# Hydrogen bond symmetrization in high-pressure ice clathrates

Lorenzo Monacelli, Maria Rescigno, Alasdair Nicholls, Umbertoluca Ranieri,

Simone Di Cataldo, Livia Eleonora Bove

► To cite this version:

Lorenzo Monacelli, Maria Rescigno, Alasdair Nicholls, Umbertoluca Ranieri, Simone Di Cataldo, et al.. Hydrogen bond symmetrization in high-pressure ice clathrates. Physical Review B, In press, 112 (21), pp.214106. 10.1103/1cgl-mklx. hal-05404182

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# Hydrogen bond symmetrization in high-pressure ice clathrates

Lorenzo Monacelli,¹ Maria Rescigno,¹,² Alasdair Nicholls,³ Umbertoluca Ranieri,⁵,⁴

Simone Di Cataldo,¹,⁶ and Livia Eleonora Bove¹,¹,³,⁷

¹Dipartimento di Fisica, Università di Roma Sapienza, Piazzale Aldo Moro 5, 00185 Roma, Italy

²Institut Laue-Langevin, 71 Avenue des Martyrs, Cedex 9, Grenoble, France

³Laboratory of Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

⁴Centre for Science at Extreme Conditions and School of Physics and Astronomy, University of Edinburgh,
EH9 3FD Edinburgh, Scotland, United Kingdom

⁵Centro de Física de Materiales (CFM-MPC), CSIC-UPV/EHU, Donostia/San Sebastián 20018, Spain

⁶Institut für Festkörperphysik, Technische Universität Wien, Wiedner Hauptstraße 8-10, Vienna 1050, Austria

⁷Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, Sorbonne Université, Centre National de la Recherche Scientifique
UMR 7590, MNHN, 4, place Jussieu, Paris, France

<figure><img src="image_2.png" /></figure>

(Received 17 July 2025; revised 24 October 2025; accepted 28 October 2025; published xxxxxxxxx)

Hydrogen bond symmetrization is a fundamental pressure-induced transformation in which the distinction between donor and acceptor sites vanishes, resulting in a symmetric hydrogen-bond network. While extensively studied in pure ice—most notably during the ice VII to ice X transition—this phenomenon remains less well characterized in hydrogen hydrates. In this work, we investigate hydrogen bond symmetrization in the high-pressure phases of hydrogen hydrates (H₂-H₂O and D₂-D₂O) through a combined approach of Raman spectroscopy and first-principles quantum atomistic simulations. We focus on the C2 and C3 filled-ice phases, using both hydrogenated and deuterated water frameworks. Our results reveal that quantum fluctuations and the interaction between the engaged H₂ molecules and the host lattice play a crucial role in driving the symmetrization process. Remarkably, we find that in both C2 and C3 phases hydrogen bond symmetrization occurs via a continuous crossover at significantly lower pressures than in pure ice, without any change in the overall crystal symmetry. These findings provide insight into the quantum-driven mechanisms of bond symmetrization in complex hydrogen-bonded systems under extreme conditions.

DOI: 10.1103/1cgl-mklx

## I. INTRODUCTION

Hydrogen bond symmetrization (HBS) is a fundamental phenomenon in hydrogen-rich materials, profoundly altering their structural, vibrational, elastic, and electronic properties under extreme conditions. It has been extensively studied in pure water ice [1–4], where it drives the transition from hydrogen-disordered ice VII to the symmetric hydrogen-bonded ice X phase [5]. Similar transitions have been identified in other hydrogen-bonded systems. In salt-filled ice structures, both experimental and computational studies have shown that ionic species can induce symmetrization of hydrogen bonds by compressing the water network and modifying the proton potential energy landscape [6,7]. Methane-filled ice phases also exhibit HBS, with Raman evidence indicating that symmetrization governs the transformation from the MH-III to the MH-IV phase [8,9]. In hydrous minerals such as δ-AlOOH, high-pressure neutron diffraction has revealed a universal onset of symmetrization at a critical O···O distance of 2.443(1) Å [10]. Despite these advances, HBS in hydrogen hydrates remains largely unexplored [11,12], even though these materials provide an ideal setting to study the interplay between hydrogen bonding and guest-host interactions under pressure.

Hydrogen hydrate is known to crystallize in five distinct phases: the low-temperature clathrate structure sII, and four

high-pressure filled-ice or framework phases labeled C0, C1, C2, and C3. The sII phase consists of a water framework encapsulating hydrogen molecules within large polyhedral cages [13–15]. C0, the first high-pressure phase, is a chiral gas-hydrate structure distinct from conventional ice polymorphs, and is stable between 0.35 and 0.7 GPa [16–18]. C1, stable from 0.7 to 2.7 GPa (and metastable up to 5.2 GPa) [19–21], adopts an ice II framework, while C2 and C3 are based on an expanded ice Ic lattice with 1:1 and 1:2 water-to-hydrogen ratios, respectively [11,13]. The C2 phase becomes stable above 2.7 GPa and transforms into C3 at 40 GPa upon heating [11]. C3 remains stable up to 90 GPa at least and is metastable down to 13 GPa upon decompression [22].

Among these, the C2 phase exhibits notable structural similarities to ice VII. Its water framework mirrors one of the two interpenetrating diamond sublattices of ice VII, while the second sublattice is occupied by H₂ molecules. The result is an expanded cubic ice lattice [23], offering a compelling model to explore HBS under the competing influences of quantum effects and guest-host interactions. The recently discovered C3 phase [11], built on the same water lattice but incorporating twice as many H₂ molecules, enables an investigation of symmetrization in an even denser hydrogen environment. These systems are not only of fundamental interest—shedding light on how symmetrization occurs in an expanded and dynamically perturbed ice lattice—but also have implications for
understanding high-pressure processes in hydrogen-rich planetary interiors, where such materials are likely to exist [22,24].

While symmetrization in ice VII has been observed experimentally [1–4] and studied through quantum simulations [5,25,26], discrepancies remain regarding the precise transition pressure, reflecting the challenges of locating hydrogen atoms under extreme conditions. In hydrogen hydrates, these challenges are compounded by the presence of the H₂ sublattice, which introduces additional interactions and dynamical effects. Quantum tunneling, already central in pure ice, may play an even more prominent role here due to the low mass and high mobility of the H₂ molecules.

In this work, we extend the study of hydrogen bond symmetrization to the C2 and C3 phases of hydrogen hydrates. Using quantum atomistic simulations and Raman spectroscopy, we identify the onset of symmetrization and characterize its structural and vibrational signatures. We find that in both phases symmetrization occurs through a smooth crossover, without modifying the crystal symmetry. In the C2 phase, in particular, HBS occurs at a much lower pressure than in pure ice. These results highlight the key role of quantum fluctuations and guest-host coupling in shaping HBS in complex hydrogen-bonded frameworks.

## II. RESULTS AND DISCUSSION

Hydrogen bond symmetrization in ice can be probed experimentally through Raman spectroscopy, where the OH stretching mode exhibits softening, broadening, and loss of intensity under increasing pressure. In ice VII, this mode becomes undistinguishable from background in the spectra before reaching full symmetrization as a result of severe broadening and intensity reduction. In pure ice, the appearance of a T²g mode, indicative of the highly symmetric cubic ice X lattice, serves as an alternative signature of the transition.

Hydrogenated (H₂O-H₂), partially deuterated (D₂O-H₂), and fully deuterated (D₂O-D₂) hydrogen hydrate samples were prepared in separate diamond anvil cells (DACs) for the Raman spectroscopy measurements of this study. They were prepared by either cryoloadi ng sII hydrogen hydrate or from the reaction between H₂O and MgH₂ in a DAC (see the Supplemental Material [27] for further details).

Raman spectra for hydrogen-filled ice C2 at selected pressures are shown in Fig. 1, for both D₂O-D₂ (top and bottom right panels) and H₂O-H₂ (bottom left panel). The hydrate stretching mode, marked by a red tick, undergoes a progressive redshift and broadening with pressure, reflecting the weakening of the donor-acceptor distinction in the hydrogen bond. A shaded region indicates the spectral range in which diamond anvil Raman activity obscures the sample signal. A higher frequency peak corresponds to the OD/OH stretching mode of ice VII, formed during the C1 to C2 phase transition [11]. The isotopic comparison reveals distinct pressure-dependent shifts and dispersions for D₂O and H₂O, enabling a direct evaluation of the effects of quantum mass.

Compared to ice VII, the OH-stretching mode in the C2 phase softens more strongly, suggesting that hydrogen bond symmetrization occurs at lower pressures. However, signal attenuation, peak broadening, and overlap with the second-order diamond band above 10 GPa limit its spectroscopic

<figure><img src="image_3.png" /><figcaption>FIG. 1. Raman spectra measured in the D₂O-D₂ (top and bottom right panel) and H₂O-H₂ (bottom left panel) filled ice C2 structure at different pressures. The red tick indicates the position of the hydrate stretching mode. The shaded area indicates the region where the diamond Raman activity covers the sample signal.</figcaption></figure>

visibility. No clear secondary Raman features emerge to indicate symmetrization directly. To complement these limitations, we performed first-principles simulations.

We used the stochastic self-consistent harmonic approximation (SSCHA) [28–31] to account for quantum fluctuations and the anharmonicity of the light atoms involved. SSCHA approximates the quantum ionic wave function as a Gaussian that is optimized to minimize the free energy. The method has been successfully applied to describe hydrogen bond symmetrization in pure ice [5], as it enables the modeling of temperature and quantum ionic fluctuations, which are crucial for capturing the correct physics of hydrogen-rich materials. The interatomic forces were modeled with an equivariant neural network potential based on NEQUIP [32], trained on density functional theory (DFT) data from previous studies [11,12]. The simulations were performed using a proton-ordered model of the ice sublattice, which contains two symmetry-inequivalent H₂O molecules in the primitive cell, as in ice VIII and XI.

The computed anharmonic Raman spectra for H₂O and D₂O C2 are shown in Fig. 2. A qualitative change in the spectral response is observed between 24 and 26 GPa, corresponding to the disappearance of all hydrate-related stretching modes.
<figure><img src="image_4.png" /><figcaption>FIG. 2. Anharmonic Raman spectra computed within the SCHA in the H₂O-H₂ (left panel) and D₂O-H₂ (right panel) filled ice C2 structure at different pressures. The shaded area indicates the region where the diamond Raman activity covers the sample signal. We checked selected pressures around the symmetrization that substituting H₂ molecules with D₂ does not significantly alter the ice lattice modes (including stretching).</figcaption></figure>

The simulations reveal three main Raman-active modes before symmetrization between 1000 and 3000 cm⁻¹: two from mixed symmetric and antisymmetric stretching, and a third bending mode near 1500 cm⁻¹ that becomes enhanced via mode coupling around 22 GPa. After symmetrization, all three disappear, consistent with the experiment. The H₂ molecule remains spectroscopically silent in this range, as it does not participate in lattice vibrations relevant to the transition, progressively hardening above 4000 cm⁻¹ upon compression. The small peak slightly below 1000 cm⁻¹ represents the H₂ rotations, which, however, are not accurately described within the SSCHA framework [33,34] and require a different treatment to be accounted for [12].

These spectra are derived from SSCHA auxiliary phonons, which account for anharmonic frequency renormalization but not finite phonon lifetimes [35–37], thereby explaining the absence of the pronounced broadening observed in experiments. However, this method enables accurate mode tracking with pressure.

Figure 3 compares the simulated and experimental stretching frequencies for H₂O and D₂O. A rigid frequency shift of 270 cm⁻¹ corrects known DFT underestimations of the OH stretching [38]. Experimental data show a redshift consistent with bond strengthening and eventual mode disappearance above 12 GPa for H₂O and 17 GPa for D₂O. Despite these limitations, the pressure dependence is well reproduced, validating the simulations.

Despite the spectral changes near 24 GPa, no analog of the ice X T²g mode is observed. This absence can be attributed to structural differences: unlike ice VII/VIII, which consists of two interpenetrating hydrogen-bonded networks, the C2 structure replaces one with H₂, reducing volume and stabilizing symmetrization at lower pressures. The decreasing Raman intensity and broadening prevent tracking of the stretching mode down to 0 cm⁻¹, a hallmark of the phase transition. For this reason, we computed the average ionic positions from the quantum ionic wave function and evaluated the bond lengths. As shown in Fig. 4, the OH covalent bond sharply

changes between 22 and 25 GPa, coinciding with the loss of Raman activity in the simulations. Such observations place the hydrogen bond symmetrization at lower pressure compared to previous studies [39–41]. In contrast to ice VIII-X [5], symmetrization in this system occurs gradually.

The isotope effect is evident: replacing H with D shifts the crossover by approximately 5 GPa, as shown in Fig. 4. This pressure shift and the suppression of symmetrization in

<figure><img src="image_5.png" /><figcaption>FIG. 3. Comparison of experimental (blue) and calculated (orange) frequencies of the OH (diamonds) and OD (dots) stretching modes as a function of pressure. The shaded area indicates the region where the diamond Raman activity covers the sample signal. Symbol sizes of simulation data are proportional to intensities and only the higher intensity peak is reported. In Fig. S3 of the Supplemental Material [27], we show the intensity behavior of the experimental data. A global rigid shift in frequencies of 270 cm⁻¹ is applied to simulation data to correct the known mismatch of semilocal DFT functionals on the OH stretching frequency of ice [38].</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>FIG. 4. OH bonds as a function of pressure. Comparison of static (no quantum or thermal fluctuations), deuterium ($D_2O$), and protium ($H_2O$) symmetrization of the OH bond. Inequivalent molecules in the proton-ordered model we employed have been averaged.</figcaption></figure>

static calculations reflect strong quantum nuclear effects in hydrogen bonding, a hallmark of ice physics.

To shed light on the crossover character of the hydrogen bond symmetrization in hydrates, we performed a symmetry analysis removing the $H_2$ guest molecules. The slight orthorhombic distortion of the lattice creates two inequivalent triplets of alternating covalent-hydrogen bonds OH···O within the ice sublattice. Unlike ice VII-X [5], the symmetrization of these two independent sets of bonds does not co-occur, as shown by their respective order parameters $\Delta_1$ and $\Delta_2$, defined as the bond length alternation between subsequent covalent and hydrogen bonds across different molecules ($\Delta = d_{OH} - d_{H...O}$, Fig. 5). A sharp drop of $\Delta_1$ occurs between 22 and 24 GPa, coinciding with the loss of the Raman signal. In contrast, $\Delta_2$ symmetrizes more smoothly, without evidence of a sudden transition, indicating that the overall

<figure><img src="image_7.png" /><figcaption>FIG. 5. Symmetrization of the OH···O bond alternation of the two symmetry inequivalent sets of bonds in the ice host lattice as a function of pressure.</figcaption></figure>

hydrogen bond symmetrization of the host lattice is gradual with pressure. This behavior contrasts with the second-order transition found in pure ice near 55 GPa between phases VIII and X [5], highlighting the interactions between the water network and the host molecular hydrogen.

We compare the symmetrization of hydrogen in the C2 phase with that observed in pure ice and the high-pressure C3 structure. In the C3 phase, the host $H_2O$ lattice is structurally equivalent to that of C2, but the molar ratio between $H_2$ and $H_2O$ is 2:1—the highest ever reported [11]. Despite the structural similarity of the ice cages in C2 and C3, the stretching mode in C3 is strongly mixed with low-frequency molecular translational modes. This results in negligible Raman activity for the mode associated with symmetrization, making the detection of hydrogen bond symmetrization via vibrational spectroscopy experimentally unfeasible.

Symmetrization in C3 occurs at significantly higher pressures than in C2 due to the larger volume of the ice sublattice, caused by the greater number of hydrogen molecules confined within the ice channels, as shown in Fig. 6(a). Also in this case, quantum fluctuations play a crucial role, lowering the transition pressure from 110 to 70 GPa. As in C2, the symmetrization proceeds through a smooth crossover rather than a sharp phase transition.

To analyze the environmental influence on hydrogen bond symmetrization, we plot the OH bond length as a function of the average O···O distance [Fig. 6(b)]. In both C2 and pure ice [5], symmetrization occurs at an O···O distance of approximately 2.41 Å. Based on their experimental equation of state (EoS), Machida *et al.* [41] suggested that such critical O···O distance is reached at about 40 GPa in the C2 phase. More recent studies [42], however, report a revised EoS in which the critical distance is reached at a pressure consistent with our simulations. In contrast, C3 symmetrizes at a shorter O···O distance of 2.39 Å, despite sharing the same ice sublattice. This deviation underscores the impact of the local environment: the denser population of $H_2$ molecules in C3 reduces out-of-plane hydrogen fluctuations within the ice lattice, thereby requiring a shorter O···O distance to achieve symmetrization.

A universal picture of hydrogen bond symmetrization emerges when we examine the OH bond as a function of O···O distance in pure ice, C2, and the C3 phase. While C2 and pure ice symmetrize at the same critical geometry, C3 requires a shorter O···O distance by 0.01 Å, despite sharing the same water framework [Fig. 6(c)]. This deviation highlights the importance of the local environment: in C3, the dense hydrogen guest network suppresses the transverse quantum fluctuations of hydrogen atoms, as shown in Fig. 6(c), thereby delaying symmetrization and altering its spectroscopic fingerprint. In particular, the OH stretching is almost completely suppressed at such high pressures in the C3, preventing the usual spectroscopic identification of the hydrogen bond symmetrization.

### III. CONCLUSIONS

We have investigated hydrogen bond symmetrization in the hydrogen-filled ice C2 phase and compared it with both pure ice and the (higher $H_2$ density) C3 phase. Despite
<figure><img src="image_8.png" /><figcaption>FIG. 6. (a) OH bond symmetrization in the ice sublattice in the C3 phase. Comparison between the simulation accounting for nuclear quantum fluctuations (dashed light blue line) with the static result neglecting quantum ionic fluctuations (solid navy line). (b) Comparison of the OH bond symmetrization vs O-O distance in different systems: pure ice VII [5], C2, and C3. While the C2 and pure ICE symmetrize at the same O-O distance, the C3 requires a smaller O-O distance to symmetrize, shedding light on the role played by the environment in hydrogen bond symmetrization. (c) Quantum average displacement of the H-bonded atom perpendicular to the oxygen-oxygen distance vector. The C3 perpendicular displacement of H atoms is inhibited by the presence of dense H₂ molecules for the same size of the H₂O cage compared to the C2 phase.</figcaption></figure>

the structural similarity of the water frameworks in C2 and C3, the spectroscopic signatures of symmetrization differ substantially.

In the C2 phase, Raman spectroscopy, supported by quantum anharmonic simulations, reveals a clear softening and eventual disappearance of the OH stretching mode, consistent with a continuous symmetrization process occurring around 24 GPa. The transition is gradual and strongly influenced by quantum nuclear effects, as shown by the isotope-dependent shift and the inability to induce symmetrization in static calculations. This behavior contrasts with the well-defined second-order transition observed in pure ice near 55 GPa, but occurs at the same critical O···O distance of approximately 2.41 Å.

In the C3 phase, however, vibrational spectroscopy fails to detect symmetrization due to strong mixing of the OH stretching mode with low-frequency translational modes. This suppresses Raman activity, making the transition spectroscopically silent. Nonetheless, simulations confirm that symmetrization does occur—at significantly higher pressures than in C2—due to the larger volume of the ice lattice caused by the higher hydrogen loading (2:1 H₂:H₂O). Quantum effects again reduce the transition pressure dramatically, from 110 to 70 GPa, but even here the process remains a smooth crossover rather than a sharp phase change.

These findings underscore the intricate interplay between lattice geometry, guest molecule density, and quantum nuclear effects in controlling hydrogen bond symmetrization. Our results demonstrate that structural similarity alone does

not determine the symmetrization mechanism: the host-guest interaction and quantum dynamics must be considered to fully understand and predict the behavior of hydrogen-bonded systems under pressure.

## ACKNOWLEDGMENTS

L.M. acknowledges the CINECA award under the ISCRA initiative for the availability of high-performance computing resources and support. L.E.B. acknowledges financial support by European Union NextGenerationEU (PRIN Project No. 2022NRBLPT), ANR Grant No. ANR-23-CE30-0034 EXOTIC-ICE, and Swiss National Fund Grant No. 212889. We thank Werner F. Kuhs, Andrzej Falenty, and Dirk Wallacher for help during sample preparation. We also thank Lorenzo Ulivi, Leonardo del Rosso, and Milva Celli for providing some of the samples used in this study. We thank Richard Gaal for assistance in sample loading.

## DATA AVAILABILITY

The data that support the findings of this article are not publicly available. The data are available from the authors upon reasonable request.

## APPENDIX: NUMERICAL METHODS

To compute the thermodynamic properties of the hydrate phases, we solved the nuclear quantum Hamiltonian at finite
temperature in the Born-Oppenheimer approximation within the SSCHA. The SSCHA performs a Gaussian ansatz on the ionic density matrix $\hat{\rho}$, which depends on the average nuclear position $\mathcal{R}$ and the covariance matrix of quantum/thermal fluctuations. This covariance matrix is defined through an auxiliary force constant matrix $\mathbf{\Phi}$ which, at convergence, satisfies the self-consistent relation

$$ \Phi_{ab} = \left\langle \frac{d^2 V}{dR_a dR_b} \right\rangle_{\hat{\rho}(\mathcal{R}, \mathbf{\Phi})}, \quad (A1) $$

where $\langle \circ \rangle_{\hat{\rho}(\mathcal{R}, \mathbf{\Phi})}$ indicates the average on an ensemble extracted according to the Gaussian density matrix and $V(\mathcal{R})$ is the Born-Oppenheimer energy landscape with the nuclei fixed in positions $\mathcal{R}$. We can diagonalize the auxiliary dynamical matrix and extract eigenvalues and eigenvectors of phonons as

$$ \sum_b \frac{\Phi_{ab} e_{\mu}^b}{\sqrt{m_a m_b}} = \omega_{\mu}^2 e_{\mu}^a \quad (A2) $$

where $\omega_{\mu}$ are the anharmonic dressed phonon frequencies and $e_{\mu}$ is the associated polarization vector. The Raman intensity of phonons is evaluated by contracting the respective polarization vector with the Raman tensor $\Xi$:

$$ \Xi_{abc} = \frac{d^3 V}{dE_a dE_b dR_c}, \\ I_{\mu} = \frac{1 + n_{\mu}}{\omega_{\mu}} \left| \sum_{abc} \Xi_{abc} e_{\mu}^c \epsilon_a \epsilon_b \right|^2 \quad (A3) $$

where $\epsilon$ is the polarization of the light. More details are reported in Refs. [28,29]. The real anharmonic frequencies should be evaluated from the time-dependent response

function [36]; however, the anharmonic dressed frequencies $\omega_{\mu}$ already account for anharmonicity and are a good approximation far from phase transitions, even in strongly anharmonic crystals [43]. To obtain the average structure, we performed a variable cell relaxation at fixed pressure $P$ by minimizing the Gibbs free energy:

$$ G(\mathbf{\Phi}, \mathcal{R}, \{a_i\}) = \langle \hat{H} \rangle_{\hat{\rho}(\mathcal{R}, \mathbf{\Phi})} - TS[\mathbf{\Phi}] + P\Omega(\{a_i\}) $$

where $\hat{H}$ is the Born-Oppenheimer Hamiltonian, $S$ is the entropy (which only depends on $\mathbf{\Phi}$), $\Omega$ is the volume, and $\{a_i\}$ are the lattice parameters. Further details on the algorithm to relax the primitive cell are discussed in Ref. [44].

The Born-Oppenheimer energy landscape $V(\mathcal{R})$ is approximated via an equivariant machine-learning force field based on NEQUIP [32]. The model is trained on a set of 800 first-principles calculations on a 2 × 2 × 1 supercell of the C2 and C3 phase randomly selected from equilibrium SSCHA relaxation performed within density functional theory, as performed in a previous work [11] (see there for all the details on the DFT calculation). The data are sampled from pressures uniformly distributed through 5 and 40 GPa of the C2 phase and from 40 and 80 GPa of the C3 phase. The dataset is divided into 650 configurations for training and 150 for testing. The root mean squared error on the energy in the test set is 1.3 meV per atom, 90.9 meV/Å for oxygen forces, and 58.7 meV/Å for hydrogen forces. We benchmark the final SSCHA result obtained with the trained potential with the one obtained using DFT at 20 and 30 GPa, finding that all the relevant observables (OH bond length, equation of state, and anharmonic vibrational frequencies) are well converged and reproduce consistently the DFT results. We then proceed to run SSCHA calculations every 2 GPa in a 3 × 3 × 2 supercell.

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