Dual quantum locking: Dynamic coupling of H₂ and H₂O sublattices in hydrogen-filled ice

Significance

Hydrogen-rich compounds under extreme pressure are central to understanding planetary interiors and developing quantum materials. In hydrogen-filled ices, hydrogen molecules are trapped in a dense ice framework, creating a unique environment where nuclear-quantum effects influence both trapped molecules and the host lattice. We reveal how pressure and temperature drive structural and orientational transitions at remarkably low pressures compared to pure hydrogen, due to strong host–guest interactions. These findings refine the high-pressure phase diagram of hydrogen hydrates, identify the appearance of a nematic phase, clarify its link with hydrogen-bond symmetrization, and highlight their role as model systems for studying quantum phenomena in confined molecular hydrogen. Our results have implications for planetary science, hydrogen storage, and quantum material design.

Abstract

Hydrogen hydrates (HH) are a unique class of materials composed of hydrogen molecules confined within crystalline water frameworks. Among their multiple phases, the filled ice structures, particularly the cubic C2 phase, exhibit exceptionally strong host–guest interactions due to ultra-short H₂-H₂O distances and a 1:1 stoichiometry leading to two interpenetrated identical diamond-like sublattices, one comprised of water molecules, the other of hydrogen molecules. At high pressures, nuclear-quantum effects involving both hydrogen molecules and the water lattice become dominant, giving rise to a dual-lattice quantum system. In this work, we explore the sequence of pressure- and temperature-driven phase transitions in HH, focusing on the interplay between molecular rotation, orientational ordering, lattice symmetry breaking, and hydrogen bond symmetrization. Using a combination of computational modeling based on classical and path-integral molecular dynamics, quantum embedding, and high pressure experiments, including Raman spectroscopy and synchrotron X-ray diffraction at low temperatures and high pressures, we identify signatures of quantum-induced ordering and structural transformations in the C2 phase. Our findings reveal that orientational ordering in HH occurs at much lower pressures than in solid hydrogen, by inducing structural changes in the water network and enhancing the coupling of water and hydrogen dynamics. This work provides insights into the quantum behavior of hydrogen under extreme mechanochemical confinement and establishes hydrogen-filled ices as a promising platform for the design of hydrogen-rich quantum materials.

The rotational dynamics of guest gas molecules confined in porous materials are profoundly influenced by their local environment (1). In large, nearly spherical cages, guest species often behave as nearly free rotors (2–4). In contrast, low-symmetry or anisotropic environments impose steric constraints that hinder rotation, leading to librational motions or even complete orientational freezing (5–7).

Such effects have been extensively studied in clathrate hydrates, where the host–guest potential not only quantizes the center-of-mass motion but can also induce partial orientational ordering at low temperatures and high pressures (5, 8–10). Zeolite frameworks, with their inherent anisotropy and stronger host–guest interactions, exhibit similar behavior: Small guest molecules often display restricted rotational freedom and develop orientational preferences within the channels or cages (11–15). Endohedral fullerenes offer a particularly clean example of quantum confinement: Molecules such as H₂ and CH₄ confined inside C₆₀ exhibit discrete quantum rotational states that are exquisitely sensitive to cage symmetry and weak van der Waals interactions (6, 16, 17). These systems provide compelling evidence of how confinement and symmetry shape quantum rotational behavior across a wide range of host architectures.

In gas-filled ice structures under high pressure (18), the extreme confinement imposed by the ice hosting lattice leads to the suppression of molecular rotations even at high temperatures, enabling the onset of orientational ordering of guest molecules (19–21). In certain cases, this ordering is accompanied by molecular distortion (20), and can even trigger symmetry-lowering transitions in the host lattice itself (22, 23). These phenomena underscore the unique interplay between quantum confinement and structural flexibility in filled-ice systems—an interplay rarely observed in classical porous materials.

Among these, hydrogen hydrates (HH)–where H₂ molecules are encapsulated within an extended H₂O lattice-stand out as a particularly rich system for exploring host–guest interactions under extreme thermodynamic conditions. Five well-characterized phases comprise the hydrogen hydrates. The first phase has the sII clathrate structure in which H₂ molecules are trapped in large polyhedral cages within a hydrogen-bonded H₂O network (3, 24). The other four-C0, C1, C2, and C3-are filled ices in which H₂ molecules occupy interstitial voids of the structures of various ice phases. Vos et al. (25) discovered and named both C1, an H₂-filled phase of ice II, and C2, a denser phase in which one of the two interpenetrating ice Ic sublattices of ice VII is replaced by H₂ molecules. Later to be identified were C0 (26), a phase in which H₂ occupies the open channels of ice XVII, and C3 (27), which is a higher pressure phase derived from cubic C2.

While sII is only stable at cryogenic temperatures, both C1 and C2 persist at room temperature under high pressure (28). C1 is stable between 0.9 and 2.7 GPa (and metastable up to 5.2 GPa), whereas C2 stabilizes above 2.7 GPa and transforms into C3 at temperatures above 670 K and pressures above 30 GPa. C3 remains stable up to at least 90 GPa and can be recovered metastably down to 7 GPa (23).

Among these phases, C2 is particularly intriguing. It adopts a topology analogous to that of ice VII, consisting of two interpenetrating but noninterconnected diamond-like sublattices. However, unlike ice VII–where both sublattices are composed of water molecules–in C2, one sublattice is entirely formed by hydrogen molecules. This breaks the structural equivalence, lowers the overall symmetry, and results in a unique dual-framework architecture. Remarkably, C2 exhibits the shortest known H₂-H₂O distances among hydrogen hydrates, and its 1:1 stoichiometry gives rise to exceptionally strong host–guest interactions.

Under such extreme densities confinement effects predominate, affecting not only the hydrogen molecules within their sublattice but also the protons constituting the ice framework. As pressure increases, the distinction between host and guest progressively diminishes, and the system is more appropriately described as two equivalent, interpenetrating quantum lattices of distinct species: H₂ and H₂O.

The rotational dynamics of hydrogen in hydrogen-filled ices are strongly pressure- and temperature-dependent. Increasing pressure and lowering temperature progressively restrict guest rotation, inducing stepwise orientational ordering (29), analogous to effects seen in solid methane (30, 31) and solid hydrogen (32–34), where intermolecular interactions produce roton band splittings. Similar but stronger effects occur in hydrogen clathrates (10).

In the C2 phase, first-principles calculations predicted a pressure-induced cubic-to-tetragonal distortion at low temperature via hydrogen alignment along the c-axis (35), confirmed in D₂O-D₂ by XRD peak splitting and roton band splitting above 20 GPa at 10 K (36). Additional high-pressure phases (HH-HP1 at 40 GPa and HH-HP2 at 60 GPa) involve further orientational ordering and possible hydrogen-bond symmetrization (36–38).

Our recent work refined this phase diagram: We observed tetragonal distortion of C2 above 30 GPa at room temperature, amorphization above 60 GPa, and transformation to C3 upon laser heating (23, 27). We also determined that hydrogen-bond symmetrization occurs at 26 GPa-much lower than in ice VII- consistent with early predictions (39) and due to shorter O–O distances in the cubic ice Ic framework (40).

In the present work, we go beyond these macroscopic observations to elucidate their microscopic origin: We demonstrate that hydrogen-bond symmetrization in the water lattice triggers the nematic alignment of the H₂ sublattice, and that this collective orientational ordering, in turn, drives the tetragonal distortion of the host framework. By disentangling this sequence of coupled events, we establish the microscopic mechanism linking proton symmetrization, guest orientational order, and host lattice deformation in C2. We probe these phase transitions via combined ab initio simulations, low-temperature Raman spectroscopy, and synchrotron XRD (41–45), highlighting how enhanced host–guest coupling under pressure suppresses rotational freedom and stabilizes orientational order at much lower pressures than in pure hydrogen.

Results and Discussion

Fig. 1 presents the pressure–temperature phase diagram constructed from our combined computational and experimental approaches, with the thermodynamic paths probed experimentally summarized in Table. 1. Within the C2 phase, H₂ exhibits three distinct regimes, dictated by temperature and the degree of coupling with the surrounding water lattice.

Fig. 1.

Pressure–temperature phase diagram of the C2 hydrogen hydrate as derived by our quantum embedded calculations and experimental data. Some experimental patterns are indicated in Figure: XRD data are shown as circles, colored red or black depending on whether they belong to the temperature- or pressure-ramp series; Raman measurements are indicated by squares. The summary of experimental thermodynamic path is reported in Table. 1. Symbols are filled in the cubic phase and open in the tetragonal phase, while half-filled symbols indicate the transition between them. The background color represents S, the orientation factor (defined in Materials and Methods) of H₂ computed in the quantum embedded framework. The white triangles with their associated error bars are derived from our molecular dynamics simulations, utilizing the orientation factor as detailed in Materials and Methods. The dashed lines serve as a guide to the eye to locate the cubic to tetragonal transition. The three Insets display the optimized geometries obtained via DFT (0 K) for the quantum plastic phase (Upper Left), herringbone phase (Lower Left), and nematic phase (Right) of H₂ orientational states.

Table 1.

Summary of experimental thermodynamic paths for X-Ray Diffraction (XRD) and Raman

ECF4FF	XRD	Raman
Isobar	2.45(3) GPa	3.5 GPa
	300 → 30 K	300 → 100 K
		14.7 GPa
		300 → 5 K
		35 GPa
		300 → 20 K
Isotherm	300 K	300 K
	2 → 46 GPa	10 → 40 GPa

At high temperatures and pressures below 30 GPa (above the dashed line), the H₂ sublattice in H₂ hydrate realizes a quantum plastic crystal. In this state, the H₂ molecules are ordered on a diamond-type lattice, defining well-localized translational equilibrium sites, yet their orientations remain dynamically disordered (46). Each H₂ behaves as a nearly free quantum rotor, i.e., the rotational degrees of freedom are not frozen but remain active down to relatively low temperatures due to quantum fluctuations. This combination of translational order and orientational quantum mobility is the defining feature of a quantum plastic phase. In contrast, the H₂O sublattice is proton-disordered but not plastic, since the orientations of H₂O molecules are constrained by the ice rules and do not undergo free rotational dynamics. The overall symmetry of the crystal is cubic, consistent with previous DFT predictions and experimental observations (23, 27, 35).

Upon crossing the dashed boundary, either through temperature reduction or compression, H₂ molecules undergo orientational ordering. At high pressure this results in a phase where H₂ are aligned along the c axis of the crystal. This ordering transition can be quantified by the orientation factor S (Materials and Methods) computed based on our quantum embedded calculations as described in the following and visualized as a color gradient in Fig. 1. The factor S, which ranges from 0 (complete orientational disorder) to 1 (perfect alignment), measures the collective alignment of the hydrogen molecules along the c axis of the crystal, capturing both the orientational order imposed by the lattice geometry and the intrinsic thermal-quantum fluctuations encoded in the density matrix.

Within the orientationally ordered phase-occurring below 30 GPa and 100 K, H₂ molecules adopt a herringbone-like arrangement, as shown in the Inset of Fig. 1. This configuration favors alignment along the crystallographic c axis. The tilt angle of the molecules relative to the c axis varies continuously with pressure, reflecting the energetically favorable response to lattice compression at low temperatures. As pressure increases, the tilt angle gradually decreases, approaching full alignment along the c axis, i.e., the nematic phase. This progressive reorientation is also captured by the pressure dependence of the orientation factor S in Fig. 1.

The low-temperature, high-pressure phases (<50 GPa) of pure H₂ and D₂ are unique molecular systems in which free rotations are governed by both quantum effects and temperature (47). Molecular hydrogen occurs in two nuclear-spin isomers, both partially occupied at ambient conditions. When the proton spins couple to the singlet state with total nuclear-spin quantum number $\mathcal{J}=0$, defined by $J=J_1+J_2$, where ${\mathcal{J}}_1={\mathcal{J}}_2=\frac{1}{2}$ are the proton resolved nuclear spins, the resulting para-hydrogen possesses an antisymmetric spin wavefunction and, to satisfy overall fermionic antisymmetry, its spatial part is restricted to even rotational quantum numbers l. Coupling the same proton spins to the triplet state ($\mathcal{J}=1$) produces ortho-hydrogen, whose symmetric spin wavefunction permits only odd l values.

At low pressures, Raman spectra of solid hydrogen and clathrates (4, 10, 27, 48, 49) display free-rotor peaks with energies $E_l=\frac{{ħ}^2}{2I}l(l+1)$ where I is the moment of inertia of the rotor and l the angular momentum, with selection rule $\mathrm{\Delta }l=\pm 2$. In the free molecule, rotational levels are $(2l+1)$-fold degenerate, and the main gas-phase Raman bands are $S_0(0)$ ($l=0\to 2$) (354 cm⁻¹) and $S_0(1)$ ($l=1\to 3$) (587 cm⁻¹) (48, 50). In the clathrate phase and in low-pressure filled ice C0 and C1 phases, the rotor bands remain close to the free-rotor values and their degeneracy is preserved. In the high-pressure C2 phase, however, the anisotropic crystal field lifts this degeneracy and shifts the main ortho–para transition band by 4 meV-about 30% of its value (51). With increasing pressure, these bands broaden, making individual rotational levels increasingly difficult to resolve.

Fig. 2 A and B shows the evolution of the Raman intensity distribution among the $S_x(0)$ transitions in a gas-loaded hydrogen hydrate sample (Materials and Methods) as a function of temperature and pressure. XRD confirmed phase-pure C2 after gas loading, whereas cryoloaded samples contained 83 mol% H₂O due to decomposition of the sII substrate (6:1 H₂O:H₂ stoichiometry).

Fig. 2.

Raman spectra as a function of temperature at P = 14.7(1) GPa (A) and as a function of pressure at T = 300 K (B). Representative X-ray diffraction data as a function of temperature at P = 2.45(3) GPa (Left panel, C–E) and as a function of pressure at T = 300 K (Right panel, F–H). The extracted evolution of the lattice parameters(in the tetragonal setting) and the degree of asymmetrization upon cooling (I and J) and compression (K and L), are plotted in the Bottom panels, respectively.

At low temperatures, the $S_0(2)$ and $S_0(3)$ bands weaken and vanish below 150 K due to thermal depopulation of higher rotational levels; in pure hydrogen, these disappear at 120 K (3 GPa) (52). In the compressed hydrate, the Raman evolution is more complex: A sudden broadening and asymmetry around 30 GPa suggest a change in the rotational state of encapsulated H₂ (48, 53), likely driven by modifications to the crystalline field from the surrounding water framework during its transition to a less compressible phase (23). The instability of C2 above 20 GPa toward decomposition into ice VII and molecular hydrogen (38) may also contribute to inhomogeneous Raman signals from free H₂. The intensity oscillations reported near 20 GPa (38) were not observed here; spectra were consistently collected from the same spot after each pressure step, with a 30-min equilibration, ensuring that the observed $S_0(1)$/$S_0(0)$ intensity ratio upon cooling (14.7 GPa) and compression (300 K) reflects ortho–para ratio, which is almost constant and close to the $3:1$ expected ratio at ambient temperature, which is metastably kept at low temperature, as discussed below. Both high pressure and low temperature are known to facilitate this process in solid hydrogen (48, 53).

XRD measurements reveal that either cooling below 150 K at 2.45 GPa or compressing above 27 GPa at RT induces peak splitting (Fig. 2C–H), signaling a cubic ($Fd\overline{3}m$) to tetragonal ($I{4}_1/amd$) transition. Although the splitting pattern is similar in both cases, the observed tetragonal distortion is much more pronounced when compressing at room temperature than during isobaric cooling, likely indicating a different driving mechanism, as discussed below. The pressure-driven transition begins abruptly at around 30 GPa, coinciding with the onset of Raman anomalies-evidence of coupling between H₂ rotational dynamics and framework distortion. In both scenarios (cooling and pressurization), the c axis stiffens while the orthogonal directions contract (Fig. 2 I and K). The distortion, quantified as $c/\sqrt{2}a$, increases smoothly, reaching 0.5% at 30 K (2.45 GPa) and 4.6% at 48 GPa (300 K) (Fig. 2 J and L).

To probe the molecular-scale mechanism linking proton alignment to the lattice distortions seen in XRD, we performed simulations of a crystal-embedded H₂ molecule by solving the Schrödinger equation with an effective potential corresponding to the crystal field generated by the water framework and surrounding H₂ molecules. Fig. 3 A and B shows the evolution of H₂ density isosurfaces with temperature and pressure, along with the angular component $P_{r_{\mathit{eq}}}(\theta ,\varphi )$ of the external potential $V_{\mathit{ext}}(r,\theta ,\varphi )=R(r)+P_r(\theta ,\varphi )$ at the equilibrium H₂ bond length $r=r_{\mathit{eq}}$. θ and ϕ are respectively the polar and azimuthal angles of the molecular orientation in a cubic lattice reference. As described in Materials and Methods and SI Appendix, densities at a given T are obtained from the para and ortho thermal density matrices mixed together with a fixed 1:3 para-ortho ratio, justified by the extremely weak coupling between the nuclear spin and spatial variables, which kinetically hinders the ortho-to-para conversion, as verified experimentally via Raman spectroscopy throughout the investigated temperature range.

Fig. 3.

Isosurfaces of the density $\rho (\overrightarrow{r})$ and angular potential $P_{r_{\mathit{eq}}}(\theta ,\varphi )$ evaluated at the equilibrium H₂ bond length, shown as a function of temperature at $P=3$ GPa (A), and as a function of pressure at $T=300$ K (B). The density is projected onto the three planes (xy, yz, and xz) and normalized with respect to the maximum density at 0 K, 3 GPa for panel (A), and at 300 K, 50 GPa for panel (B). Pressure dependence of the first nine eigenvalues at $T=0$ K. Ortho and para levels are shown as circles and squares, respectively (C). Temperature dependence of the populations for the first five eigenvalues at P = 50 GPa, weighted by nuclear spin statistics (ortho:para = 3:1). Note that ${\lambda }_2$ and ${\lambda }_3$ eigenvalues are degenerate at this pressure. (D). Isosurfaces of the wavefunctions squared modulus for the ground state (para, $l=0$) and the first three excited states (ortho, $l=1$) of the H₂ molecule at P = 50 GPa (E).

At low pressure ($P=3$ GPa, Fig. 3A), the external angular potential exhibits minima that force the molecules into a herringbone-like arrangement. As T increases within the C2 stability range, para- and ortho-wavefunctions mix and the density approaches spherical symmetry. With increasing pressure, intermolecular interactions intensify and potential barriers rise, while the minima shift toward $\theta =0 [\pi ]$, ultimately favoring full nematicity. At room temperature (Fig. 3B), the density evolves from nearly spherical shells at 10 GPa-characteristic of the quantum-plastic phase-to well-defined lobes aligned with the c-axis at 50 GPa.

The eigenvalue evolution shown in Fig. 3C demonstrates how the anisotropic crystal field in the C2 phase lifts the free-rotor degeneracy and induces a pressure-dependent giant splitting, in agreement with earlier findings (51). This quantum mechanical picture explains the emergence of nematic ordering at high pressures and its persistence to elevated temperatures. At low pressure, within the C2 stability range, the small energy gap between the fundamental para state ($l=0$) and the three ortho states ($l=1,{\lambda }_1,{\lambda }_2,{\lambda }_3$) enables thermal mixing above ∼100 K, yielding nearly spherical densities. Increasing pressure drives a sharp rise in the splitting between ${\lambda }_1$ and ${\lambda }_{2,3}$, bringing the ${\lambda }_1$ level close to $l=0$. This restricts significant mixing to $l=0$ and $l=1,{\lambda }_1$, both oriented along the c axis, producing a nematic density distribution. The fixed 1:3 para-ortho ratio further enhances the statistical weight of ${\lambda }_1$, stabilizing orientational order up to ∼500 K (Fig. 3 D and E).

From these computed eigenvalues, we can determine the Stokes rotational transitions $S_0(0)$ and $S_0(1)$, plotted in Fig. 4A as orange and blue curves, respectively. The comparison with the experimental Raman data in the same panel shows good agreement, particularly in the pressure evolution of the main features, thus validating the simulation framework. The change in slope of both $S_0(0)$ and $S_0(1)$ near 30 GPa, which is more pronounced in the simulations due to the PBE overestimating the distortion and from the absence of lattice fluctuations in the static PES, coincides with the cubic-tetragonal distortion observed in XRD and with the cubic-tetragonal distortion observed in XRD and with the calculated onset of dominant ${\lambda }_1-l=0$ mixing (Fig. 3 C and D), supporting the interpretation that the structural transition is driven by a pressure-induced orientational ordering of the guest molecules.

Fig. 4.

(A) Comparison of the experimental Raman points and all simulated S₀(0) and S₀(1) rotons. The blue and orange lines are theoretical while the red and green point indicate the S₀(0) and S₀(1) rotons respectively. The rotons were modeled with three components each, as indicated by the square, crosses, and diamonds. (B) Pressure dependence of the S₀(1) frequency ratio ${\omega }_{{\text{H}}_2}/{\omega }_{{\text{D}}_2}$ derived from the experimental Raman data. The rigid rotor ratio is indicated by the dotted line at ${\omega }_{{\text{H}}_2}/{\omega }_{{\text{D}}_2}=2$, while the quantum harmonic oscillator ratio is indicated by the dashed line at ${\omega }_{{\text{H}}_2}/{\omega }_{{\text{D}}_2}=\sqrt{2}$. HH crosses into the harmonic oscillator regime at around 27 GPa, the same pressure a structural change is previously reported (23). (C) Experimental Raman frequencies derived from the fitting procedure described in Materials and Methods (squares) and computed Raman frequencies obtained via DFPT (circles) for the hydrogen vibron upon compression of the hydrogen hydrate. The change in slope, highlighted by the shadow area, corresponds to the pressure where the hydrogen molecules order.

Fig. 4B shows the pressure dependence of the ratio between the $S_0(1)$ Raman frequencies measured for a H₂-hydrate (${\omega }_{{\mathrm{H}}_2}$) and for a D₂-hydrate (${\omega }_{{\mathrm{D}}_2}$). For further information, SI Appendix, Fig. S5 shows a visualization of the evolution in pressure of all the rotons of both H₂- and the D₂-hydrate, while SI Appendix, Fig. S7 displays the single-peak fitting of the $S_0(1)$ roton for each hydrate, which has been used to calculate this ratio. At low pressure, the ratio remains close to the free-rotor limit of 2, reflecting the $1/I$ scaling of rotational energies with the molecular moment of inertia. Upon compression, the increasingly strong crystalline field exerted by the rigid water lattice progressively hinders the molecular rotations, effectively locking the H₂/D₂ orientation relative to the host cages. As rotational freedom is quenched, the excitations acquire librational character and the ratio approaches the harmonic mass-scaling limit $\sqrt{2}$, characteristic of an oscillator with frequency $\propto m^{-1/2}$. A qualitatively similar evolution has been reported for pure solid hydrogen and deuterium at much higher pressures (above ∼140 GPa) (48, 50), and a direct comparison between the dispersion curves for the hydrate and for the elemental solids is provided in SI Appendix, Fig. S6. In analogy with the dense elemental phases, if the hindering potential in the hydrate is sufficiently anharmonic—i.e., weaker than purely harmonic at large angular displacements—the ratio can transiently overshoot below $\sqrt{2}$ before converging, reflecting an enhanced softening of the librational mode. Notably, the roton–libron cross-over in the hydrate occurs at pressures an order of magnitude lower than in pure hydrogen/deuterium, highlighting the much stronger orientational constraints imposed by the host lattice. The observed trend thus constitutes direct spectroscopic evidence for the progressive locking of the guest-rotor orientation due to the strengthening of H₂/D₂-H₂O coupling under pressure.

This progressive hindering of the guest rotations revealed in Fig. 4B has a direct counterpart in the behavior of the intramolecular H-H stretching mode, since changes in rotational freedom modify the coupling between rotational and vibrational degrees of freedom and thus the vibron pressure dependence. Consequently, from the analysis of the H-H stretching mode in Fig. 4C comparing Density Functional Perturbation Theory (DFPT) calculations with experimental Raman frequencies it turns out that both simulation and experiment display a clear change in slope near 30 GPa. This is consistent with the signatures seen in the rotational modes and further linking the vibron behavior to the onset of pressure-induced orientational ordering predicted by our quantum embedded model.

The structure of the C2 hydrogen hydrate at low pressure is shown in Fig. 5A. The two interpenetrating H₂ and H₂O cubic sublattices are fully orientationally disordered: The protons in the water framework are randomly distributed between equivalent hydrogen-bond configurations, and the H₂ molecules adopt random orientations within their cages. Bottom Fig. 5B reports the evolution with pressure of the positions of the two oxygen atoms closest to a given water proton Hw-the intramolecular oxygen covalently bound to Hw and the oxygen of the neighboring molecule to which Hw is hydrogen-bonded-shown, as extracted from our Path-Integral Molecular Dynamics (PIMD) trajectories (Materials and Methods). With increasing pressure, these two nearest oxygen atoms approach to each other and their corresponding distances to Hw converge at ∼26 GPa (40), indicating proton symmetrization and the formation of symmetric $O-H\cdots O$ bonds. In the Top Fig. 5B, the red dashed line represents the orientation factor $S_{\text{PIMD}}$ of the H₂ molecules, computed from path-integral molecular dynamics as an ensemble average over beads and time steps, as defined in Materials and Methods. This quantity implicitly incorporates both the geometrical arrangement of the molecules and their thermal-quantum fluctuations relative to the c axis. The open blue symbols represent the global order parameter $M_{\mathit{tot}}^{\mathit{PIMD}}$ computed from PIMD trajectories as defined in SI Appendix, section S2. Before symmetrization, compression primarily shortens the hydrogen bonds, with little ordering of the H₂ sublattice. Proton symmetrization abruptly stiffens the $O-H\cdots O$ network, producing a sharp increase in the bulk modulus B (23, 27) and making the H₂O sublattice much less compressible. Beyond this point, further isotropic compression is energetically disfavored; instead, it becomes more favorable to reorient the H₂ molecules with their bond axes parallel to the c direction (Fig. 5C), reducing the volume along a and b owing to the increased compressibility along these axes. This collective alignment of the H₂ sublattice drives the tetragonal elongation of the H₂O framework and produces a second, more gradual rise in B (23, 27). In other words, the lattice first complies with increased pressure through hydrogen-bond rigidification in the water network, and then through anisotropic reorientation of the hydrogen network, rather than by further reducing the $O-O$ separation. Fig. 5 thus reveals a two-stage process in which the behavior of the H₂O and H₂ sublattices becomes tightly interlocked, their structural and dynamical degrees of freedom evolving in unison and merging into a single, quantum-stabilized lattice.

Fig. 5.

(A) Crystal structure of C2 hydrogen hydrate at low pressures ($P\lesssim 30$ GPa), showing the two interpenetrating cubic sublattices of H₂O and H₂. The water-proton network is disordered, while the H₂ molecules are orientationally disordered at high temperature and adopt a herringbone-like arrangement at low temperature. (B) Bottom: Pressure dependence of the first (solid lines) and second (dotted lines) peak positions in the radial distribution functions $g(r)$ for the $H_w-O$ pairs, from PIMD simulations at $T=300$ K. Top: The red dashed line represents the orientation factor $S_{\text{PIMD}}$ and the open blue symbols represent the global order parameter $M_{\mathit{tot}}^{\mathit{PIMD}}$ computed from PIMD trajectories. We display via a blue shaded area the uncertainty of the transition pressure region as evaluated from our quantum embedded model at 300 K. (C) Crystal structure of C2 hydrogen hydrate in the symmetrized (water-proton) state at high pressures ($P\gtrsim 30$ GPa). At the highest pressures, H₂ molecules become nematic along the c-axis, coinciding with the tetragonal distortion direction.

Our results demonstrate that C2 hydrogen hydrate undergoes a pressure-driven, two-stage transformation in which the hindering of the rotational dynamics of the H₂ sublattice becomes intertwined with the distortion of the H₂O sublattice. The first stage—proton symmetrization in the host lattice—abruptly stiffens the hydrogen-bond network, while the second-anisotropic alignment of the H₂ molecules-relieves further compression and induces a tetragonal distortion. This collective alignment is directly manifested in the structural transition of the lattice, marking the point where the guest reorientation and host deformation proceed in lockstep. The coupling is evident across structural, elastic, rotational, and vibrational observables, linking the roton–libron cross-over of the guest molecules to the proton ordering of the host. In this regime, the motions of the host lattice and the guest molecules become strongly interconnected, forming collective quantum modes rather than distinct subsystems. This provides an exquisite example of coupled nuclear-quantum dynamics in a crystalline solid, with potential implications for other hydrogen-rich materials under high pressure.

Materials and Methods

Experimental Details.

Samples were prepared in two different ways, depending on the specific experiment, cryoloaded or gas-loaded in the diamond anvil cell (DAC). In the first case, polycrystalline hydrogen hydrate samples with the sII clathrate structure were prepared by exposing H₂O ice (Ih) to gas H₂ at 0.28 GPa for 20 to 30 min, following the method described in ref. 3. The ice was made of spheres with typical diameters of a few tens of micrometers. After preparation, the samples were recovered at ambient pressure and stored at liquid nitrogen temperature. Then, a small amount of sample was loaded at liquid nitrogen temperature into a diamond anvil cell partially immersed in a liquid nitrogen bath and in an atmosphere saturated with nitrogen vapor to avoid water or oxygen condensation on the culet. The loaded samples were then compressed to pressures of a few GPa before being warmed to room temperature. This is a similar loading procedure to what we had previously done for methane hydrate (20, 22). This procedure inevitably produces a mixture of C2 hydrogen hydrate and excess pure ice, due to the difference in molar ratios between the clathrate sII phase and the C2 phase. Clathrate sII samples prepared following our method have a molar ratio H₂O:H₂ between 4 and 5, as we have verified by neutron diffraction in the past (3), and the C2 phase is characterized by a molar ratio of 1. Two samples were prepared directly in the DAC by loading room-temperature liquid H₂O and high-pressure H₂ using a gas loading setup. Briefly, liquid water was loaded together with an air bubble, consisting of approximately 70% of the sample chamber volume. The air bubble was replaced with 99.99+% H₂ at 1,400 bar, and the DAC was subsequently pneumatically closed and pressurized. These samples were measured by both Raman spectroscopy and by XRD. Some of the cryoloaded samples contained a small amount of pure N₂ and/or pure O₂, both of which can easily be identified from their respective vibron Raman peaks. They were trapped in the sample chamber during cryoloading and would have acted as a pressure-transmitting medium. In the gas-loaded samples, H₂ itself was acting as the pressure transmitting medium. Culet diameters ranging between 200 to 500 μm for the IIas quality diamond anvils were used. Re foil gaskets were used to contain the sample while ruby and/or gold spheres were used as pressure calibrants in Raman and X-ray experiments respectively.

Angle-dispersive X-ray powder diffraction patterns were acquired at ID15b at ESRF (Grenoble, France) using a monochromatic X-ray beam ($\lambda =0.41$ Å) and an Eiger 2 9M CdTe flat-panel detector, with a typical acquisition time of a ten seconds (45). The beam spot size was $6\times 6 \mathrm{\mu }$m, and the DAC was rocked 5^° during the acquisition. The 2D diffraction patterns were treated to mask the Bragg peaks of the diamond anvils, then integrated into one-dimensional patterns, and a smooth polynomial background was subtracted. Le Bail refinements were performed using Topas Academic version 6 (54). In the XRD measurements, the pressure was determined using the equation of state of gold from ref. 55. Raman spectra at ambient temperature were acquired using a commercial Horiba Jobin-Yvon LabRam HR800 Raman spectrometer in a backscattering geometry, equipped with a COBOLT SambaTM green 532 nm laser, with a nominal power of 1,000 mW. Low-temperature Raman spectra were collected using an Argon laser (120 mW output power) tuned to 514.32 nm and a Peltier-cooled HR460 spectrograph with 1,500 gr/mm grating. The acquisition time was typically about a minute. Reference spectra were recorded at each pressure from the gasket close to the sample chamber, and particular care was taken to check the effect of the background subtraction on the fit results. In the Raman measurements, pressure was determined by the shift of the R1 ruby fluorescence line (56), from the edge of the stressed diamond signal (above 40 GPa) (57), or from the vibron frequency of pure solid H₂ (58). A helium cryostat was used for low-temperature measurements.

Computational Details.

Ab initio molecular dynamics (AIMD) simulations were carried out on a $2\times 2\times 2$ supercell of the C2 phase (space group Pna2₁) over a grid of pressures from 3 to 50 GPa and temperatures from 25 to 300 K. Calculations employed the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional (59) and were performed with both VASP (60) and Quantum Espresso (61, 62) packages. The Path-Integral molecular dynamics approach (PIMD) (63–67) was used to efficiently sample nuclear-quantum effects under the desired thermodynamic conditions. Simulations were conducted for temperatures above 25 K, by using the Path-Integral Ornstein–Uhlenbeck Dynamics (PIOUD) algorithm implemented in Quantum Espresso (68). After performing convergence tests on both the virial and primitive estimators of the quantum kinetic energy, the number of beads ($N_b$) was set to 40 at 300 K and increased to 160 at 25 K for PIMD runs. For both AIMD and PIMD, a plane-wave cutoff of 100 Ry and a $2\times 2\times 2$ k-point mesh were sufficient to achieve convergence in the DFT solution of the electronic problem. Trajectories were obtained with a time-step of 0.5 fs and a total duration of approximately 10 ps for each (P, T) condition.

Spin isomers such as ortho- and para- hydrogen are not taken into account in molecular dynamics (MD) simulations. Indeed, to access the true low-temperature physics of the C2 phase while retaining quantum effects and accounting for spin isomers, we solved the Schrödinger equation at 0 K for an H₂ molecule in an effective potential modeling the induced interactions between H₂ and the water lattice. This approach can be viewed as the quantum embedding model of a system of quantum rigid rotor in an effective potential representing an isolated H2 surrounded by a rigid water lattice. As detailed in SI Appendix, section S1, going beyond previous studies (69–72), we evaluated the full, nonseparable potential $V_{\text{ext}}(r,\theta ,\varphi )$$=R(r)+P_r(\theta ,\varphi )$ by performing self-consistent DFT calculations (Quantum Espresso, 300 eV cutoff, $5\times 4\times 5$ k-mesh) for each configuration generated on a dense spherical grid with a fixed center of mass varying the H₂ bond lengths r and angular orientation θ and ϕ (72). These 0 K DFT energies directly capture the interaction of H₂ with the classical equilibrium geometry of the surrounding C2 structure (water lattice and neighbor H₂ molecules). We fitted the radial part of the potential $R(r)$ using a Morse function and interpolated linearly on our grid the angular part $P_r(\theta ,\varphi )$, for each r value.

The Schrödinger equation has been solved numerically using the Lanczos algorithm in shift inverted mode, using a basis set of the form ${\psi }_n^{\mathit{lm}}(r,\theta ,\varphi )={\chi }_n(r)Y_l^m(\theta ,\varphi )$, where ${\chi }_n(r)$ is the radial basis of the Morse potential and $Y_l^m(\theta ,\varphi )$ are the spherical harmonics.The matrix elements computation is explained in detail in SI Appendix, section S1.

The spatial density at finite temperature T was obtained by treating the two nuclear-spin manifolds of ${\mathrm{H}}_2$ independently. Assuming that ortho $(\mathcal{J} = 1)$ and para $(\mathcal{J} = 0)$ populations do not interconvert on the time scale of the simulation, the total density reads

$$\begin{array}{cc}\hfill \rho (\mathbf{r})& \hfill ={\displaystyle \sum _{i\in \text{ortho}}}\frac{f_{\mathrm{o}}{e}^{-\beta (E_i-E_0^{\mathrm{o}})}}{Z_{\mathrm{o}}}{|{\psi }_i(\mathbf{r})|}^2\hfill \\ \hfill & \hfill + {\displaystyle \sum _{j\in \text{para}}}\frac{f_{\text{p}}{e}^{-\beta (E_j-E_0^{\text{p}})}}{Z_{\text{p}}} {|{\psi }_j(\mathbf{r})|}^2,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill Z_{\mathrm{o}}={\displaystyle \sum _{i\in \text{ortho}}}{e}^{-\beta (E_i-E_0^{\mathrm{o}})},\\ \hfill Z_{\mathrm{p}}={\displaystyle \sum _{j\in \text{para}}}{e}^{-\beta (E_j-E_0^{\mathrm{p}})},\end{array}$$

Here, $\beta ={(k_{\mathrm{B}}T)}^{-1}$, $|{\psi }_n{|}^2$ and $E_n$ are the eigenfunctions and eigenvalues obtained from the Schrödinger equation evaluated in the relaxed hydrate lattice; $E_0^{\mathrm{o}}$ and $E_0^{\mathrm{p}}$ are the ortho and para ground-state energies, and $f_{\mathrm{p}}:f_{\mathrm{o}}=1:3$ is the fixed statistical spin ratio. This model entails specific approximations: It is a static, mean-field description that neglects cage deformation effects, as the potential is evaluated on the fixed lattice geometry. Furthermore, the density at finite temperature is derived by mixing the para (J = 0) and ortho (J = 1) manifolds in a fixed 1:3 ratio, assuming no spin interconversion on the simulation timescale. This assumption is based on the experimentally observed negligible rate of ortho–para conversion in the absence of coupling between spatial and spin variables, combined with sufficiently rapid cooling to preserve the ortho population in a metastable state, even at low temperatures. A key advantage of the embedded quantum model is that it naturally incorporates the correct ortho/para mixture of H₂. In contrast, PIMD simulations describe only the para state (the ground rotational state), while in reality the ortho species is more populated than the para one in a 3:1 ratio. Because the ortho states are more localized than the para state, this difference has a major impact on orientational properties, particularly at the lowest temperatures. Accounting for the ortho/para mixture is therefore essential for a quantitatively accurate description of the orientational regimes and phase transitions. Another clear advantage of the quantum embedded model is the possibility to explore low-temperature regimes at constant computational effort, compared to PIMD whose effort increases dramatically as temperature is lowered. We therefore adopted the quantum embedded model as viable approach to study the quantum phase diagram of hydrogen-filled ice as a function of pressure and temperature. We used several exchange-correlation functionals to evaluate $V_{\mathit{ext}}$ and their comparison is shown in SI Appendix, Fig. S1. We found that the use of 6 the vdW-DF functional improves the accuracy of energy levels estimation. Accounting for proton disorder by averaging over multiple proton-disordered water lattices obeying ice rules reduces the anisotropy of the effective potential. This leads to a reduction in the rotational energy level splitting, bringing the computed eigenvalues into closer quantitative agreement with experimental inelastic neutron scattering observations (72). For consistency with the MD calculations, the main results are obtained using PBE. Despite the use of a static, proton-ordered PBE framework in this study, the model correctly predicts the pressure-induced orientational ordering transition and successfully reproduces the experimental Raman $S_0(0)$ and $S_0(1)$ rotons spreading with pressure.

To characterize the three distinct orientational regimes for H₂ [quantum plastic, herringbone-like configurations, and nematicity (27, 35)] from the quantum embedded model, we introduce the orientation factor S as follows (73, 74).

In analogy with diffraction theory, where the scattering amplitude can be factorized into a lattice contribution and a local form factor, we define the orientation factor as the product of a collective geometric term and an intrinsic quantum factor:

$$\begin{array}{c}\hfill S(P,T)=S_g(P) S_q(P,T).\end{array}$$ [1]

The geometric contribution $S_g$ is evaluated on the 0 K equilibrium geometry obtained from DFT calculations at a given pressure P. It is defined from the orientation of each molecule i, through the angle ${\theta }_i$ between the crystallographic c axis (preferential axis of alignment) and the molecular axis. Its definition coincides with the nematic order parameter (75, 76):

$$\begin{array}{c}\hfill S_g=\frac{1}{N}{\displaystyle \sum _{i=1}^N}\left(\frac{3}{2}{cos}^2{\theta }_i-\frac{1}{2}\right).\end{array}$$ [2]

This factor thus encodes the collective orientational arrangement of the atomic sites. By construction, $S_g=1$ when all H₂ molecular axes are perfectly aligned along the crystallographic c axis, while $S_g=0$ corresponds to a completely random, isotropic distribution of the molecular orientations in the simulation cell.

We then apply a quantum dressing to this geometric factor, obtained from the probability density of the quantum embedded model at pressure P and temperature T. For this purpose, we define the alignment tensor

$$\begin{array}{c}\hfill {\mathbf{Q}}_q=\frac{3}{2}\int \rho (\mathbf{r}) {\mathbf{u}}_{\mathbf{r}}{\mathbf{u}}_{\mathbf{r}} {\mathrm{d}}^3\mathbf{r}-\frac{1}{2}\mathbf{I},\end{array}$$ [3]

where ${\mathbf{u}}_{\mathbf{r}}$ is the unit vector along $\mathbf{r}$, and $\rho (\mathbf{r})$ is the diagonal part of the thermal density matrix. We define the quantum contribution $S_q$ as the largest eigenvalue of ${\mathbf{Q}}_q$. By construction, $S_q=1$ when the quantum distribution of the H₂ molecular axes is perfectly aligned along a given direction while $S_q=0$ corresponds to an isotropic quantum distribution. This quantum dressing ensures that the final orientation factor incorporates both the orientation linked to the underlying geometry ($S_g$) and the intrinsic thermal-quantum fluctuations encoded in the density matrix ($S_q$), in the same way as diffraction separates lattice geometry from the internal structure of the scatterer.

With this definition, the orientation factor $S(P,T)$ can be mapped across the phase diagram, as shown in Fig. 1.

$S(P,T)$ can also be computed from PIMD simulations at pressure P and temperature T, whenever available, instead of the quantum embedded model. In this case, the orientation factor reads $S_{\mathrm{PIMD}}=\int dX \rho (X) \frac{1}{N{N}_b}\sum _{i=1}^{N{N}_b}\left(\frac{3}{2}{cos}^2{\theta }_i(X)-\frac{1}{2}\right)$, where the integral runs over the path configurations X, and the sum runs over all N H₂ molecules and the beads. Here, $\rho (X)$ denotes the normalized configurational probability density sampled during the path-integral molecular dynamics, so that the integral represents an ensemble average over beads and time steps. Notice how the latter equation contains implicitly both a geometrical contribution, given by the orientational arrangement of the H₂ molecules in the supercell, and thermal-quantum fluctuations, provided by the path-integral formalism. The orientation factor obtained from the PIMD simulations $S_{\mathrm{PIMD}}$ at 300 K is plotted in Fig. 5, by neglecting spin isomers effects and relying on the similar density distribution of the $\mathcal{J}=0$ and $\mathcal{J}=1$ states at room temperature.

To locate the transition between the quantum plastic and orientationally ordered phases, we employ a complementary global order parameter $M_{\mathrm{tot}}$. It is defined as a structure-factor measure of orientational coherence, constructed from angular phases chosen such that symmetry-related orientations (phase/antiphase configurations, e.g. in the herringbone pattern) interfere constructively. Randomly oriented H₂ molecules then cancel out ($M_{\text{tot}}\approx 0$), while herringbone or nematic H₂ molecules add coherently ($M_{\text{tot}}\approx 1$). The precise mathematical definition, including the angular cumulative distribution function transforms and rank-m order parameters, is given in SI Appendix, section S2. In molecular dynamics trajectories sampled over the $(P,T)$ grid, orientational transitions are identified by the vanishing of $M_{\mathrm{tot}}$, which signals the loss of orientational order in the H₂ sublattice. For each pressure, the temperature at which $M_{\mathrm{tot}}$ vanishes is referenced as the transition point.

Since $M_{\mathrm{tot}}$ cannot be evaluated within the quantum embedded model, which describes individually embedded H₂, we instead compute the global order parameter from classical MD simulations. At the classical transition points, we evaluate the corresponding orientation factors, where this time thermal-quantum fluctuations in Eq. 3 are replaced by a thermal average performed over the classical canonical ensemble in the simulation cell. Within the approximation of dominant electrostatic effects, the resulting ensemble-averaged value of S at the classical transition points provides a threshold, $S^{\mathrm{th}}$, for the critical value of S computed with the quantum embedded model. Within the embedded framework, the transition temperature $T_P^{\ast }$ at pressure P is then obtained from the condition

$$S(P,T_P^{\ast })=S^{\mathrm{th}}$$

Plotting $T_P^{\ast }$ as a function of pressure gives the points of the white dashed transition line shown in Fig. 1. The error bars on $T_P^{\ast }$ are obtained by propagating the uncertainty in $S^{\mathit{th}}$: We evaluate the inverse relation at the two bounds $S=S^{\mathit{th}}\pm {\sigma }_S^{\mathit{th}}$. The temperature uncertainty is then defined as $\mathrm{\Delta }T=\frac{1}{2}|T_{\mathrm{high}}^{\ast }-T_{\mathrm{low}}^{\ast }|$, where $T_{\mathrm{high}}^{\ast }$ ($T_{\mathrm{low}}^{\ast }$) corresponds to the temperature extracted from the inverse map at the $S^{\mathit{th}}+{\sigma }_S^{\mathit{th}}$ ($S^{\mathit{th}}-{\sigma }_S^{\mathit{th}}$) bound (SI Appendix, Fig. S4).

Finally, because $S_g$ is computed with respect to the c axis in Eq. 2, the orientation factor also provides direct information on the nematic orientation of the system, the largest values of S being taken in the nematic aligned phase.

The phonon spectrum and Raman cross-section of the system were computed using density functional perturbation theory (DFPT) as implemented in the Quantum ESPRESSO package (77, 78), employing a unit cell and a $4\times 4\times 4$ k-point grid. This is the only calculation performed entirely within a classical nuclear framework, as no implementation currently exists for computing Raman cross sections from PIMD trajectories.

To investigate the relationship between H₂ orientational order, H₂O-H₂ interactions, and proton symmetrization within the H₂O sublattice, the radial distribution functions $g(r)$ as a function of pressure were computed from our MD trajectories for Hw–O, Hh–O, and O–O pairs. Here, Hw denotes protons belonging to the H₂O sublattice, Hh refers to protons from H₂ molecules, and O represents oxygen atoms.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Experimental data collected during session HC-5914 are publicly available via the European Synchrotron Radiation Facility data repository (79). All files required to reproduce the theoretical results, including input parameters for MD simulations, the Potential Energy Surface (PES) map, and the custom Schrödinger solver code, have been deposited in Zenodo (80). This database also includes analysis scripts, numerical data used to generate all figures in the main text and SI, and a set of representative output trajectories.