## Abstract

The quantum dynamics of a hydrogen molecule encapsulated inside the cage of a C_{60} 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.

## 1. Introduction

In recent years, advances in organic synthesis have led to a practical realization of a ubiquitous concept in quantum mechanics, namely a quantum particle entrapped by a confining potential. The synthetic techniques pioneered by the groups of Komatsu, Murata and Rubin [1–4] begin with the opening and stabilization of an orifice in a C_{60} fullerene molecule by a series of chemical reactions, whereupon a small molecule such as H_{2} or H_{2}O is physically inserted inside the fullerene cage through the opening. The synthesis is completed with a further series of controlled chemical reactions that seal the orifice resulting in the small molecule being permanently trapped inside a C_{60} cage. The dextrous and highly evolved synthetic procedures have been described by the evocative phrase ‘molecular surgery’. Good yields with cage filling factors approaching 100 per cent have now provided samples in sufficient volume to make inelastic neutron scattering (INS) experiments a practical reality, thus enabling the quantum dynamics of the entrapped molecule to be studied in detail. Alongside nuclear magnetic resonance (NMR) [5–7] and infrared (IR) spectroscopy [8–11], INS is one of a range of experimental techniques that have been employed in recent years to investigate the quantum rotation and translation of the confined molecules. Previous INS studies have been conducted on the model complex H_{2}@C_{60} [11,12] and its isotopomer HD@C_{60} [11,13] which have highly symmetric confining potentials, as well as H_{2}@ATOCF [14] which is an open-cage derivative with an anisotropic cage potential. Recently, the first collaborative investigation of H_{2}O@C_{60} was conducted using NMR, IR and INS [15] where the quantum rotation of the isolated water molecule and its nuclear spin isomers was studied in detail. This revealed a splitting of the ground state of *ortho*-water which is attributed to a symmetry breaking of the water environment, possibly associated with interactions of the electric dipole moment.

NMR involves a change of nuclear spin state and conventionally this transition is driven by applying electromagnetic radiation in the radiofrequency region of the spectrum. However, in the case of symmetrical quantum rotors, the requirements of the Pauli exclusion principle (PEP) mean there is an entanglement of the spatial and nuclear spin wave functions that characterize the system. Therefore, in addition to pure Zeeman transitions, there exist transitions that involve a simultaneous change in spatial and spin states of the quantum rotor. These are strongly forbidden for excitation by photons, but they become allowed when interactions are present that connect space and spin.

In this paper, we shall be considering symmetrical rotors that comprise hydrogen atoms with ^{1}H nuclear isotopes. Here, the nuclei are indistinguishable fermions (protons) so the antisymmetry principle, which underpins the PEP, determines the allowable space–spin eigenstates. When spatial degrees of freedom become entangled with nuclear spin in this way, splittings between quantum rotor eigenstates involving a nuclear spin flip will invariably be substantially larger than the nuclear Zeeman energy. This is a result of an exchange interaction arising from the fundamental indistinguishability of the nuclei that constitute the quantum rotor. It effectively gives rise to a zero-field splitting. There are a number of examples, the best known of which is the spin-symmetry of molecular hydrogen, H_{2}, or dihydrogen. This, the simplest of molecules, is characterized by two spin isomers, *ortho*-hydrogen (*o*-H_{2}) with total nuclear spin *I*=1 and *para*-hydrogen (*p*-H_{2}) with total nuclear spin zero. The splitting between the ground states of *o*-H_{2} and *p*-H_{2} is determined by the rotational energy and has magnitude *ΔE*=2*B*, where is the rotational constant and *I*_{m} is the moment of inertia. Another example is that of the methyl rotor, CH_{3}, where the spin-symmetry of the space–spin eigenstates is isomorphous with the irreducible representations of the C_{3} symmetry group [16,17]. In the condensed phase, CH_{3} rotation is usually hindered by a potential barrier and the rotational dynamics is characterized by coherent quantum tunnelling between wells with the discrete frequency *ν*_{t}. As a result, the energy splitting between the spin-symmetry species (or nuclear spin isomers) of CH_{3} is simply *ΔE*=*hν*_{t}, where *ν*_{t} is an inverse exponential function of the barrier height. Hence, the exchange splitting can be much smaller than the rotational constant, but invariably larger than the ^{1}H Zeeman splitting. Then, magnetic dipole–dipole interactions (nuclear–nuclear or electron–nuclear) may have sufficient magnitude relative to *hν*_{t} to mediate space–spin transitions with significant probability and hence make them accessible to NMR spectroscopy [17–20]. Likewise, hindered rotation of dihydrogen has been shown to give rise to tunnelling splittings that reveal themselves in NMR spectra when coherent tunnelling and an associated exchange interaction are present [21]. These are often termed ‘quantum exchange J-couplings’ since the effect of tunnelling is equivalent to that of a J-coupling in NMR, but the magnitudes are usually much larger.

For H_{2} entrapped inside a fullerene molecule, the cage potential has high symmetry (icosahedral) which for the lowest lying rotational and translational states is effectively isotropic. In this case, there is no barrier to rotation, tunnelling is absent and the nuclear spin isomers are separated in energy by multiples of *B* which is meV in magnitude. Electromagnetic photons have the wrong symmetry to drive transitions that involve simultaneous changes in space and spin eigenstates. So, for example, far-IR, vibrational and magnetic resonance spectroscopies are able to interrogate the eigenstates of H_{2} so long as there is no change in spin-isomer species; this necessarily limits the information content of such experiments. However, in scattering from an atomic nucleus, the interaction felt by a neutron is the one that is mediated by spin and is therefore magnetic in character. It transpires that the technique of INS has the capacity to change space and spin states simultaneously and is therefore ideally suited to the study of quantum rotors; the interaction is sufficiently strong to mediate changes in energy of order *B* while at the same time flipping a nuclear spin. In this sense, one may view the INS experiment as a form of magnetic resonance but one which is capable of mediating energy exchanges that are substantially larger than Zeeman splittings in practically accessible magnetic fields. Indeed, since the splittings do not require the intervention of an external magnetic field, it might be viewed as a form of magnetic resonance that can be conducted in zero field. In driving transitions that involve spin-isomer exchange and because it accesses a different energy window, INS is also highly complementary to photon spectroscopies and can provide important insights as part of a collaborative investigation [11,15].

Extending earlier INS investigations of hydrogen endofullerenes, in this paper, we revisit H_{2}@C_{60} but investigate in more detail the temperature dependence of the INS spectrum. This provides further insight into both the translation–rotation (TR) states of entrapped dihydrogen and the practical aspects of INS.

## 2. Theoretical

### (a) Translation–rotation eigenstates of entrapped dihydrogen

The eigenstates of a caged H_{2} molecule have been described in detail in a variety of papers in the past few years [22–24]. Therefore, it is only necessary to summarize the main features here. For an isolated dihydrogen molecule, the rotational energy levels are given by *E*_{rot}=*BJ*(*J*+1), where *J*=0,1,2,… is the rotational quantum number and *B*=7.37 meV. The rotational degeneracy is *g*_{r}=2*J*+1 and the corresponding sublevels are characterized by the quantum number *m*_{J}=−*J*,−*J*+1,…,+*J*.

The translation of the H_{2} molecule inside its fullerene cage approximates to a three-dimensional harmonic oscillator. The translational energy is characterized by the principal quantum number *n*=0,1,2,… and the orbital angular momentum quantum number *l*=*n*,*n*−2,…, 1 (0) for odd (even) *n*. The cage symmetry (icosahedral) is close to isotropic so in the absence of further interactions, the low-lying translational levels retain a high degree of degeneracy, *g*_{t}=2*l*+1.

The dihydrogen molecule has translational and rotational angular momenta and these couple in such a way that we can characterize the combined TR states by the quantum number *λ*=*l*+*J*, *l*+*J*−1,…,|*l*−*J*|. This TR coupling raises some degeneracies and gives rise to splittings. These TR states are labelled by *λ* and have degeneracy *g*_{λ}=2*λ*+1.

Owing to the PEP and the antisymmetry principle, the spatial eigenstates are entangled with the spin states. This leads to the identification of the nuclear spin isomers *p*-H_{2} (*I*=0, *J*=0,2,… even) and *o*-H_{2} (*I*=1, *J*=1,3,… odd). Therefore, the ground states of *p*-H_{2} and *o*-H_{2} are separated in energy by *ΔE*_{rot}=2*B*. For pure H_{2}@C_{60}, there are no significant paramagnetic interactions that can efficiently interconvert the *p*-H_{2} and *o*-H_{2} species. Therefore, *o*-H_{2} appears as a metastable moiety and the two spin isomers retain their distinct and independent identities.

The TR levels of H_{2}@C_{60} are represented in figure 1. States below about 70 meV have been determined by INS in an earlier paper [12]. States above this energy have been inferred from the computational study by Xu *et al.* [24] which was shown in our earlier paper to provide a good representation of the TR energy levels (corrections have been applied for the appropriate value of *B* in the ground vibrational state). The states are labelled (*J*, *n*, *l*, *λ*) but for clarity the value of *λ* is not shown explicitly in figure 1.

### (b) Inelastic neutron scattering

The INS theory has been discussed in some detail in published papers [12,25–28], so again we shall only summarize the main features. The differential scattering cross section defines the INS spectrum and is given by
2.1where *E*_{i}, **k**_{i} and *E*_{f}, **k**_{f} are the energies and wavevectors of the incident and scattered neutrons, respectively. There is inelastic scattering of the neutron by H_{2}, so we define the energy transfer relative to the neutron as and the momentum transfer as .

The interaction potential *V* defines the matrix element in equation (2.1),
2.2where *b*_{c} and *b*_{i} are scattering lengths of the ^{1}H nucleus which are proportional to the square roots of the coherent and incoherent scattering cross sections, respectively. There is a scalar coupling between neutron spin ** σ** and

^{1}H spin

**I**

_{α}, where

*α*=1,2 labels the two

^{1}H nuclei in the H

_{2}molecule which have position vectors

**R**

_{α}=

**R**

_{0}+(−1)

^{α}

**/2. Here,**

*ρ***R**

_{0}and

**are vectors defining the centre of mass of H**

*ρ*_{2}and the inter-proton distance, respectively. In equation (2.1), |

*i*〉=|

*σ*

_{i}〉|

*ψ*

_{i}〉 is the initial state of the neutron–H

_{2}system and |

*f*〉 is its final state, where |

*σ*〉 is the neutron spin wave function and |

*ψ*〉 is the molecular wave function that characterizes the energy levels described in §2a. Finally,

*p*

_{i}is the statistical weight of the initial state which is determined by the population of the molecular wave function |

*ψ*

_{i}〉. However, it should be noted that the

*o*-H

_{2}and

*p*-H

_{2}do not interconvert on the experimental time scale (at least one week), so they are distinguishable species that are not in equilibrium. Therefore, Boltzmann populations within the manifold of

*ortho*states are acquired separately from

*para*states.

A special feature of neutron interactions with H_{2} is that the scattering from the two ^{1}H nuclei is correlated. This results from the highly quantum nature of dihydrogen and cross terms in the summation of equation (2.2) are non-zero. Then for each molecule
2.3This is highly significant and worthy of deeper investigation. For ^{1}H, the coherent cross section is only 2% of the incoherent cross section. In equation (2.3), the first term scales as *b*_{c} but given *b*_{c}≪*b*_{i} its contribution to the INS spectrum is largely negligible. Examining the spin operators, we conclude that the second term in equation (2.3) can only drive transitions that involve no net change in total nuclear spin. Therefore, owing to the spin-symmetry, only transitions that solely entail changes in translational energy (and no change of rotational energy) can arise from the second term. Furthermore, since the total nuclear spin of *p*-H_{2} is zero, only the translational transitions of *o*-H_{2} can have finite amplitude.

From further examination of the spin operators, we conclude that the third term in equation (2.3) has the capacity to drive transitions between *o*-H_{2} and *p*-H_{2}. These involve simultaneous changes in spatial and spin states of the H_{2} molecule. They can be purely rotational transitions, or a combination of translation and rotation. Since they necessarily involve a change of nuclear spin, they may be viewed as analogous to NMR but where the splitting is determined by interactions of the spatial wave function as opposed to purely those of the spin wave function.

Since both the second and third terms in equation (2.3) both scale as *b*_{i}, we conclude that the spectral features that arise from the two will have broadly similar amplitude.

## 3. Experimental details

The experiments were conducted using two time-of-flight INS spectrometers at the Institut Laue-Langevin, Grenoble. IN4C gave access to a range of energy transfer in both neutron energy (NE) gain and NE loss regions of the spectrum. This spectrometer is situated on a thermal neutron beam and combines high neutron flux with medium resolution. The primary spectrometer comprises a curved highly ordered pyrolytic graphite monochromator from which mono-energetic neutrons were selected by Bragg diffraction. Incident neutron wavelengths could be selected in the range 1.1≤*λ*_{n}≤1.8 Å, the choice determining the resolution and energy transfer range accessible. The pulsed structure of the neutron beam is provided by a Fermi chopper. The secondary spectrometer comprises a 2 m flight chamber with an array of detectors. The energy transfer is determined by the time-of-flight between arrival of the neutron pulse at the sample and the detection of a scattered neutron in a detector.

IN5 operates on a cold neutron beam line and in the set-up used that optimizes spectral resolution, the NE gain region of the spectrum is accessed. The primary spectrometer comprises six pairs of counter rotating choppers and the incident wavelength in the range 1.8≤*λ*_{n}≤20 Å is selected by setting the rotation rate and phase of the different choppers. The flight chamber is 4 m and neutron arrival is detected in an array of pixilated position sensitive detectors. The large surface area (30 m^{2}) occupied by the detector bank enables neutrons scattered over a wide solid angle to be detected, enhancing the sensitivity of the instrument.

Spectra are plotted as a function of energy transfer *ΔE* which is defined relative to the sample. This sign convention means that a scattering event with negative energy transfer involves NE gain corresponding to a transition in the sample from a high energy state to one of the lower energy states. Conversely, positive energy transfer involves NE loss and absorption of energy by the sample.

For both IN4C and IN5, the resolution function is well represented by a Gaussian. However, for both instruments the full width at half maximum (FWHM) of the Gaussian varies systematically with NE transfer. Spectra were recorded as a function of energy transfer and momentum transfer, *κ*. The latter is not the subject of this paper so to maximize the counting statistics in the energy transfer domain, the data were integrated over the available range of *κ*. For the IN5 experiments (*λ*=8 Å), this corresponded approximately to 2≤*κ*≤3.2 Å^{−1} and for the IN4C (*λ*=1.6 Å) 1.5≤*κ*≤6 Å^{−1}.

The 250 mg powdered H_{2}@C_{60} sample used in these experiments is the same as that reported in an earlier publication [12] where further details on the sample preparation are recorded. As will be reported in a later section, during the course of these INS experiments, this endofullerene sample was found to have lower crystal symmetry than that adopted by pure forms of C_{60} which is cubic. This suggests that there may be solvent inclusions remaining in the crystal as an exohedral impurity. However, this material does not contribute significantly to the INS spectrum.

By recording a mass-weighted INS spectrum of a 1 g sample of C_{60} powder that has empty cages, scattering from the C_{60} cage was shown to be unimportant in the energy transfer ranges considered in this investigation.

## 4. Results and discussion

### (a) The low temperature inelastic neutron scattering spectrum

At the lowest temperatures, the spectrum is determined by transitions originating in the ground states of either *p*-H_{2} or *o*-H_{2} as these are the only occupied states. In figure 2, INS spectra recorded at *T*<1.6 K are presented. Figure 2*a*,*b* were recorded on IN4C with incident wavelengths *λ*_{n}=1.6 Åand 1.2 Å, respectively. Figure 2*c* was recorded on IN5 with *λ*_{n}=8 Å. The spectrum is sparse, comprising discrete peaks that reveal quantization of both rotational and translational energy. Figure 2 shows how the accessible energy range and resolution are dependent on the spectrometer and its configuration.

The assignment of the peaks in figure 2 has been described in an earlier paper [12]; however to interpret the temperature dependence spectra, it is expedient to revisit that here. The sole peak in NE gain (*ΔE*_{0−1}=−14.7±0.1 meV) arises from the transition from the ground state of *o*-H_{2} (*J*,*n*,*l*,*λ*)=(1,0,0,1) to the ground state of *p*-H_{2} (0,0,0,0); this is a rotational transition connecting *J*=0 and *J*=1 and we refer to it as the principal rotational peak. As has been reported in earlier publications, the presence of this peak in NE gain shows how *o*-H_{2} exists as a metastable species that is not in equilibrium with *p*-H_{2}. There is no mechanism that is sufficiently strong to induce *ortho*–*para* conversion. The lifetime of *o*-H_{2} is very long since no conversion is evident over the time scale of the experiments which is many days. The transition in the opposite direction from (0,0,0,0) to (1,0,0,1) appears at *ΔE*_{0−1}=+14.7 meV in NE loss.

The first translational transition of *o*-H_{2} from (1,0,0,1) to (1,1,1,{*λ*=0,1,2}) appears as a triplet centred on approximately *ΔE*=+22.5 meV. The *λ*=1 and *λ*=2 components have energy transfers of 22.17 and 22.88 meV, respectively. As was originally noted [12], the *λ*=0 component with *ΔE*=24.3±0.1 meV has much lower INS amplitude than the other two.

The peak centred on +8.08±0.03 meV is a transition that entails a simultaneous change in rotational and translational state. It converts *p*-H_{2} (0,1,1,1) to *o*-H_{2} (1,0,0,1) with an upward change in rotational energy and a downward change in translational energy. Finally, the peak centred on +29.2±0.2 meV is of particular interest to the temperature dependence experiments to be described later; this is a pure rotational transition from *o*-H_{2} (1,0,0,1) to *p*-H_{2} (2,0,0,2). Peaks at higher energy transfer have the following assignments: +32.0±0.3 meV is from *o*-H_{2} (1,0,0,1) to *p*-H_{2} (0,2,2,2); +37.8±0.1 meV is from *p*-H_{2} (0,0,0,0) to *o*-H_{2} (1,1,1,{*λ*=0,1,2};+46.6±0.1 meV is from *o*-H_{2} (1,0,0,1) to *o*-H_{2} (1,2,2,{*λ*=1,2,3}) [12]. In all the above, the error bars represent the systematic uncertainty.

### (b) The temperature dependence of the neutron energy gain spectrum

In an earlier paper [12], we considered in depth the temperature dependence of the INS spectrum in the region of the −14.7 meV rotational line. In this paper, we shall investigate the temperature dependence of the spectrum in the range 32≥|*ΔE*|≥18 meV, in particular emphasizing the energy transfer region of the first translational peak.

INS spectra have been recorded on IN5 in the temperature range 65≤*T*≤280 K with incident wavelength *λ*_{n}=8 Å. These are presented in figure 3 for −32≤*ΔE*≤−18 meV. The translational multiplet centred on approximately −22.5 meV begins to emerge at 40 K and systematically grows in intensity with increasing temperature until approximately 200 K when it begins to slowly diminish. This is the energy gain partner of the translational peak observed at low temperature in the NE loss spectra of figure 2. Other peaks which are first to emerge with increasing temperature are centred on approximately −24.3 and −29.2 meV. The latter is the NE gain partner of the second pure rotational peak of figure 2 where *J*=2→1. At the higher temperatures, additional overlapping peaks appear in between the −24.3 and −29.2 meV peaks.

Studying the energy level diagram of figure 1, we can identify a significant number of transitions that will potentially give rise to peaks in the energy transfer range of figure 3. Owing to the TR fine structure, this number amounts to approximately 115. This is too large to consider any ‘profile refinement’ analysis of the INS bands since the information content of the spectrum is insufficient to consider fitting with such a large number of components. Therefore, we shall adopt a more pragmatic approach and fit with seven Gaussian components with fixed energy transfer. Two of these will be the *λ*=1, 2 components of the translational peak centred on −22.17 and −22.88 meV. A third is centred on −29.2 meV corresponding with the energy transfer of the *J*=2→1 pure rotational peak. The remaining four Gaussians are centred on −24.3, −25.5, −26.4 and −27.5 meV which occupy approximately equal spacing in energy transfer in the region in between.

The aim of the fitting procedure is to determine the temperature dependence of the amplitudes of the Gaussian components, so it is expedient to fix the peak widths as well as the energy transfers in the fitting model. The −22.17 and −22.88 meV Gaussians are modelled with widths of 0.9 meV (FWHM) which is our best estimate of the resolution function at this energy transfer; this is the first pure translational mode and there are few additional components likely to contribute to the scattering in this energy transfer region. The remaining five Gaussians are modelled with broader peaks seeking to model two effects: first, the resolution function of IN5 increases systematically with increasing |*ΔE*|; second, and more importantly, because of the TR fine structure, multiple transitions covering a range of energy transfer will contribute to each component. Taking into consideration the expected pattern of TR fine structure, the −24.3 meV peak was modelled with width 1.5 meV (FWHM), whereas the −25.5, −26.4 and −27.5 meV peaks were modelled with widths 1.0 meV and the −29.2 meV peak with width 1.7 meV.

For each measured temperature, the INS spectra have been fitted with the above seven Gaussian model in the energy transfer range −32≤*ΔE*≤−18 meV. The fitted envelope of the scattering amplitude (solid line, figure 3) gives a good representation of the observed spectra at all temperatures. The peak amplitudes are the only floated parameters, and in figure 4 they are plotted as a function of sample temperature; here, the values are normalized with the amplitude of the elastic line to eliminate other physical factors, including the Debye–Waller factor, from the temperature dependence. In figure 4, the peak amplitudes are observed to vary smoothly and systematically with temperature.

Assuming that the transition energies remain temperature independent, the normalized amplitude of an INS peak depends only on the statistical weight of the initial state, *p*_{i}, equation (2.1). At equilibrium, this will correspond to the Boltzmann population of that state. Accordingly, the populations have been calculated with
4.1where *E*_{m} and *g*_{m} are, respectively, the energy and degeneracy of the levels labelled *m*, which correspond with those plotted in figure 1. *Z* is the partition function and the superscript signifies either the *ortho* or *para* species which acquire their separate equilibriums. The calculated populations will now be compared with the fitted Gaussian amplitudes.

In figure 4, the amplitudes of the −22.17 and −22.88 meV Gaussians follow a similar trajectory, initially appearing at ≈50 K, rising steeply at ≈80 K and peaking in amplitude at ≈200 K. This is consistent with states that are approximately 22 meV (≡255 K) above their ground state, as indeed is the case for the first translationally excited levels of *J*=1 *o*-H_{2}, labelled (1,1,1,{*λ*=0,1,2}). The −22.78 meV peak has larger amplitude than the −22.17 meV peak, and this can partly be attributed to their respective degeneracies, 5 and 3 [24]. Scaling by numerical factors, the Boltzmann populations, equation (4.1), of the *λ*=1,2 components of (1,1,1) are plotted in figure 4, providing a good representation of the experimental data. The amplitude of the −22.88 meV peak (*λ*=2) is systematically larger than that of the −22.17 meV peak (*λ*=1) and the relative scaling factors suggest that the transition probability of the *λ*=2 component of the multiplet, as measured by the square magnitude of the matrix element of *V* in equation (2.1), is approximately 1.16 times that of the *λ*=1 component. It is already known from the low temperature spectrum that both have significantly larger transition probability than the *λ*=0 component of the same multiplet.

As noted in earlier papers [12–14], the temperature dependence of the INS peak amplitudes provides a highly effective tool for assigning the corresponding transitions. Indeed, in figure 4, it is evident that the Gaussian amplitudes have distinct and distinguishable behaviour in this respect. The amplitudes of the −24.3 and −29.2 meV peaks each grow at similar rates with increasing temperature, displaying steeply rising behaviour at ≈120 K. This suggests that the initial states for both must lie approximately 46 meV (535 K) above their respective ground states. The −24.3 meV peak grows at a rate that is consistent with a second excited translational state; either the (0,2,2,2) and (0,2,0,0) levels of *p*-H_{2} with energies of 47.1 and 49.5 meV, respectively, or the (1,2,2,{*λ*=1,2,3}) levels of *o*-H_{2} with energy ≈61 meV. Given the only downward transitions with the required energy from either of the stated *p*-H_{2} states lead only to another *p*-H_{2} state, these alternatives must be ruled out since they cannot have significant intensity (§2b). Therefore, we conclude that the observed peak at −24.3 meV must correspond to (1,2,2,{*λ*=1,2,3})→(1,1,1,{*λ*=0,1,2}), representing a downward transition in translational state for *o*-H_{2}. Owing to the TR fine structure, there are nine possible components and since these are not resolved, it is not possible to identify the individual contributions these make to the overall amplitude. However, using equation (4.1), a typical temperature profile has been calculated from the Boltzmann populations and when scaled by an appropriate factor (short-dashed line in figure 4), the curve shows good correspondence with the amplitude of the fitted Gaussian.

The temperature dependence of the −29.2 meV peak cannot be explained as a single transition. A significant proportion of the scattering intensity comes from the pure rotational transition (2,0,0,2)→(1,0,0,1), and in figure 4, a representative scaled Boltzmann factor is indicated with a dashed-dotted line. To fully explain the amplitude of this peak, an additional transition is required. Inspection of the energy level diagram, figure 1, shows the principal candidate is a multiplet of transitions with energy transfer in the region of −29 meV characterized by (2,2,2,*λ*)→(1,2,2,*λ*). These represent the second excited translational state undergoing a change in rotational state from *J*=2 to *J*=1. However, the multiplicity means there may be as many as 28 contributing transitions. A good representation of the observed curve is shown as a double dashed-dotted line in figure 4; this represents an admixture of Boltzmann populations appropriate to the transitions (2,0,0,2)→(1,0,0,1) and (2,2,2,*λ*)→(1,2,2,*λ*). It is clear that both types of transition are closely related, since they both involve pure rotational transitions *J*=2 to *J*=1.

The temperature dependence of the Gaussian components with energy transfer −25.5 and −26.4 meV grow with similar trajectories, beginning to appear at significantly higher temperature than those considered above. Elementary calculations show that the temperature profile of these curves is consistent with states that lie approximately 90 meV above their ground states. Candidate states are (3,1,1,*λ*) of *o*-H_{2} and (2,2,2,*λ*) of *p*-H_{2}. However, with increasing energy, the multiplicity of the states grows and there are an increasingly large number of possible transitions that could contribute to NE gain peaks with the requisite energy transfer. Therefore, a definitive assignment of these peaks is not currently possible other than in these very general terms.

The −27.5 meV peak begins to appear with significant amplitude at higher temperatures than peaks discussed previously and therefore originates in yet higher energy states. An explicit assignment is not possible, given the spectrometer resolution available, and detailed simulations of the INS spectra, such as those conducted by the group of Bačić and co-workers [27,28], would be required to gain deeper insight into the spectra recorded at the higher temperatures.

### (c) The temperature dependence of the neutron energy loss spectrum

The NE loss spectrum has been studied as a function of temperature using the IN4C spectrometer with incident wavelength *λ*_{n}=1.6 Å. In NE loss, this configuration provides access to energy transfers below 28.2 meV. Our interest lies principally with the region of the first translational line so we shall consider the range 18≤*ΔE*≤28.2 meV. Spectra recorded at seven temperatures in the range 1.65≤*T*≤160 K are presented in figure 5. The principal component is the first translational triplet centred on approximately 22.5 meV; this is (1,0,0,1)→(1,1,1,*λ*). With increasing temperature, this feature loses amplitude while additional peaks begin to appear at approximately 24 and 26 meV.

The spectra have been fitted with four Gaussians centred on 22.17, 22.88, 24.3 and 25.5 meV, representing the same fitting model as for the NE loss spectrum in §4b, except for the two components which do not contribute to the temperature range studied and one which is outside the energy range. The Gaussian widths (FWHM) are fixed with values of 1.2 meV so the only floated variables are the Gaussian amplitudes. The overall fit and the fitted Gaussian components are shown in figure 5.

The spectral assignments of the Gaussians have been made in §4b. The first translational transition (1,0,0,1)→(1,1,1,*λ*) is a triplet, but the third component centred on 24.3 meV has very small intensity. For clarity, the two principal components of this feature are shown as a single peak in figure 5. This transition loses amplitude with increasing temperature as excited states of *ortho*-H_{2} become populated. Superimposed on the 24.3 meV member of the translational triplet is another peak which was assigned to the multiplet (1,2,2,*λ*)→(1,1,1,*λ*) on the basis of the NE gain spectrum. This is a pure translational transition within *J*=1 that connects the first and second excited states. It is interesting to compare the temperature dependence of the amplitudes of this peak in NE loss and NE gain, figure 6. The behaviours are quite distinct. In NE loss, the amplitude begins to rise at approximately 60 K, whereas in NE gain, the amplitude begins to rise at a significantly higher temperature, approximately 100 K. This arises because the INS cross section is proportional to the statistical weight *p*_{i} of the initial state. Therefore, for the upward transition (NE loss), *p*_{i} is determined by the population of (1,1,1,*λ*), whereas, as has already been shown for the downward transition (NE gain), *p*_{i} is determined by the population of (1,2,2,*λ*). The solid lines in figure 6 are proportional to the calculated populations, equation (4.1), which give a good representation of the observed behaviour. (The amplitudes have arbitrary units and there are unknown scaling factors relating the amplitudes of the IN4C and IN5 spectra.) This reinforces and extends the idea that the temperature dependence of the spectrum provides a valuable tool for assignment of the INS peaks. In this particular case, it confirms the assignment made in the previous section, but more generally, it shows how measuring the INS amplitudes in NE loss and NE gain provides a signature of both contributing states involved in a transition.

### (d) Rotational fine structure: the rotational peak studied at low temperature

Using heat capacity measurements at low temperature, Kohama *et al.* [29] have suggested there is evidence for a splitting of the order of 0.1–0.2 meV of the principal rotational peak centred on |*ΔE*_{0−1}|=14.7 meV. The C_{60} cage is icosahedral so the potential energy surface has too high a symmetry to be responsible for such a splitting. Therefore, one has to look to the symmetry of the crystal field and inter-fullerene interactions to identify a possible mechanism for such a splitting. A tentative model envisages inhomogeneities in the molecular environment arising from the distribution of *ortho*-H_{2} and *para*-H_{2}. Alternatively, the crystal symmetry may be responsible: at low temperature, the space group of C_{60} is , and Kohama *et al.* [29] suggest there may be ‘two types of orientation of C_{60}’ and that ‘these orientations can lead to two different crystal fields’. We have sought direct spectral evidence for such a splitting using INS.

The splitting of order *ΔE*_{fs}≈0.2 meV is too small to be detected directly given the available resolution on IN5. Nevertheless, a splitting of this order may still lead to detectable changes in the INS peak properties when the Boltzmann populations of its different components are responding sensitively to changes in temperature, namely where *ΔE*_{fs}≈*k*_{B}*T*. Accordingly, the rotational peak has been studied with the highest resolution available on IN5 in the temperature range 1.6≤*T*≤40 K (incident neutron wavelength *λ*_{n}=8 Å). The spectra indeed display a small shift with temperature, and the peak centres have been determined by fitting with a Gaussian. In figure 7, the shift, (|*ΔE*_{0−1}|−14.7) meV, is plotted showing the decrease in |*ΔE*_{0−1}| with decreasing temperature.

The temperature range in which the shift is observed is consistent with fine structure of the order of tenths of meV. However, further more detailed and quantitative analysis is problematic, because it emerged from an analysis of the neutron diffraction detectors on IN5 that the sample has crystal symmetry that is lower than the cubic class adopted by pure forms of C_{60}. This may arise from solvent inclusions. Therefore, owing to its dependence on the statistical weight of the initial state, the INS technique has shown itself to be capable of detecting small splittings in the rotational line that may reveal underlying fine structure arising from the crystal field. However, in order to attribute these to inter-fullerene interactions in a detailed and systematic way requires a sample of the necessary volume with higher purity than has been available to date.

## 5. Concluding remarks

We have shown how neutron scattering interactions with the ^{1}H nuclei have the capacity to simultaneously drive changes in spatial and spin states. This is particularly advantageous in the field of quantum rotors where the symmetry of the system naturally gives rise to nuclear spin isomers that are characterized by entangled space–spin states. In H_{2}@C_{60}, many of the observed INS transitions involve *ortho*–*para* conversion entailing simultaneous nuclear spin flips and in the introduction the analogy with NMR was emphasized. The magnitudes of the splittings are of course very much larger than pure Zeeman splittings in conventional magnetic fields because of the intimate involvement of the spatial degrees of freedom, in particular the exchange interactions connected with particle indistinguishability and the PEP. Furthermore, these splittings are present in zero magnetic field.

This investigation has reinforced the view that the temperature dependence of the INS spectrum provides an effective tool for assigning spectral lines. In particular, as exemplified in figure 6, studying NE gain and NE loss spectra together enables the absolute energies of both participating states in a transition to be identified as well as their splitting. This is of interest not only to the particular case of endofullerenes but also to the study of molecular dynamics by INS more generally.

The extent to which crystal field effects and inter-fullerene interactions influence the potential energy surface experienced by the H_{2} rotors is an interesting question. The small shift in energy transfer observed at low temperature of the principal rotational peak (figure 7) suggests that these effects are small. Therefore, to a good approximation, the hydrogen endofullerenes behave as isolated complexes and the H_{2} dynamics are dominated by the C_{60} cage potential. Nevertheless, a small shift has been observed at the lowest temperatures which indicates that this is an area that merits further investigation and that INS spectroscopy could be an effective tool. However, due to the uncertain nature of interactions with solvent inclusions, a quantitative and systematic assessment of effects such as those shown in figure 7 can only become viable once samples with higher purity become available in sufficient volume. It is interesting to note that the first investigations of H_{2}O@C_{60} [15] have identified a splitting of a rotational line that is attributable to symmetry breaking in the water environment. However, in this case, the permanent electric dipole moment of the H_{2}O molecule is likely to provide a much stronger inter-fullerene interaction than any present in H_{2}@C_{60}.

## Funding statement

This work is supported in the UK by the Engineering and Physical Sciences Research Council, in particular K.G. is in receipt of a postgraduate scholarship. The authors at Columbia University thank the NSF for its generous support through grant no. CHE 07 17518. M.C. is grateful for support from the Royal Society.

## Footnotes

One contribution of 13 to a Theo Murphy Meeting Issue ‘Nanolaboratories: physics and chemistry of small-molecule endofullerenes’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.