Two state model for the ML-BOP potential

The coarse-grained machine-learning derived ML-BOP model [Chan et al., Nature Commun. 10, 379 (2019)] provides a monoatomic representation of the water-water interaction potential in which orientational interactions are included as three-body contributions. Despite its simplicity, the model reproduces the phase diagram of water and its anomalies. Here, we show that a two-state Gibbs free energy expression - fitted simultaneously on the temperature and pressure dependence of the density and internal energy - predicts the existence of a liquid-liquid critical point, with critical parameters consistent with previous estimates. We also show that in this model: (i) while the low density liquid is pre-empted by crystal nucleation, the high-density liquid and its spinodal are accessible in numerical studies down to 100 K; (ii) crystallisation requires the presence of a local low density region. Thus, for densities larger than the critical density, spinodal decomposition (or nucleation of the low-density liquid) is a pre-requisite for ice nucleation.