Quantum Anharmonic Effects in Hydrogen-Bond Symmetrization of High-Pressure Ice

Abstract

The nuclear quantum effects of hydrogen play a significant role in determining the phase stability of water ice. Hydrogen-bond symmetrization occurs as hydrogen atoms tunnel in a double-well potential, ultimately occupying the midpoint between oxygen atoms and transforming ice VIII into ice X under high pressure. Quantum fluctuations lower this transition from classical predictions of over 100 GPa to 60 GPa. We reveal that the Perdew-Burke-Ernzerhof functional underestimates the hydrogen double-well barrier, thus resulting in a transition pressure over 10 GPa lower than the strongly constrained and appropriately normed functional, which is validated against quantum Monte Carlo calculations. Nuclear quantum anharmonicity, treated via neural canonical transformation (NCT), reveals that this transition pressure is temperature-independent and observes a 2 GPa reduction when comparing the non-Gaussian flow models based wavefunction compared to the self-consistent harmonic approximation. Despite increasing pressure typically shortens chemical bonds and hardens phonon modes, NCT calculations reveal that hydrogen bond softens hydrogen-oxygen stretching in ice VIII upon pressure.

Figures (6)

• Figure 1: (a) The sketch of ice VIII, visualized using VESTA momma2011vesta. The illustration shows only part of the VIII unit cell, which contains oppositely polarized configurations for net-zero polarization. (b) Ice X. (c) The potential energy surface is calculated using various density functional theory functionals. All hydrogen atoms are moved along the O-O direction, with their positions defined in (a) while the oxygen atoms remain fixed. (d) The energy difference of ice X and ice VIII computed with various density functionals are compared with QMC results.
• Figure 2: Numerical results of the NCT method for high-pressure ice. Helmholtz free energy differences $\Delta F$ between ice VIII and X as a function of volume at (a) $T=0~\text{K}$, (b) $50~\text{K}$, and (c) $100~\text{K}$. The free energy of ice VIII is set to zero as the reference. (d) The O-H bond lengths $d_{\text{OH}}$ plotted against the O-O distances $d_{\text{OO}}$. (e) The order parameter $\Delta = |d'_\text{OH} - d_\text{OO}/2|$ for ice VIII, where $d'_\text{OH}$ is the projection of $d_{\text{OH}}$ onto the O-O direction. (f) The equation of state. Experimental results were obtained at room temperature wolanin1997equationsugimura2008compressionasahara2010thermoelastic, while the NCT results were calculated at $100~\text{K}$. The inset highlights the transition pressure.
• Figure 3: Phonon density of states for ice VIII under pressure. Anharmonic phonon frequencies from Neural Canonical Transformations are derived from single-phonon excitation energies. Experimental data are taken from Refs. goncharov1999raman, with molecular O-H vibrational modes denoted as $\nu_1$ and $\nu_3$.
• Figure 4: Numerical results of NCT with non-Gaussian and Gaussian normalizing flow model. (a) Energy differences between ice VIII and X structures at zero temperature, with ice VIII (Gaussian) set to zero as a reference. (b) O-H bond lengths $d_{\text{OH}}$ versus O-O distances $d_{\text{OO}}$, with the inset showing the order parameter $\Delta = |d'_{OH} - d_{OO}/2|$.
• Figure S1: Comparison between MACE predictions and density functional theory calculations with SCAN functional on the test dataset: (a) energy, (b) force.