Rotational sublevels of an ortho-hydrogen molecule encapsulated in an isotropic C60 cage

Quantum Criticality and Universal Behavior in Molecular Dipolar Lattices of Endofullerenes

Fullerene cages allow the confinement of single molecules and the construction of molecular assemblies whose properties strongly differ from those of free species. In this work, we employ the density-matrix renormalization group method to show that chains of fullerenes filled with polar molecules (LiF, HF, and H₂O) can form dipole-ordered quantum phases. In symmetry broken environments, these ordered phases are ferroelectric, a property that makes them promising candidates for quantum devices. We demonstrate that for a given guest molecule, the occurrence of these quantum phases can be enforced or influenced either by changing the effective electric dipole moment or by isotopic substitution. In the ordered phase, all systems under consideration are characterized by universal behavior that depends only on the ratio of the effective electric dipole and of the rotational constant. A phase diagram is derived, and further molecules are proposed as candidates for dipole-ordered endofullerene chains.

On the nature of the Schottky anomaly in endohedral water

In this work, we study the heat capacity contribution of a rigid water molecule encapsulated in C₆₀ by performing six-dimensional eigenstate calculations with the inclusion of its quantized rotational and translational degrees of freedom. Two confinement model potentials are considered: in the first, confinement is described using distributed pairwise Lennard-Jones interactions, while in the second, the water molecule is trapped within an eccentric but isotropic 3D harmonic effective confinement potential [Wespiser et al., J. Chem. Phys. 156, 074304 (2022)]. Contributions to the heat capacity from both the ortho and para nuclear spin isomers of water are considered to enable the effects of their interconversion to be assessed. By including a symmetry-breaking quadrupolar potential energy term in the Hamiltonian, we can reproduce the experimentally observed Schottky anomaly at ∼2 K [Suzuki et al., J. Phys. Chem. Lett. 10, 1306 (2019)]. Furthermore, our calculations predict a second Schottky anomaly at ∼0.1 K resulting from the H configuration, a different orientational arrangement of the fullerene cages in crystalline solid C₆₀. Contributions from the H configuration to C_V also explain the second peak observed at ∼7 K in the experimentally measured heat capacity.

H2O inside the fullerene C60: Inelastic neutron scattering spectrum from rigorous quantum calculations

We present a methodology that, for the first time, allows rigorous quantum calculation of the inelastic neutron scattering (INS) spectra of a triatomic molecule in a nanoscale cavity, in this case, H₂O inside the fullerene C₆₀. Both moieties are taken to be rigid. Our treatment incorporates the quantum six-dimensional translation-rotation (TR) wave functions of the encapsulated H₂O, which serve as the spatial parts of the initial and final states of the INS transitions. As a result, the simulated INS spectra reflect the coupled TR dynamics of the nanoconfined guest molecule. They also exhibit the features arising from symmetry breaking observed for solid H₂O@C₆₀ at low temperatures. Utilizing this methodology, we compute the INS spectra of H₂O@C₆₀ for two incident neutron wavelengths and compare them with the corresponding experimental spectra. Good overall agreement is found, and the calculated spectra provide valuable additional insights.

Dynamics and magnetic properties of NO molecules encapsulated in open-cage fullerene derivatives evidenced by low temperature heat capacity

Low-temperature heat capacity analyses for an NO-encapsulated fullerene derivative revealed (i) low-energy motion and (ii) strong magnetic anisotropy of the NO molecule due to its orbital angular momentum. The low-energy motion was attributed to reorientational motions of the NO molecules, in which only a small number (n ∼ 0.04) of NO molecules were found to participate. The NO molecules were confirmed to be paramagnetic even at 1 K. Ab-initio calculation indicated that the magnetic properties of the NO unit strongly depended on its surroundings, allowing the conformation of the fullerene cage to be estimated.

Orientation of adsorbed polar molecules (dipoles) in external electrostatic field

A model is proposed in the framework of classical electrostatics to describe the behavior of an adsorbed polar molecule near the plane interface between two insulators under the action of an external electrostatic field. The molecule is considered as a permanent point dipole that polarizes the interface and interacts with it through electrostatic image forces. The latter and the applied field try to reorient the dipole in a competitive manner. The system behavior turns out to be rather complicated: it may show a bistable character with a hysteresis (a switch). Such a switch can serve as an element in a memory network made of adsorbed molecules.

The Endofullerene HF@C60: Inelastic Neutron Scattering Spectra from Quantum Simulations and Experiment, Validity of the Selection Rule, and Symmetry Breaking

Accurate quantum simulations of the low-temperature inelastic neutron scattering (INS) spectra of HF@C₆₀ are reported for two incident neutron wavelengths. They are distinguished by the rigorous inclusion of symmetry-breaking effects in the treatment and having the spectra computed with HF as the guest, rather than H₂ or HD, as in the past work. The results demonstrate that the precedent-setting INS selection rule, originally derived for H₂ and HD in near-spherical nanocavities, applies also to HF@C₆₀, despite the large mass asymmetry of HF and the strongly mixed character of its translation-rotation eigenstates. This lends crucial support to the theoretical prediction made earlier that the INS selection rule is valid for any diatomic molecule in near-spherical nanoconfinement. The selection rule remains valid in the presence of symmetry breaking but is modified slightly in an interesting way. Comparison is made with the recently published experimental INS spectrum of HF@C₆₀. The agreement is very good, apart from one peak for which our calculations suggest a reassignment. This reassignment is consistent with the measured INS spectrum presented in this work, which covers an extended energy range.

The thermodynamic properties and molecular dynamics of [Li+@C60](PF6-) associated with structural phase transitions

Calorimetric and terahertz-far-infrared (THz-FIR) spectroscopic and infrared (IR) spectroscopic measurements were conducted for [Li+@C60](PF6-) at temperatures between 1.8 and 395 K. [Li+@C60](PF6-) underwent a structural phase transition at around 360 K accompanied by the orientational order-disorder transition of Li+@C60 and PF6-. The transition occurred in a step-wise manner. The total transition entropy (ΔtrsS) of 40.1 ± 0.4 J K-1 mol-1 was smaller than that of the orientational order-disorder transition in a pristine C60 crystal (ΔtrsS = 45.4 ± 0.5 J K-1 mol-1). Thus, the orientational disorder of Li+@C60 in the high-temperature phase of [Li+@C60](PF6-) was much less excited than that of the pristine C60 owing to the Coulombic interactions, which stabilized the ionic crystal lattice of [Li+@C60](PF6-). At T < 100 K, upon cooling, Li+ ions were trapped in two pockets on the inner surface of C60, and no phase transition was observed. Finally, the Li+ ions achieved a complete order at 24 K through antiferroelectric transition. The ΔtrsS value of 4.6 ± 0.4 J K-1 mol-1 was slightly smaller than R ln 2 = 5.76 J K-1 mol-1 expected for the two-site order-disorder transition. The extent of the Li+ motion in the C60 cage was related to the selection rule in the THz-FIR and IR spectroscopy of the C60 internal vibrations, because a C60 cage should be polarized by the Li+ ion. It is shown that the local symmetry of the caged molecule can be modified by the rotational or hopping motion of the encaged ions.

Rotational Motion and Nuclear Spin Interconversion of H2O Encapsulated in C60 Appearing in the Low-Temperature Heat Capacity

The heat capacity of H₂O encapsulated in fullerene C₆₀ is determined for the first time at temperatures between 0.6 and 200 K. The water molecule in H₂O@C₆₀ undergoes quantum rotation at low temperature, and the ortho-H₂O and para-H₂O isomers are identified by labeling the rotational energy levels with the nuclear spin states. A rounded heat capacity maximum is observed at ∼2 K after rapid cooling due to splitting of the rotational J _KaKc = 1₀₁ ground state of ortho-H₂O. This anomalous feature decreases in magnitude over time, reflecting the conversion of ortho-H₂O to para-H₂O. Time-dependent heat capacity measurements at constant temperature reveal three nuclear spin conversion processes: a thermally activated transition with E_a ≈ 3.2 meV and two temperature-independent tunneling processes with time constants of τ₁ ≈ 1.5 h and τ₂ ≈ 11 h.

Interactions between a water molecule and C60 in the endohedral fullerene H2O@C60

A water molecule encapsulated inside a C60 fullerene cage behaves almost like an asymmetric top rotor, as would be expected of an isolated water molecule. However, inelastic neutron scattering (INS) experiments show evidence of interactions between the water molecule and its environment [Goh et al., Phys. Chem. Chem. Phys., 2014, 16, 21330]. In particular, a resolved splitting of the 101 rotational level into a singlet and a doublet indicates that the water molecule experiences an environment of lower symmetry than the icosahedral symmetry of a C60 cage. Recent calculations have shown that the splitting can be explained in terms of electrostatic quadrupolar interactions between the water molecule and the electron clouds of nearest-neighbour C60 molecules, which results in an effective environment of S6 symmetry [Felker et al., Phys. Chem. Chem. Phys., 2017, 19, 31274 and Bačić et al., Faraday Discussions, 2018, 212, 547-567]. We use symmetry arguments to obtain a simple algebraic expression, expressed in terms of a linear combination of products of translational and rotational basis functions, that describes the effect on a water molecule of any potential of S6 symmetry. We show that we can reproduce the results of the electrostatic interaction model up to ≈12 meV in terms of two unknown parameters only. The resulting potential is in a form that can readily be used in future calculations, without needing to use density functional theory (DFT) for example. Adjusting parameters in our potential would help identify whether other symmetry-lowering interactions are also present if experimental results that resolve splittings in higher-energy rotational levels are obtained in the future. As another application of our model, we show that the results of DFT calculations of the variation in energy as a water molecule moves inside the cage of an isolated C60 molecule, where the water molecule experiences an environment of icosahedral symmetry, can also be reproduced using our model.

Effects of symmetry breaking on the translation-rotation eigenstates of H2, HF, and H2O inside the fullerene C60

Splittings of the translation-rotation (TR) eigenstates of the solid light-molecule endofullerenes M@C60 (M = H2, H2O, HF) attributed to the symmetry breaking have been observed in the infrared (IR) and inelastic neutron scattering spectra of these species in the past couple of years. In a recent paper [Felker et al., Phys. Chem. Chem. Phys., 2017, 19, 31274], we established that the electrostatic, quadrupolar interaction between the guest molecule M and the twelve nearest-neighbor C60 cages of the solid is the main source of the symmetry breaking. The splittings of the three-fold degenerate ground states of the endohedral ortho-H2, ortho-H2O and the j = 1 level of HF calculated using this model were found to be in excellent agreement with the experimental results. Utilizing the same electrostatic model, this theoretical study investigates the effects of the symmetry breaking on the excited TR eigenstates of the three species, and how they manifest in their simulated low-temperature (5-6 K) near-IR (NIR) and far-IR (FIR) spectra. The TR eigenstates are calculated variationally for both the major P and minor H crystal orientations. For the H orientation, the calculated splittings of all of the TR levels of these species are less than 0.1 cm-1. For the dominant P orientation, the splittings vary strongly depending on the character of the excitations involved. In all of the species, the splittings of the higher rotationally excited levels are comparable in magnitude to those for the j = 1 levels. For the levels corresponding to purely translational excitations, the calculated splittings are about an order of magnitude smaller than those of the purely rotational eigenstates. Based on the computed TR eigenstates, the low-temperature NIR (for M = H2) and FIR (for M = HF and H2O) spectra are simulated for both the P and H orientations, and also combined as their weighted sum (0.15H + 0.85P). The weighted sum spectra computed for M = H2 and HF match quantitatively the corresponding measured spectra, while for M = H2O, the weighted sum FIR spectrum predicts features that can potentially be observed experimentally.

Perspective: Accurate treatment of the quantum dynamics of light molecules inside fullerene cages: Translation-rotation states, spectroscopy, and symmetry breaking

In this perspective, I review the current status of the theoretical investigations of the quantum translation-rotation (TR) dynamics and spectroscopy of light molecules encapsulated inside fullerenes, mostly C₆₀ and C₇₀. The methodologies developed in the past decade allow accurate quantum calculations of the TR eigenstates of one and two nanoconfined molecules and have led to deep insights into the nature of the underlying dynamics. Combining these bound-state methodologies with the formalism of inelastic neutron scattering (INS) has resulted in the novel and powerful approach for the quantum calculation of the INS spectra of a diatomic molecule in a nanocavity with an arbitrary geometry. These simulations have not only become indispensable for the interpretation and assignment of the experimental spectra but are also behind the surprising discovery of the INS selection rule for diatomics in near-spherical nanocavities. Promising directions for future research are discussed.

Quantum fluctuations of a fullerene cage modulate its internal magnetic environment

To investigate the effect of quantum fluctuations on the magnetic environment inside a C₆₀ fullerene cage, we have calculated the nuclear magnetic shielding constant of protons in H₂@C₆₀ and HD@C₆₀ systems by on-the-fly ab initio path integral simulation, including both thermal and nuclear quantum effects. The most dominant upfield from an isolated hydrogen molecule occurs due to the diamagnetic current of the C₆₀ cage, which is partly cancelled by the paramagnetic current, where the paramagnetic contribution is enlarged by the zero-point vibrational fluctuation of the C₆₀ carbon backbone structure via a widely distributed HOMO-LUMO gap. This quantum modulation mechanism of the nuclear magnetic shielding constant is newly proposed. Because this quantum effect is independent of the difference between H₂ and HD, the H₂/HD isotope shift occurs in spite of the C₆₀ cage. The nuclear magnetic constants computed for H₂@C₆₀ and HD@C₆₀ are 32.047 and 32.081 ppm, respectively, which are in reasonable agreement with the corresponding values of 32.19 and 32.23 ppm estimated from the experimental values of the chemical shifts.

Explaining the symmetry breaking observed in the endofullerenes H2@C60, HF@C60, and H2O@C60

Symmetry breaking has been recently observed in the endofullerenes M@C₆₀ (M = H₂, HF, H₂O), manifesting in the splittings of the three-fold degenerate ground states of the endohedral ortho-H₂, ortho-H₂O and the j = 1 level of HF.

Rotational dynamics of Li+ ions encapsulated in C60 cages at low temperatures

Li⁺ ions encapsulated in fullerene C₆₀ cages (Li⁺@C₆₀) are expected to be suitable as molecular switches that respond to local electric fields. In this study, the rotational dynamics of Li⁺ ions in C₆₀ cages at low temperatures are experimentally revealed for the first time using terahertz absorption spectroscopy. In crystalline [Li⁺@C₆₀](PF₆^-), the Li⁺ ion rotates in the carbon cage even at 150 K. The rotational mode gradually changes into a librational mode below 120 K, which is associated with the localization of Li⁺ ions due to the electrostatic interactions with its screening image charge on the C₆₀ cage as well as with the neighboring Li⁺@C₆₀ and PF₆^- ions. A simple rotational/librational energy scheme for the Li⁺ ions successfully explains the spectroscopic results, and the potential of Li⁺@C₆₀ as a molecular switch is discussed based on the energy scheme.

Symmetry-breaking in the H2@C60 endofullerene revealed by inelastic neutron scattering at low temperature

The fine structure of the rotational ground state of molecular ortho-hydrogen confined inside the fullerene cage C₆₀ is investigated by inelastic neutron scattering (INS).

Symmetry-breaking in the endofullerene H2O@C60 revealed in the quantum dynamics of ortho and para-water: a neutron scattering investigation

The splitting of the ortho-H₂O ground state is clearly revealed by inelastic neutron scattering.

Spectroscopy of light-molecule endofullerenes

Molecular endofullerenes are supramolecular systems consisting of fullerene cages encapsulating small molecules. Although most early examples consist of encapsulated metal clusters, recently developed synthetic routes have provided endofullerenes with non-metallic guest molecules in high purity and macroscopic quantities. The encapsulated light molecule behaves as a confined quantum rotor, displaying rotational quantization as well as translational quantization, and a rich coupling between the translational and rotational degrees of freedom. Furthermore, many encapsulated molecules display spin isomerism. Spectroscopies such as inelastic neutron scattering, nuclear magnetic resonance and infrared spectroscopy may be used to obtain information on the quantized energy level structure and spin isomerism of the guest molecules. It is also possible to study the influence of the guest molecules on the cages, and to explore the communication between the guest molecules and the molecular environment outside the cage.

Infrared spectroscopy of small-molecule endofullerenes

Hydrogen is one of the few molecules that has been incarcerated in the molecular cage of C₆₀ to form the endohedral supramolecular complex H₂@C₆₀. In this confinement, hydrogen acquires new properties. Its translation motion, within the C₆₀ cavity, becomes quantized, is correlated with its rotation and breaks inversion symmetry that induces infrared (IR) activity of H₂. We apply IR spectroscopy to study the dynamics of hydrogen isotopologues H₂, D₂ and HD incarcerated in C₆₀. The translation and rotation modes appear as side bands to the hydrogen vibration mode in the mid-IR part of the absorption spectrum. Because of the large mass difference of hydrogen and C₆₀ and the high symmetry of C₆₀ the problem is almost identical to a vibrating rotor moving in a three-dimensional spherical potential. We derive potential, rotation, vibration and dipole moment parameters from the analysis of the IR absorption spectra. Our results were used to derive the parameters of a pairwise additive five-dimensional potential energy surface for H₂@C₆₀. The same parameters were used to predict H₂ energies inside C₇₀. We compare the predicted energies and the low-temperature IR absorption spectra of H₂@C₇₀.

Quantum rotation and translation of hydrogen molecules encapsulated inside C₆₀: temperature dependence of inelastic neutron scattering spectra

The quantum dynamics of a hydrogen molecule encapsulated inside the cage of a C60 fullerene molecule is investigated using inelastic neutron scattering (INS). The emphasis is on the temperature dependence of the INS spectra which were recorded using time-of-flight spectrometers. The hydrogen endofullerene system is highly quantum mechanical, exhibiting both translational and rotational quantization. The profound influence of the Pauli exclusion principle is revealed through nuclear spin isomerism. INS is shown to be exceptionally able to drive transitions between ortho-hydrogen and para-hydrogen which are spin-forbidden to photon spectroscopies. Spectra in the temperature range 1.6≤T≤280 K are presented, and examples are given which demonstrate how the temperature dependence of the INS peak amplitudes can provide an effective tool for assigning the transitions. It is also shown in a preliminary investigation how the temperature dependence may conceivably be used to probe crystal field effects and inter-fullerene interactions.

Comparative NMR properties of H2 and HD in toluene-d8 and in H2/HD@C60

Spin-lattice relaxation times, T(1), have been measured from 200-340 K for the protons in H(2) and HD molecules dissolved in toluene-d(8) and incarcerated in C(60). It is found that HD relaxes more slowly than H(2) in both environments and at all temperatures, as expected from the smaller values of the spin-rotation and dipole-dipole coupling in HD compared to H(2). More detailed analysis using models developed to describe relaxation in both condensed media and the gas phase indicates that transitions among the rotational states of H(2) occur at a rate similar to those of HD in both toluene-d(8) solution and in C(60), in contrast to the situation in gas phase collisions between hydrogen and He or Ar, where the lifetimes of rotational states of HD are markedly shorter than those for H(2). Measurements of the relative (1)H chemical shifts of H(2) and HD, the coupling constant J(HD), and the widths of the HD peaks at various temperatures revealed only small effects with insufficient accuracy to warrant more detailed interpretation.