The density isobar of water: A comparative study of vdW-DF-cx and RPBE-D3

Accurately modeling volume-dependent properties of water remains a challenge for density functional theory (DFT), with widely used functionals failing to reproduce key features of the water density isobar, including its shape, density and temperature of the density maximum. Here, we compare the performance of the RPBE-D3 and vdW-DF-cx functionals using replica exchange molecular dynamics (MD) driven by machine-learned force fields. Our simulations reveal that vdW-DF-cx predicts the water density more accurately than RPBE-D3 and reproduces the isobar closely between 307-340 K. In contrast, RPBE-D3 underestimates the density across the entire temperature range. However, vdW-DF-cx predicts the maximum density temperatures to be approximately 30 K higher than experiment. Using the local structure index, we attribute this shift to an onset of low-density, ice-like structures in the vdW-DF-cx-based MD at too high temperatures. Static DFT calculations on water dimers and representative high- and low-density water structures, reveal that key features of the density isobars are reflected in the static energy-volume curves. Specifically, the equilibrium intermolecular distance and curvature correlate with the maximum density and curvature around the maximum of the density isobar. Likewise, the early onset of the low-density structure is connected to the energetic preference for more structured, low-density water over a less ordered, high-density water structure. Decomposing the exchange-correlation energy reveals that the non-local dispersion energy decisively influences the predicted equilibrium intermolecular distances, whereas the semi-local part governs the balance between low- and high-density liquid structures.