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Data and code for: Understanding the density maximum of water with machine learned potentials

Song, Yizhi1; Liu, Renxi1; Zhang, Chunyi21; Li, Yifan3; Santra, Biswajit1; Chen, Mohan4; Klein, Michael1; Wu, Xifan1

Published May 27, 2026 on Dryad. https://doi.org/10.5061/dryad.bvq83bkqm

Data files

May 27, 2026 version files 4.70 GB

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Abstract

After melting, at ambient pressure, the density of water continues to increase with temperature until it reaches a maximum around 4 °C. For nearly a century, this phenomenon has been qualitatively attributed to a mixture of ordered and disordered structures. Herein, we employ a deep neural network to train a machine learned (ML) interatomic potential for water using electronic structure data from advanced density functional theory. Notably, molecular dynamics simulations with the ML potential reproduce both the experimental water density anomaly and the thermal expansion coefficient. Detailed structural analysis of the computed hydrogen-bond network reveals that the density anomaly arises from an emergent liquid structure that retains nearly ideal tetrahedral coordination at short range but collapses at intermediate range. Our findings point to a more delicate mechanism causing the density maximum than the conventional picture, emphasizing the collective roles of structural orderings at different length scales.