The 2021 Deep Potential water model — a neural-network potential energy surface trained on SCAN density-functional theory data — reproduced most of water's phase diagram up to about 2400 K and 50 GPa, but left a precisely defined set of gaps: the vapor stability region and the ice-X pressure regime were excluded from the training domain; phases whose free energies differ by less than roughly 1 meV per H2O molecule — notably the experimentally stable ice III and ice XV, predicted metastable by the model — could not be reliably ranked; and the nuclei were treated classically, so isotope-dependent phase boundaries were out of reach. Subsequent work has addressed part of this agenda at reduced scope: quantum simulations with the MB-pol potential, Deep Potential studies of liquid-vapor equilibrium and of superionic ice, atomic-cluster-expansion phase diagrams on revPBE-D3 and hybrid references, and path-integral molecular dynamics with Deep Potentials across several functionals. Each of these leaves a specific part of the target open: the MB-pol quantum simulations reach only the low-pressure region and do not treat ice XV, the vapor or ice-X regimes, or isotope-dependent coexistence lines, and rest on MB-pol rather than a DFT-referenced potential; the liquid-vapor and superionic-ice Deep Potential studies use classical nuclei and cover only their own sub-region; the atomic-cluster-expansion diagrams leave ice III metastable under the revPBE-D3 reference; and the path-integral Deep Potential studies find that DFT-referenced models mispredict the sign of the isotope shift of the melting temperature and stop short of the full isotope-dependent diagram. No published work simultaneously spans vapor-to-ice-X pressures, includes nuclear quantum effects with correct isotope-dependent boundaries, and resolves the sub-meV degeneracies under an ab initio-referenced potential; the problem remains open, with the error budget — density-functional accuracy versus potential-representation fidelity versus classical nuclei — now sharply posed but unquantified.

Background

Water exhibits one of the richest phase diagrams of any pure substance: around twenty crystalline polymorphs of ice distinguished by their oxygen sublattices and by proton ordering (many are proton disordered, with hydrogen positions statistically distributed subject to the Bernal-Fowler ice rules), plus a molecular liquid and vapor, an ionic fluid in which molecules partially dissociate, and superionic phases at high pressure and temperature in which oxygen stays ordered while protons diffuse. Predicting this diagram from quantum mechanics rather than by fitting to experiment is a long-standing challenge. Kohn-Sham density functional theory (DFT) approximates the electronic exchange-correlation energy; the SCAN (strongly constrained and appropriately normed) meta-GGA functional describes water reasonably well but is known to overestimate hydrogen-bond strength. Direct ab initio molecular dynamics is far too expensive for the long, large simulations that free energies require, so recent work interpolates the DFT potential energy surface (PES) with machine-learning potentials, such as the Deep Potential (DP) neural-network representation, trained iteratively on DFT energies and forces through active learning (the DP-GEN procedure). Phase boundaries are then obtained by free-energy methods: thermodynamic integration from an Einstein-crystal reference for the solids, integration along isotherms or isobars for the fluid, and Gibbs-Duhem integration to trace coexistence lines from coexistence points. The accuracy scale is severe: competing ice polymorphs can differ in free energy by about 1 meV per H2O molecule (roughly 0.1 kJ/mol), which is comparable to the residual errors of both DFT functionals and fitted PES representations, so any single ingredient can invert a stability ranking. Finally, hydrogen nuclei are quantum objects: nuclear quantum effects (NQE), i.e. zero-point motion and tunneling, treatable with path-integral molecular dynamics (PIMD, in which each nucleus is represented by a ring polymer of beads), shift thermodynamic properties and produce the observed isotope differences between H2O and D2O (for example, ordinary water melts about 4 K below heavy water). Against this backdrop, Zhang, Wang, Car and E (Phys. Rev. Lett. 126, 236001 (2021)) trained a DP model on SCAN data and computed the phase diagram up to about 2400 K and 50 GPa: all stable ice polymorphs and the fluid phases are predicted correctly except ice III and ice XV, which are stable experimentally but metastable in the model; the ice Ih-fluid melting line is shifted by about 40 K to higher temperature, consistent with SCAN's overestimated hydrogen-bond strength; the vapor stability region and the ice X (symmetric, non-molecular) pressure regime lie outside the training domain; and the nuclei are treated classically. An earlier quantum-mechanical phase diagram by Reinhardt and Cheng, based on hybrid-DFT machine-learning potentials, covered only 150-300 K and 0.01-1 GPa. What remains unresolved is the closing of all these gaps as one coherent objective: a first-principles-based phase diagram of water that is quantitatively right, isotope-resolved, and error-controlled.

Problem Statement

Can the phase diagram of water be computed from a potential energy surface of ab initio quality with sub-meV free-energy control per H2O molecule and nuclear quantum effects included, such that the observed stability of ice III and ice XV, the vapor and ice-X regimes, and the isotope-dependent phase boundaries are all reproduced? Concretely, (i) per-phase free energies are converged and uncertainty-quantified well below the scale of about 1 meV per H2O molecule at which competing phases are degenerate, so that closely competing phases — in particular the experimentally stable ice III and ice XV — are correctly ranked rather than left effectively degenerate; (ii) isotope-dependent coexistence lines (H2O versus D2O) are predicted in quantitative agreement with experiment, i.e. resolved on the scale of the observed isotope separations such as the roughly 4 K difference between the H2O and D2O melting points; and (iii) the overall phase diagram agrees with experiment throughout the full vapor-to-ice-X domain targeted here, from the vapor stability region to the ice-X pressure regime, with the same sub-meV per H2O free-energy control required in (i). Criteria (i), (ii) and (iii) are independently required and jointly necessary: satisfying only a subset of them does not resolve the task.

References

  1. Linfeng Zhang, Han Wang, Roberto Car, Weinan E. Phase Diagram of a Deep Potential Water Model. Physical Review Letters 126, 236001 (2021). DOI: 10.1103/PhysRevLett.126.236001; arXiv:2102.04804, https://arxiv.org/abs/2102.04804
  2. Aleks Reinhardt, Bingqing Cheng. Quantum-mechanical exploration of the phase diagram of water. Nature Communications 12, 588 (2021). DOI: 10.1038/s41467-020-20821-w, https://www.nature.com/articles/s41467-020-20821-w
  3. Sigbjørn Løland Bore, Francesco Paesani. Realistic phase diagram of water from “first principles” data-driven quantum simulations. Nature Communications 14, 3349 (2023). DOI: 10.1038/s41467-023-38855-1, https://www.nature.com/articles/s41467-023-38855-1
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  6. Eslam Ibrahim, Yury Lysogorskiy, Ralf Drautz, Pablo M. Piaggi. Water Phase Diagram from a General-Purpose Atomic Cluster Expansion Potential. Journal of Chemical Theory and Computation 22, 4758-4766 (2026). DOI: 10.1021/acs.jctc.6c00094; arXiv:2601.12947, https://arxiv.org/abs/2601.12947
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