**footnotetext: These authors contributed equally to this work.

# Giant splitting of the hydrogen rotational eigenenergies in the C2 filled
ice

Simone Di Cataldo  0000-0002-8902-0125<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
00187 Roma, Italy  Institüt für Festkörperphysik, TU Wien, Wiedner Hauptstraße
8-10, Vienna 1050, Austria  Maria Rescigno  0000-0002-5976-4449
<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università di
Roma, Piazzale Aldo Moro 5, 00187 Roma, Italy  Laboratory of Quantum
Magnetism, Institute of Physics, École Polytechnique Fédéerale de Lausanne,
CH-1015 Lausanne, Switzerland  Lorenzo Monacelli  0000-0002-6381-3741
Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5,
00187 Roma, Italy  Umbertoluca Ranieri  0000-0003-2026-6132 Centre for Science
at Extreme Conditions and School of Physics and Astronomy, University of
Edinburgh, EH9 3FD Edinburgh, UK  Richard Gaal  0000-0001-5181-423X Laboratory
of Quantum Magnetism, Institute of Physics, École Polytechnique Fédéerale de
Lausanne, CH-1015 Lausanne, Switzerland  Stefan Klotz  0000-0002-5682-4960
Sorbonne Université, UMR CNRS 7590, Institut de Minéralogie, de Physique des
Matériaux et de Cosmochimie (IMPMC), 5 Place Jussieu, 75005 Paris, France
Jacques Ollivier  Institut Laue-Langevin, 71 Avenue des Martyrs, Cedex 9,
Grenoble, France  Michael Marek Koza  0000-0002-5133-8584 Institut Laue-
Langevin, 71 Avenue des Martyrs, Cedex 9, Grenoble, France  Cristiano De
Michele  0000-0002-8367-0610 Dipartimento di Fisica, Sapienza Università di
Roma, Piazzale Aldo Moro 5, 00187 Roma, Italy  Livia Eleonora Bove
0000-0003-1386-8207<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza
Università di Roma, Piazzale Aldo Moro 5, 00187 Roma, Italy  Laboratory of
Quantum Magnetism, Institute of Physics, École Polytechnique Fédéerale de
Lausanne, CH-1015 Lausanne, Switzerland  Sorbonne Université, UMR CNRS 7590,
Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC),
5 Place Jussieu, 75005 Paris, France

###### Abstract

Hydrogen hydrates present a rich phase diagram influenced by both pressure and
temperature, with the so-called C2 phase emerging prominently above 2.5 GPa.
In this phase, hydrogen molecules are densely packed within a cubic ice-like
lattice and the interaction with the surrounding water molecules profoundly
affects their quantum rotational dynamics. Herein, we delve into this
intricate interplay by directly solving the Schrödinger’s equation for a
quantum H2 rotor in the C2 crystal field at finite temperature, generated
through Density Functional Theory. Our calculations reveal a giant energy
splitting relative to the magnetic quantum number of $\pm$3.2 meV for $l=1$.
Employing inelastic neutron scattering, we experimentally measure the energy
levels of H2 within the C2 phase at 6.0 and 3.4 GPa and low temperatures,
finding remarkable agreement with our theoretical predictions. These findings
underscore the pivotal role of hydrogen–water interactions in dictating the
rotational behavior of the hydrogen molecules within the C2 phase and indicate
heightened induced-dipole interactions compared to other hydrogen hydrates.

The phase behavior of hydrogen hydrates represents a captivating realm of
inquiry at the intersection of physics and quantum chemistry, offering
insights into the complex interplay of molecular interactions under extreme
conditions and quantum motions under extreme confinements. At pressures above
0.1 GPa and below $\sim$0.5 GPa, mixtures of water and hydrogen molecules
crystallize in a non-stoichiometric clathrate compound, which is characterized
by polyhedral cavities formed by hydrogen-bonded water molecules encaging a
variable amount of non-bonded H2 molecules [1, 2, 3, 4, 5]. The dynamics of
the (freely rotating) hydrogen molecules within the cages stabilizes the
highly porous ice structure [6]. At low temperatures, the quantum motion of
the “encaged” guest H2 molecules has been characterized by means of inelastic
neutron scattering and Raman spectroscopy experiments, mostly on samples
recovered to ambient pressure [7, 8, 9, 10], and reveals to be similar to the
one observed for H2 trapped in other nanocavities [11, 12, 13, 14, 15, 16,
17]. The anisotropy in the hydrogen–water interaction potential has been shown
to be effective in splitting the quantized translational and rotational energy
levels; however, such splittings are rather small, indicating a weak
guest–host interaction. Bačić and co-workers led the way in developing the
theoretical framework for describing this cage potential, resulting in a more
precise assessment of the energy levels that define the quantum dynamics of
H2. These computational studies shed light on a crucial aspect of dynamics:
the interaction between translational and rotational angular momentum. This
interaction is mediated by the 5D-cage potential, which includes terms
influenced by the orientation of the axis of the H2 molecule in relation to
the normal direction of the cage wall [18, 19, 20, 17]. The quantum dynamics
of the H2 molecule confined in a clathrate hydrate structure was also studied
as a function of pressure up to 0.5 GPa [21] showing only a small impact on
the energy levels and splittings, despite the varying water-hydrogen distances
with pressure.

At pressures above $\sim$0.5 GPa, the hydrates of hydrogen adopt so-called
filled ice phases, in which the water sublattice is structurally similar to
ice [22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31] and hydrogen molecules occupy
defined positions inside the ice-like channels. In the filled ice C1 phase,
which is constituted by an ice II-like host frame and has a 1:6 hydrogen to
water molecular ratio [24, 25, 26], the first rotational energy level of the
confined H2 was found (by inelastic neutron scattering experiments at high
pressure and low temperature) to be only marginally different from that of a
free H2 rotor [21]. Similar results [32] were observed for the quantum
dynamics of H2 confined in the C0 structure of hydrogen hydrate, where the
host water lattice is not based on any stable ice phase [33, 22, 23]. Among
the diverse filled ice phases, the C2 phase (stable above $\sim$2.5 GPa)
stands out for its distinctive characteristics arising from the interplay
between hydrogen and water molecules, which form two identical and
interpenetrated ice I$c$-like (or diamond-like) sublattices with a 1:1
hydrogen to water molecular ratio. As a result, hydrogen molecules are very
densely packed in this structure and the hydrogen–water distances are very
short [26, 27, 28, 29, 34]. One of the remarkable features of the C2 phase is
indeed the profound influence of the surrounding water molecules on the
quantum dynamics of the encapsulated hydrogen molecules. Unlike in other
phases, in the C2 phase, the crystal field imposed by the water molecules is
expected to strongly affect the quantum rotational dynamics of the H2
molecules. To our knowledge, the quantum mechanical behavior of molecular
hydrogen within the C2 lattice has not been explored yet. Above $\sim$40 GPa,
the C2 phase transforms into the C3 phase upon laser heating [35, 36], which
has an identical cubic water sublattice but is even richer in H2 (2:1
molecular ratio), resulting in significant unit cell expansion and consequent
larger hydrogen–water distances. Therefore, the C2 phase represents the most
extreme case of confinement for hydrogen molecules in a water frame and thus
provides an ideal benchmark to study the quantum motion of hydrogen in strong
confinement.

In this work, we employed state-of-the-art computational and experimental
techniques to fully characterize the rotational quantum dynamics of the H2
molecules in the C2 phase at low temperatures. In particular, we solve the 3D
Schrödinger’s equation for a hydrogen molecule embedded in the effective
crystal field of the C2 phase: the H2 effective potential at finite
temperature computed within _ab initio_ molecular dynamics (AIMD) accounting
for van der Waals corrections to the exchange-correlation functional. Our
calculations reveal that the H2O–H2 interaction is such to induce a giant
splitting of $\pm 3.2$ meV in the $l$=1 rotational eigenenergies. To validate
our theoretical predictions and further probe the quantum rotational dynamics
of H2 within the C2 phase, we employ inelastic neutron scattering (INS), a
powerful experimental technique capable of providing detailed insights into
the energy levels of hydrogen molecules. Through pioneering INS experimental
measurements conducted at 6 GPa and low temperature (4 K $<$ $T$ $<$ 85 K), we
corroborate our theoretical findings, thereby affirming the accuracy of our
predictions regarding the quantum rotational dynamics of H2 within the C2
phase.

Our results not only deepen the understanding of the fundamental physics
governing hydrogen hydrates and shed light on the hydrogen–water interaction
in dense systems, but also carry broader implications for molecular hydrogen
under extreme conditions. Furthermore, the observed heightened induced-dipole
interactions within the C2 phase underscore the significance of hydrogen
hydrates in diverse fields ranging from materials science to astrochemistry.

Figure 1: Panel (a): Radial part of the potential ($V_{rad}$), DFT-calculated
data points are shown as blue circles, the fit with the Morse potential is
shown as a blue line. The energy zero is taken as the last calculated point.
Panels (b) and (c) show the calculated and fitted angular part of the
potential ($V_{ang}(\theta,\phi)$), respectively, at a temperature $T$ = 10 K.
See Supplementary Material [37] for further details.

The H2 molecule is the textbook example of a quantum rigid rotor, with
rotational peaks equally spaced and separated by 2$B$ = $\hbar^{2}/I$, where
$I$ is the moment of inertia $\mu R^{2}=\frac{mR^{2}}{2}$, with $\mu$ the
reduced mass, $m$ the mass of H2, and $R$ the H–H distance. The Schrödinger’s
equation extends to the case of an external potential $V_{ext}$ (Cartesian
coordinates, atomic units):

$\left(-\frac{\nabla^{2}}{2\mu}+V_{ext}(\vec{r})\right)\psi_{n}(\vec{r})=\epsilon_{n}\psi_{n}(\vec{r})$
(1)

For a H2 molecule in the C2 crystal, the induced-dipole–dipole interactions
between H2 and the water sublattice can be modeled precisely as such an
effective potential $V_{ext}$. We neglect quantum fluctuations in the hosting
lattice. Within this approximation, at $0\text{\,}\mathrm{K}$, $V_{ext}$ is
the total energy of the crystal at various hydrogen orientations. Temperature
can also be accounted for by replacing $V_{ext}$ with the free energy
landscape averaging over different snapshots from molecular dynamics at the
chosen temperature (see Supplementary Material (SM) for further details [37]).
The calculations were performed using the DFT-relaxed structure of C2 having
the correct experimental volume (see Tab. S2 in SM [37] for full structural
details). The result is the effective potential shown in Fig. 1, separated
into its radial ($|r|$) and angular ($\theta,\phi$) components. The radial
component is very well fitted by a Morse potential, while the angular part is
characterized by a complex landscape well reproduced by a combination of sines
and cosines (see SM [37]).

The Schödinger’s equation 1 with the calculated $V_{ext}$ was solved
numerically using the Lanczos algorithm in shift-invert mode to obtain the
lowest-energy eigenvalues and eigenfunctions. In Fig. 2(a), we show the energy
states compared to the free rotor. We remark that H2 molecules being
constituted of two identical fermionic hydrogens form two spin isomers, namely
para-hydrogen ($p$-H2) with nuclear spin number I=0 and ortho-hydrogen
($o$-H2) with nuclear spin number I=1. Due to total wavefunction symmetry
constraints, $p$-H2 can only be associated with even rotational quantum number
states ($l$=0,2,4…) and $o$-H2 with odd states ($l$=1,3…). To benchmark the
solver, Fig. 2(a) shows that our method is consistent with the analytic result
in the absence of $V_{ext}$. When the host lattice is accounted for, the
magnetic quantum number $m$ degeneracy is broken with a splitting of $\pm$3.2
meV for $l=1$.

Figure 2: Panel (a): Calculated energy levels for the quantum rotations of the
H2 molecule in the C2 phase (black) and for the free rotor (blue). The
eigenenergies are divided into even-$l$ (p-H2) and odd-$l$ ones (o-H2). Arrows
indicate the transition from ortho-H2 ($l$=1) to para-H2 ($l$=0). Panel (b):
Isosurfaces showing the eigenfunctions of the ground state and the first three
excited states of the H2 molecule in the C2 crystal field (isolevel: $7e-5$),
the color indicates the $z$ axis and is used as a visual aid. The total
angular momentum $l$, index $\lambda$, and eigenenergies are given.

In Fig. 2(b), we show the wavefunctions corresponding to the ground state and
the first three excited levels. The host crystal field breaks the rotational
symmetry of the molecule, and both $l$ and $m$ are no longer good quantum
numbers. However, the low energy states still keep a dominant $l$ value that
can be used to group states, while the $m$ states with a similar $l$ are mixed
to form oriented orbitals that we number with an index $\lambda$ (see Fig. 2b
and Sect. S2B of SM [37] for more details). It is easier to interpret the
wavefunctions in comparison with the spherical harmonics, i.e., the result for
the unperturbed rotor (shown in Fig. S6): the ground-state wavefunction is
shaped like a hollow hemisphere, i.e., localized around $|\vec{r}|\sim R$.
Unlike the unperturbed rotor (where the wavefunction is a perfect sphere),
this wavefunction exhibits a broken angular symmetry in the direction of the
minimum of the potential. The first excited state wavefunction is oriented
roughly in the same direction as the ground state, while the other two point
in orthogonal directions, like the $p_{x}$, $p_{y}$, and $p_{z}$ angular
orbitals of the hydrogen atom.

Transitions between quantum rotational levels can be directly measured with
inelastic neutron scattering. In particular, as transitions between the ground
state ($l$=0) and the first excited state ($l$=1) involve changes in the
nuclear spin, INS is the only experimental technique able to directly measure
the fundamental rotational transition of molecular hydrogen. In addition,
thanks to the long-lived metastable nature of the ortho state [38], the ortho-
para transition can be measured in the neutron energy gain side of the
dynamical structure factor even at low temperature (see paragraph S5 E in
[37]) [16]. Previous studies focusing on hydrogen molecules trapped in
molecular cages of different sizes showed that rotational transitions can
combine with low-energy rattling modes [9, 21], leading to many possible
combinations. In the C2 hydrate, however, rattling modes are out of the probed
energy window due to the short average H2–H2O distances, leading to a much
simpler INS spectrum, as shown in Fig. 3(a).

Two INS experiments were performed on the IN5 time-of-flight spectrometer at
ILL in Grenoble, France (raw data available at [39] and [40]). The H2:D2O
sample was loaded cryogenically in a Paris-Edinburgh press (see SM [37] for
further details on the sample preparation and environment). In the first run,
the sample was compressed to 6.0 GPa (see Fig. S9) and cooled down to 4.4 K;
in the second, it was cooled to 9 K and the incoherent dynamical structure
factor $S(Q,\omega)$ was measured using an incident wavelength of 4.8 Å, thus
mainly measuring in the neutron energy gain side. The results from the first
run are shown in Fig. 3, while those from the second run are shown in SM
(Figure S11) [37].

Figure 3: INS experimental data acquired on IN5 at $P$=6.0 GPa and $T$=4.4 K.
Panel (a): $S(Q,\omega)$ after data reduction. A strong inelastic feature is
observed at about -11.7 meV. Panel (b): $Q$ binned spectra and their Gaussian
fits for $Q$ = 2.05 Å-1 (red line) and $Q$ = 3.25 Å-1 (blue line). Panel (c):
FWHM of the Gaussian fits to the $Q$-sliced INS spectra as a function of $Q$.
Panel (d): Integrated intensity of the Gaussian fits to the $Q$-sliced INS
spectra as a function of $Q$ and its best fit (see Supplementary Material [37]
for further details). Figure 4: $T$ dependence of the INS experimental
results. Panel (a): INS spectra at 6 GPa and 8 K, 33 K, 85 K, compared to
their best fits (blue areas). The gray shaded area is the peak from the pure
solid hydrogen in the sample. Panel (b): Energies of the three rotational
peaks at 6.0 and 3.4 GPa (blue circles and squares respectively) as a function
of temperature compared with numerical results (yellow diamonds). Panel (c):
Normalised intensities of the three peaks at 6.0 and 3.4 GPa (blue circles and
squares respectively) compared to the statistical weights of the initial
states calculated using the Boltzmann distribution. Plots with error bars are
reported in Fig. S13 in Supplementary Material [37]. The different blue shades
are consistently associated to the transitions from the $\lambda_{1}$,
$\lambda_{2}$, $\lambda_{3}$ levels to the ground state as shown in the
legend.

In both runs, we observed the first ortho-para ($l$=1, $\lambda_{1}$ to $l$=0)
transition of the hydrogen molecules confined in the C2 phase to have an
energy of 11.7 meV, i.e. about 3 meV lower compared to the ortho-para
transition of free hydrogen (14.6 meV). The rotational nature of the
excitation peak observed in the $S(Q,\omega)$ is confirmed from the exchanged
wavevector $Q$ analysis of both peak width and intensity. After $Q$-slicing
the $S(Q,\omega)$, the spectra (Figure 3 panel (b)) can be fitted with
Gaussian functions, whose FWHM are found to be constant with $Q$ (panel (c) of
Fig. 3) and whose intensities are well described by a squared first-order
spherical Bessel function (panel (d) of Fig. 3). Both of these trends are
indicative of a rotational nature for the excitation. As the temperature was
increased up to 85 K, we observed a progressive population of the higher
energy levels ($l$=1, $\lambda_{2}$ and $l$=1, $\lambda_{3}$) and the
consequent associated transition to the ground state (see Fig. S10 for a
schematic representation of the observed transitions). The occupation of the
levels is in good agreement with the Boltzmann distribution and the splitting
of the $l=1$ level is of $\pm$ 2.2 meV, 30% smaller than in our calculations
(yellow diamonds in Fig. 4 panel (b)). Examples of INS spectra (obtained by
integrating the $S(Q,\omega)$ over the whole measured $Q$ range), compared
with their best fits, and fit results are shown in Fig. 4. The $Q$ analysis of
the data shows a consistent behavior also at the highest investigated
temperatures (see Fig. S15). The sample was then decompressed to 3.4 GPa at 85
K and measured at 5 K and 33 K (squares in Fig. 4 panel (b), spectra are
reported in S16), where we observed a 7.5% reduction of the energy splitting.
In addition, we observed that the rotational transitions show an intrinsic
broadening much larger than our experimental resolution (0.8 meV), which can
be attributed to structural defects formed upon compression (see Fig. S17).

Overall, we find the calculated and experimental results for the energy levels
in excellent agreement in terms of both the absolute values and the energy
splitting (see Fig. 4 panel (b)). The largest discrepancy is for the high-
energy mode, which decreases with increasing temperature. This is expected as
the temperature dependency of our model comes from the thermal activation of
the phonons of the host water frame, altering the effective potential on the
H2 molecule. At very low temperatures, the zero-point motion of the ice
skeleton kicks in, which is neglected by our classical MD sampling. Indeed, as
temperature increases, the classical sampling of the ice phonons improves as
the agreement between simulations and experimental data.

In summary, we studied the quantum dynamics of the H2 molecule in the C2
filled ice phase at high pressure and low temperature. The overall excellent
agreement between our calculations and the experimental results offers a
simple interpretation of the data. In the C2 phase, the H2 and water interact
quite strongly, leading to a strong angular anisotropy in the effective
crystalline potential felt by the hydrogen molecule. This removes the angular
degeneracy with respect to the magnetic quantum number $m$, resulting in an
energy splitting (for the triplet $l$=1) of $\pm$3.2 meV in our calculations
and $\pm$ 2.2 meV experimentally. To the best of our knowledge, such a large
splitting in the energy levels of a quantum rotor in a water-based clathrate
is unprecedented. As the C2 hydrate is stable (or metastable) over a wide
pressure range spanning from nearly ambient pressure (at low temperature) [2,
34] to pressures above 80 GPa at least [36], it represents an ideal playground
to further explore low-energy physics of quantum rotors in the future.

We acknowledge the Institut Laue-Langevin (ILL) for providing beamtime on IN5
and the assistance from Claude Payre with the high-pressure setup. We thank
Leon Andriambariarijaona and Alasdair Nichols for their help during the
experiments on IN5, and Thomas C. Hansen for technical assistance during the
experiment on D20. We thank Lorenzo Ulivi, Leonardo del Rosso, and Milva Celli
for providing the sample. S.D.C. acknowledges computational resources from
CINECA, proj. IsC90-HTS-TECH and IsC99-ACME-C, the Vienna Scientific Cluster,
proj. 71754 ”TEST”. L.E.B. acknowledges the financial support by the European
Union - NextGenerationEU (PRIN N. F2022NRBLPT), the ANR-23-CE30-0034 EXOTIC-
ICE, and the Swiss National Fund (FNS) under Grant No. 212889.

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