Proton dynamics in high-pressure ice-VII from density functional theory

Florian Trybel, Michael Cosacchi, Thomas Meier, Vollrath Martin Axt, and Gerd Steinle-Neumann

Phys. Rev. B 102, 184310 – Published 30 November 2020

Abstract

Using a density functional theory based approach, we explore the symmetrization and proton dynamics in ice-VII, for which recent high-pressure NMR experiments indicate significant proton dynamics in the pressure-range of $20 - 95 GPa$ . We directly sample the potential seen by the proton and find a continuous transition from double- to single-well character over the pressure range of 2 to 130 GPa accompanied by proton dynamics, in agreement with the NMR experiments.

Received 30 June 2020
Accepted 4 November 2020

DOI:https://doi.org/10.1103/PhysRevB.102.184310

Physics Subject Headings (PhySH)

Dynamical tunneling Phonons

Ice

Density functional theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Florian Trybel ^1,*, Michael Cosacchi², Thomas Meier¹, Vollrath Martin Axt², and Gerd Steinle-Neumann ¹

¹Bayerisches Geoinstitut, Universität Bayreuth, D-95440 Bayreuth, Germany
²Theoretische Physik III, Universität Bayreuth, D-95440 Bayreuth, Germany

^*f.trybel@uni-bayreuth.de

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Issue

Vol. 102, Iss. 18 — 1 November 2020

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Images

Figure 1
Cut through the $4 \times 4 \times 4$ ice-VII supercell. The sampling of the correlated hexagonal configuration is performed by moving the green protons in the ring of orange oxygen atoms. For the single proton sampling, the red proton is moved. To provide local charge neutrality, the ice-rule violating defects for the ${OH}^{-}$ and $H_{3} O^{+}$ configurations are moved to the edge of the simulations cell, along a path similar to that outlined by the golden hydrogen atoms. vesta 3 [27] is used for structural visualizations.
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Figure 2
Energy of different proton configurations for $l = 3.35$ Å (top) and $l = 2.95$ Å (bottom) and (a)–(d) visualization of possible configurations with a single-proton ice-rule (IR) violation: Oxygen is shown in red, unoccupied hydrogen positions in white, occupied hydrogen positions in yellow, and sampled hydrogen in blue: (a) $H_{3} O^{+}$ in cis configuration (cis- $H_{3} O^{+}$ ), (b) trans- $H_{3} O^{+}$ , (c) cis- ${OH}^{-}$ , and (d) trans- ${OH}^{-}$ . Potentials displayed are for the collective motion of protons (all), the collective hexagonal configuration (hexagon) and a single proton (single), all obeying IR. Potentials for the IR-breaking arrangements are trans- $H_{3} O^{+}$ (t), without and with conjugation (tc) of the associated charge defect, and cis- ${OH}^{-}$ in the conjugated configuration (cc). Conjugated trans- $H_{3} O^{+}$ and conjugated cis- ${OH}^{-}$ are shown because they bracket the potential of all conjugated IR-breaking structures.
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Figure 3
Parameters describing the structure of the double-well potential as a function of lattice parameter $l$ . (top) $δ_{c}$ , the distance between the center of the double well and the minima. (bottom) $Φ_{m}$ , the height of the barrier. Solid lines are fits of a square root for $δ_{c}$ and a quadratic function for $Φ_{m}$ to the hexagonal [green, $Φ_{m}^{Hex} = 1.42 {(x - 2.54)}^{2}, δ_{c}^{Hex} = 0.57 {(x - 2.59)}^{0.5} - 0.10]$ and trans- ${OH}^{-}$ points [orange, $Φ_{m}^{{OH}^{-} t c} = 1.46 {(x - 2.79)}^{2}, δ_{c}^{{OH}^{-} t c} = 0.90 {(x - 2.49)}^{0.5} - 0.46]$ . Open black diamonds show the result of path integral ab initio molecular-dynamics simulations at $l = 2.84$ Å [20].
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Figure 4
Probability of finding the proton in the left half of the potential $[p (x < 0)]$ as a function of time for the hexagonal configuration. Black lines indicate $p (x < 0) = 0.5$ . For $l \geq 3.25$ Å, $p (x < 0) ≳ 0.5$ by construction. With increasing compression (decreasing $l$ ) the noise level increases, but no clear underlying dynamics is found until at $l ≲ 3.2$ Å $p (x < 0)$ starts to oscillate around a value of 0.5 with decreasing wavelength. At $l ≲ 2.7$ Å the amplitude of $p (x < 0)$ suddenly collapses and no further dynamics is observed.
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Figure 5
Results of the dynamical calculation for the hexagonal configuration. (bottom) Transition frequency spectrum calculated from the eigenvalues of the Hamiltonian (black vertical lines) and the solution to the dynamic problem, represented by the Fourier transformation of the probability of finding the proton in the left side of the potential, $p (k)$ , with amplitudes normalized to $ν_{01}$ (red) for $l = 2.95$ Å. (top) $ν_{01}^{H}$ from the eigenvalue analysis (black circles) and position of $ν_{01}^{F T}$ in $p (k)$ (red crosses) as a function of $P$ . Blue squares show hydrogen jump rates calculated by Hernandez and Caracas [24] (HC 2018) on the basis of molecular-dynamics simulations.
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Figure 6
(top) Excitation frequencies $ν_{01}^{F T}$ as a function of $P$ at 300 K for the different proton configurations explored in this study. Black squares show hydrogen jump rates calculated by Hernandez and Caracas [24] (HC 2018) on the basis of molecular-dynamics simulations. (bottom) Corresponding amplitudes of $ν_{01}^{F T}$ in $p (k)$ . Amplitudes are normalized to the maximum value for the respective configuration [49]. Diamonds in magenta and purple show the result of a peak analysis of NMR experiments by Meier et al. [25] (M 2018).
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