to the implicit cutoff of the SW form, which couples the pair-interaction tail to the well depth, unlike the explicit Tersoff cutoff. (iii) The four-body dihedral term inherits the accuracy of its PBE-based training data and can be retrained once better ab initio ice energetics become available. (iv) The bulk transformation of stacking-disordered ice (Isd, ice with random interleaving of cubic and hexagonal stacking sequences) to hexagonal ice was not observed even in 1.3 µs of MD at 260 K, consistent with a climbing-image nudged elastic band (CI-NEB, a standard saddle-point search method) barrier of about 170 meV for eliminating a single stacking-fault plane; the temperature- and size-dependent kinetics of the transformation remain unquantified. Beyond these four documented limits, the models are short-ranged, one-bead, and non-polarizable, so they cannot represent explicit electrostatics, polarizability, ions, or charged species, which atomistic polarizable models can treat only at much higher cost.

Since 2019 the ML-BOP family has been taken up and validated by other groups — its equation of state, ice–water thermodynamics, amorphous-ice phenomenology, vapor–liquid equilibrium, and even a liquid–liquid transition have been characterized — but the four documented defects themselves have not been removed, and improved reference data that bear on them (calorimetry on pure cubic ice, giving a cubic-to-hexagonal enthalpy change ΔHc→h=−37.7±2.3\Delta H_{c\to h} = -37.7 \pm 2.3 J/mol; higher-level electronic-structure benchmarks; and molecular-resolution cryogenic electron microscopy of freezing) have only recently become available. The unresolved target is therefore: cure, or rigorously characterize as intrinsic, the accuracy limits of ML-parameterized short-range CG water.

Problem Statement

Resolve the four documented accuracy limits of the machine-learned short-range coarse-grained (CG) water models ML-BOP, ML-BOP_dih and ML-mW — either by constructing and validating improved ML-parameterized CG water models that remove the defects while retaining the roughly two-orders-of-magnitude efficiency advantage over atomistic models, or by rigorously establishing which defects are intrinsic to short-range one-bead CG functional forms. The common objective is a coarse-grained water description whose accuracy limits are removed or controlled; the four jointly required results are:

(1) Liquid structure: remove the over-structuring of liquid water — the oxygen–oxygen radial distribution function minimum near 3.4 Å must match the experimental depth, to within about 5% in the depth of the first minimum, in liquid water at both ambient (298 K, 1 bar) and supercooled (254 K, 1 bar) conditions, and the exchange of molecules between the first and second solvation shells — quantified as the rate of first-to-second coordination-shell exchange events per molecule, equivalently the mean first-shell residence time — must be reproduced — without degrading the already-achieved melting point, temperature of maximum density, densities, and diffusivity.

(2) Stillinger–Weber thermodynamics: determine whether a parameterization of the Stillinger–Weber functional form can simultaneously reproduce the melting point (Tm≈273T_m \approx 273 K) and the experimental ordering and relative separation of the temperature of maximum density and the melting point (TMD−Tm≈+4T_{\mathrm{MD}} - T_m \approx +4 K), where ML-mW fails (Tm=289T_m = 289 K, TMD−Tm=−10T_{\mathrm{MD}} - T_m = -10 K); or establish with controlled evidence that the implicit cutoff of that form makes the joint target unreachable.

(3) Four-body ice energetics: determine four-body dihedral energetics for ML-BOP_dih that improve on the DFT-PBE hexagonal–cubic ice energy difference (about 1.4 meV per H2O) on which the original four-body term was trained, and quantify the consequence for the relative stability of hexagonal, cubic, and stacking-disordered ice.

(4) Stacking-disordered ice kinetics: quantify the temperature- and size-dependent kinetics of the bulk Isd→IhI_{\mathrm{sd}} \to I_h transformation, whose single stacking-fault-elimination barrier was computed at about 170 meV and which was not observable in microsecond molecular dynamics at 260 K; identify the pathways and timescales, and validate them against experiment or against higher-fidelity electronic-structure benchmarks.

Each of the four documented defects must either be removed in a validated model that retains the documented ML-BOP accuracies — melting point within 273±1273 \pm 1 K, temperature of maximum density in the 276–279 K range with the experimental ordering TMD−Tm≈+4T_{\mathrm{MD}} - T_m \approx +4 K preserved, densities of ice and water within 1.4% of experiment, and room-temperature diffusivity no further from the experimental 2.3×10−52.3 \times 10^{-5} cm2^2 s−1^{-1} than ML-BOP's reported 3.0×10−53.0 \times 10^{-5} cm2^2 s−1^{-1} — or shown to be intrinsic to the class of short-range one-bead CG functional forms to which the defective model belongs.

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