## Abstract

Hydrogen is one of the few molecules that has been incarcerated in the molecular cage of C_{60} to form the endohedral supramolecular complex H_{2}@C_{60}. In this confinement, hydrogen acquires new properties. Its translation motion, within the C_{60} cavity, becomes quantized, is correlated with its rotation and breaks inversion symmetry that induces infrared (IR) activity of H_{2}. We apply IR spectroscopy to study the dynamics of hydrogen isotopologues H_{2}, D_{2} and HD incarcerated in C_{60}. 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 C_{60} and the high symmetry of C_{60} 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 H_{2}@C_{60}. The same parameters were used to predict H_{2} energies inside C_{70}. We compare the predicted energies and the low-temperature IR absorption spectra of H_{2}@C_{70}.

## 1. Introduction

A small cavity inside the fullerene cage is a potential site for trapping atoms and has attracted the attention of scientists from the moment of discovery of C_{60} [1]. The demonstration of formation of La@C_{60} after laser bombardment of La-impregnated graphite was immediate [2]. Since then the field of studies of endohedral fullerenes has been expanding. Endohedral fullerenes with noble gas (He and Ne [3]; Ar, Kr, Xe [4]), nitrogen [5] or phosphorus [6] atoms and with metal clusters [7] are made under extreme conditions using arc discharge, ion bombardment or high-pressure and high-temperature treatment.

Extreme methods are not suitable for encapsulation of small molecules. A different approach, ‘chemical surgery’, was applied by Rubin [8] when he made the first open-cage fullerene with an orifice large enough to load it with ^{3}He or H_{2} using less extreme temperature and pressure [9]. Soon Murata *et al*. [10] synthesized another open-cage derivative of C_{60} and achieved 100% yield in filling with H_{2}. Subsequently, the generation of a closed-cage H_{2}@C_{60} was observed in the process of matrix-assisted laser desorption/ionization time-of-flight mass spectrometry analysis of this open-cage complex. Chemical methods were developed to close open-cage fullerenes and H_{2}@C_{60} was produced in milligram quantities [11,12]. To accommodate two hydrogen molecules a cavity larger than C_{60} was needed. Two H_{2} were trapped in open-cage C_{70} with a yield of 3 : 97 in favour of species with one H_{2} per cage [13]. The restoration of the closed cage retains approximately the same ratio of (H_{2})_{2}@C_{70} to H_{2}@C_{70} [14]. Molecules other than hydrogen trapped in open-cage fullerenes are carbon monoxide [15], water [16,17], ammonia [18] and methane [19]. Recently, Kurotobi & Murata [20] succeeded in closing one of them and making the first closed-cage endohedral complex with a trapped polar molecule, H_{2}O@C_{60}. The rotational modes of endohedral water were observed by inelastic neutron scattering (INS), far infrared (far-IR) and nuclear magnetic resonance (NMR) at cryogenic temperatures [21].

To date, H_{2}@C_{60} has been the most studied small-molecule endofullerene. The inhomogeneous distribution of interaction parameters is expected to be small, mainly because of the crystal field effects in solid H_{2}@C_{60}. The H_{2}@C_{60} is a stable complex and can survive a short period of heating up to 500^{°}C under vacuum [12]. These properties make H_{2}@C_{60} appealing for spectroscopic and theoretical investigations of interactions between the molecular hydrogen and carbon nanosurfaces.

Three spectroscopic techniques—NMR, INS and infrared (IR)—have been used to study endohedral hydrogen. NMR studies cover spin lattice relaxation rates of H_{2}@C_{60} in organic solvents [22,23], and in the presence of paramagnetic relaxants [24,25]. NMR was used to follow the *ortho*–*para* conversion in H_{2}@C_{60} in the presence of molecular oxygen at 77 K [26] or upon photoexcitation of a C_{70} triplet state [27]. NMR study of micro-crystalline H_{2}@C_{60} samples at cryogenic temperatures shows splitting of the *J*=1 rotational state [28,29], a sign of the symmetry reduction from the icosahedral symmetry in the solid phase. Similarly, splitting of the ground *ortho* state was deduced from the heat capacity measurements [30].

An overview of the low-temperature NMR, INS and IR work on H_{2}@C_{60} is given by Mamone *et al.* [31]. The first IR study of H_{2}@C_{60} was limited to 6 K [32]. The translational and rotational transitions appeared as sidebands to the hydrogen molecule bond-stretching vibrational transition, , in the mid-IR spectral range. The direct translational and rotational transitions were not observed in the far-IR below 200 cm^{−1} [33]. The extension of IR studies to higher temperature made it possible to probe the hydrogen–C_{60} interaction potential in the ground *v*=0 and first excited *v*=1 vibrational states and a whole range of hydrogen isotopologues H_{2}, D_{2} and HD were studied [33,34]. The isotope effects and translation–rotation coupling were also studied by INS in H_{2}@C_{60} and HD@C_{60} [35]. The translational and rotational energies of H_{2}@C_{60} and HD@C_{60} in the *v*=0 state determined by IR spectroscopy are consistent with the low-temperature INS results [35]. There are no Raman data on H_{2}@C_{60}, except a report on H_{2} inside an open-cage fullerene [36].

In this paper, we will review the IR studies of hydrogen isotopologues in C_{60} and present the analysis of IR low-temperature spectra of H_{2}@C_{70}. The far-IR properties of H_{2}O@C_{60} will not be reviewed here [21].

## 2. Theory

Quantum statistics plays an important role in the dihydrogen wave function symmetry and has a pronounced effect on the rotation of the hydrogen molecule [37]. The symmetry relative to the interchange of two protons dictates that there are two forms of molecular hydrogen, called *para*- and *ortho*-H_{2}. The two proton spins () are in the antisymmetric *I*=0 total nuclear spin state in *para*-H_{2} and in the symmetric *I*=1 state in *ortho*-H_{2}. Even rotational quantum numbers *J* are allowed for *para*-H_{2} and odd *J* for *ortho*-H_{2}. The nucleus of D is a boson, nuclear spin *I*_{d}=1. Thus, the rotational state with an even quantum number *J* has D_{2} in the state where the total nuclear spin of D_{2} is either zero or two, *I*=0,2. This is called *ortho*-D_{2}, while *para*-D_{2} has the total nuclear spin *I*=1 and odd *J* values.

The ground rotational state of a homonuclear diatomic molecule with the total nuclear spin *I*=1 is the *J*=1 state. This *J*=1 rotational state is 118 cm^{−1} for H_{2}, and 58 cm^{−1} for D_{2}, above the rotational ground state *J*=0 of even-*I* species. A thermal transition must be accompanied by a change in the total nuclear spin of the molecule, a process of very low probability. The time constant of thermal relaxation between the *ortho* and the *para* manifolds is very long and the room temperature *ortho*–*para* ratio is maintained even at cryogenic temperatures. The equilibrium distribution of H_{2} nuclear spin isomers is *n*_{o}/*n*_{p}=3 and of D_{2} is *n*_{o}/*n*_{p}=2 at room temperature. To change the total nuclear spin of a molecule the two nuclei must experience different magnetic fields. The *ortho*–*para* conversion can be activated by using a paramagnetic centre as a source of the magnetic field gradient. The equilibrium *n*_{o}/*n*_{p}=1.0 is reached at 77 K by dispersing H_{2}@C_{60} on a zeolite surface and exposing it to molecular oxygen, which acts like a spin catalyst [38]. There are no *ortho* and *para* species for HD. All rotational levels of HD are in thermal equilibrium and there is one rotational ground state, *J*=0.

Quantum chemistry calculations are challenging for a hydrogen molecule in a weak van der Waals interaction with a large fullerene molecule. The availability of experimental data on endohedral H_{2} has stimulated theoretical work in this direction. Theoretical investigations currently cover the calculations of rotation–translation energies of hydrogen isotopologues in C_{60} [39,40] and H_{2} in C_{70} [41], and the stability of C_{60} or C_{70} with one or more incarcerated H_{2} [42,43]. Empirical parameters of the Morse potential between H–H and contact Dirac interaction between H–C were adjusted [44], and the density-fitting local Møller–Plesset theory was tested [45] using the experimental H_{2} vibrational frequency inside C_{60}. Classical molecular dynamics and density functional theory have been combined to reproduce accurately the NMR chemical shift of ^{1}H in H_{2}@C_{60} [46].

Both the hydrogen molecule and fullerene have closed-shell electronic structures and therefore the interaction between them is the van der Waals interaction. This simplicity makes H_{2}@C_{60} ideal for the studies of non-covalent bondings between H_{2} and carbon nanosurfaces, the knowledge needed for the design of carbon-based hydrogen storage materials. The high icosahedral symmetry of the C_{60} cavity is close to spherical; therefore, H_{2}@C_{60} represents a textbook example of a body moving in a spherical potential well [47,48]. In addition, H_{2} rotates around its centre of mass. H_{2} is not spherical and therefore its interaction with the walls of the cavity depends on its orientation, which leads to the coupling between translational and rotational motion [49]. If the translational and the rotational motions are coupled then in the spherical potential the conserved angular momentum is the sum of translational and rotational angular momenta [50,39]. H_{2}@C_{60} is a rare example in which the quantum dynamics of a diatomic rotor in a confined environment can be studied. Another, but with limited degrees of freedom, interesting example related to the fullerenes is the quantum rotor C_{2} in a metallofullerene C_{2}@Sc_{2}C_{84} [51,52]. The two scandium atoms limit the translational motion and fix the rotational axis of C_{2} relative to the fullerene cage. At low temperature, the rotation of C_{2} is hindered because it has a small rotational constant and is therefore more susceptible to the corrugations of the carbon surface. H_{2} provides examples of two-dimensional rotors, such as H_{2} on a Cu surface [53] or H_{2} in intercalated graphite [54].

High-pressure loading of solid C_{60} creates interstitial H_{2}. Exohedral H_{2} has been studied by IR [55,56], INS [57,58], NMR [59,60] and Raman [61] spectroscopies. Hydrogen is trapped in an interstitial site of octahedral symmetry and theory predicts translation–rotation coupling [50,62]. However, broadening of experimental lines has prohibited accurate determination of the H_{2}–C_{60} interaction potential.

The observed IR spectra of hydrogen encapsulated in C_{60} consist of several absorption lines. We construct a model Hamiltonian and a dipole moment operator with few adjustable parameters to describe accurately the position and intensity of such a multi-line spectrum.

### (a) Diatomic molecule in a spherical potential well

To describe the motion of a hydrogen molecule inside C_{60}, we use the following model. The C_{60} is considered to be rigid, its centre of mass is not moving and also does not rotate. We treat H_{2}@C_{60} as an isolated molecular complex and approximate the true icosahedral symmetry of an isolated C_{60} with spherical symmetry. It means that, in this approximation, H_{2} moves in a rigid spherically symmetric bounding potential provided by the cavity of C_{60}. Besides the translational movement inside C_{60} the hydrogen molecule has its internal degrees of freedom, vibration and rotation of two nuclei relative to its centre of mass. There are no coupling terms between *ortho* and *para* states in our model Hamiltonian.

The theoretical work of Olthof *et al.* [63] is a comprehensive description of the dynamics of a loosely bound molecule inside C_{60}. Olthof *et al.* [63] model the intermolecular potential as a sum of atom–atom potentials and expand it in spherical harmonics. They determined the radial part of the wave function with a discrete variable representation method. The radial part of the wave function in our approach is given by algebraic functions, which are solutions of the three-dimensional spherical oscillator [47,48,64]. The advantage is that matrix elements are calculated in algebraic form, avoiding time-consuming numerical integration.

The position and orientation of the H_{2} molecule is given by spherical coordinates **R**={*R*,*Ω*}, *Ω*={*Θ*,*Φ*} and **s**={*s*,*Ω*_{s}}, *Ω*_{s}={*θ*,*ϕ*}, where **R** is the vector from the centre of the C_{60} cage to the centre of mass of H_{2} and **s** is the internuclear H–H vector. The centre of mass translational motion of H_{2} is given by eigenfunctions of the isotropic three-dimensional harmonic oscillator [48,64]
2.1where is the radial wave function and *Y* _{LML} is the spherical harmonic. The size of the H_{2} molecule depends on its vibrational state |*v*〉. Therefore both the bounding potential and depend on the vibrational quantum number *v*. The translational quantum numbers are *N*=0,1,2,…. The orbital angular momentum quantum number takes values *L*=*N*,*N*−2,…,1 or 0, depending on the parity of *N*, and the azimuthal quantum number is *M*_{L}=−*L*,−*L*+1,…,*L*. The rotational wave functions, defined by the rotational quantum numbers *J*=0,1,… and *M*_{J}=−*J*,−*J*+1,…,*J*, are given by the spherical harmonics *Y* _{JMJ}(*θ*,*ϕ*).

We use bipolar spherical harmonics with overall spherical rank *Λ* and component *M*_{Λ}
2.2where are the Clebsch–Gordan coefficients [64]. Then the full wave function describing the motion of the H_{2} molecule is
2.3where is the vibrational wave function with the quantum number *v*.

The Hamiltonian for the trapped molecule includes coupling terms between vibrational, translational and rotational motion. For simplicity, we neglect all matrix elements non-diagonal in *v* and introduce a parametric dependence on *v*,
2.4where is the vibration–rotation Hamiltonian, *p* is the molecular momentum operator and *m* is the molecular mass of the diatomic molecule. ^{v}*V* =〈*v*|*V* (*R*,*s*,*Ω*,*Ω*_{s})|*v*〉 is the potential energy of the hydrogen molecule in the vibrational state *ψ*^{vib}_{v}(*s*). The vibration–rotation Hamiltonian is diagonal in the basis set |*vJNLΛM*_{Λ}〉 with eigenvalues given by
2.5, where *ω*_{0} is the fundamental vibrational frequency, *α*_{e} is the anharmonic correction and *D*_{e} is the centrifugal correction to the rotational constant *B*_{e} [65,66].

We start from the general expansion of the potential in multi-poles,
2.6and expand the radial part of potential in powers of *R*
2.7where *n*≥*l* and the parities of *l* and *n* are the same.

*V* (**R**,**s**) is a scalar and transforms under fully symmetric representation *A*_{g} of the symmetry group *I*_{h}. The spherical harmonics *λ*=0,6,10,… transform as *A*_{g} of the symmetry group *I*_{h} [67]. We use fully spherical approximation of the potential, *λ*=0. Because *λ*=0 and *λ*=|*l*−*j*|,|*l*−*j*|+1,…,*l*+*j* it must be that *l*=*j*.

The total potential is
2.8if we limit our expansion to *j*=*l*=2 and *n*=4. The odd-*j* terms are not allowed by symmetry for H_{2} and D_{2} and thus the coefficients and are zero for homonuclear diatomic molecules.

If we set the constant off-set to zero and write the perturbation part as 2.9and the isotropic harmonic term as , then the total Hamiltonian reads 2.10

The unperturbed Hamiltonian eigenvalues in the basis |*vJNLΛM*_{Λ}〉 are
2.11where is the frequency for translational oscillations within the cavity.

The meaning of different parts of the perturbation is explained by their influence on the energy levels of a harmonic three-dimensional spherical oscillator (table 1). The translation–rotation coupling term splits the energy of the |*vJNLΛ*〉 state into levels with different *Λ*, where *Λ*=|*L*−*J*|,|*L*−*J*|+1,…,*L*+*J*. For example, the *N*=*L*=*J*=1 state is split into three levels with different total angular momentum *Λ*=0,1,2. The ordering of levels depends on the sign of . The anharmonic correction to translation–rotation coupling is . If isotropic anharmonic correction is positive, the distance between energy levels increases with *N* and this correction is different for the levels with same *N* but different *L*. For example, for positive the *N*=2,*L*=0 level has higher energy than the *N*=2,*L*=2 level.

The length scale (dimension m^{−2}) of the radial part of a three-dimensional spherical oscillator wave function is related to the expectation value of the centre of mass amplitude in state |*N*〉 as [47]
2.12

Terms described by the translation–rotation coupling coefficients and do not appear in table 1 because the first-order correction to energies vanishes as the matrix element of is zero if diagonal in *L* or *J*. These terms mix states with different *N* and *J*. For example, the first excited rotational state *J*=1, *N*=0 (expectation value of HD centre of mass is on the cage centre) has the state *J*=0, *N*=1 (expectation value of HD centre of mass is off the cage centre) mixed in [34]. The effect is that HD is forced to rotate about its geometric centre instead of its centre of mass.

It was found by the five-dimensional quantum mechanical calculation that the rotational quantum number *J* is almost a good quantum number for the homonuclear H_{2}@C_{60} and D_{2}@C_{60} and not for the heteronuclear HD@C_{60} [40]. Indeed, mixes states with different *J* for homonuclear species as well, but the effect is reduced compared with the effect of . In the former case, *J*±2 and *L*±2 are mixed, while in the latter case the *J*±1 and *L*±1 states that have a smaller energy separation are mixed.

The states with different *Λ* are not mixed in the spherical approximation, i.e. the total angular momentum ** Λ=L+J** is conserved and

*Λ*is a good quantum number. The other consequence of the spherical symmetry is that the energy does not depend on

*M*

_{Λ}. Therefore, it is practical to use a reduced basis and reduced matrix elements [68] which are independent of

*M*

_{Λ}. This reduces the number of states by factor 2

*Λ*+1 for each

*Λ*.

### (b) Model Hamiltonian of H_{2}@C_{70}

A spherical approximation of the potential of a molecule trapped in C_{70} would be an oversimplification because of the elongated shape of C_{70}. The symmetry of C_{70} is *D*_{5h}, the distance between the centres of two capping pentagons (*z*-direction) is 7.906 Å. The diameter of the equatorial *xy* plane is 7.180 Å[69], similar to the diameter of the icosahedral sphere of C_{60}, 7.113 Å[70]. The anisotropy of the potential of H_{2} inside C_{70} is supported by the five-dimensional quantum mechanical calculation [41] that shows that the lowest translational excitation in the *z*-direction is 54 cm^{−1} and in the *xy* plane is 132 cm^{−1}, while in C_{60} it is 180 cm^{−1} and isotropic [33]. We derive from the IR spectra (see below) that the *xy* plane excitation energy is 151 cm^{−1}, somewhat larger than theoretically predicted.

Although the *z*-axis translational energy in C_{70} is three times less than in the icosahedral C_{60}, the effect of the C_{70} potential on the rotational motion is moderate. The splitting of the *J*=1 state is 7 cm^{−1}, which is relatively small compared with the rotational energy 120 cm^{−1} in this state [41].

To analyse the IR spectra of H_{2}@C_{70}, we use a simplified Hamiltonian in which the translational energy is represented in the form of the sum of two oscillators, a one-dimensional linear and a two-dimensional circular oscillator, and we do not consider anharmonic corrections and the translation–rotation coupling.

The vibration–rotation energy
2.13is the same as for H_{2}@C_{60} except the last term, which accounts for the axial symmetry of the C_{70} potential with the rotational anisotropy parameter ^{v}*κ* [71]. For example, the three-fold degenerate *J*=1 rotational state in *I*_{h} symmetry is split in *D*_{5h} symmetry and if ^{v}*κ*>0 the *J*_{z}=0 state is 3^{v}*κ* below the twice degenerate *J*_{z}=±1 rotational state.

The translational part is added to the vibration–rotation Hamiltonian, equation (2.13), and the total energy becomes
2.14Here, the translational energy is written as a sum of two oscillators, a linear oscillator along the *z*-axis with translational quantum and a two-dimensional (circular) oscillator [48,64] in the *xy* plane with translational quantum . Quantum numbers *n*_{z} and *n* are positive integers including zero and *l*=*n*,*n*−2,…,−*n*+2,−*n*.

We will show below that the frequencies of *z* and *xy* translational modes, and , can be determined from the experimental data even though the translation–rotation coupling is not known. We take advantage of translation–rotation coupling being zero in the *J*=0 rotational state. The complication arises from the fact that the potential is different in the initial and final states of the IR transitions, *v*=0 and 1. However, this complication could be resolved if the energy of the fundamental vibrational transition (without change of *n* and *n*_{z}) is known.

The *ΔJ*=0 transition from the *para*-H_{2} ground state leads to two excitation energies in the IR spectrum, first for the *z* mode and second for the *xy* mode
2.15and
2.16Defining the fundamental *para* transition energy as , we may rewrite equations (2.15) and (2.16) as
2.17From these equations, translational frequencies in the excited *v*=1 state, and , can be determined without knowing the translation–rotation coupling.

The classification of energy levels up to *J*=1 and *n*=*n*_{z}=1 by irreducible representations *Γ*_{i} of the symmetry group *D*_{5h} is given in table 2. We get the irreducible representations *Γ*_{j}: and by subducting the translational states represented by spherical harmonics *Y* _{LML} from the full rotational group *O*(3) to the symmetry group *D*_{5h}. *A*_{1}′ is the *para*-H_{2} ground state, *n*=*l*=*n*_{z}=0. The first excited state of the *z* mode is *n*_{z}=1 and *A*_{2}′′. The first excited state of the *xy* mode *n*=*l*=1 is doubly degenerate *E*_{1}′. The full symmetry when translations and rotations are taken into account is *Γ*_{i}=*Γ*_{j}⊗*Γ*^{(J)}. For example, the *ortho*-H_{2} ground state, *J*=1 and (*nln*_{z})=(000), is split into *J*_{z}=0 (*A*_{2}′′) and *J*_{z}=±1 (*E*_{1}′) (table 2).

### (c) Induced dipole moment of hydrogen in a spherical environment

IR light is not absorbed by vibrations and rotations of isolated homonuclear diatomic molecules [65]. IR activity of H_{2} is induced in the presence of intermolecular interactions, such as in the solid and liquid phases [72,73], in constrained environments [74,55,56,62], and in pressurized gases [75,76]. IR spectra of such systems are usually broad because of inhomogeneities in the system or because of random molecular collisions. As an exception, narrow lines are observed in semiconductor crystals [77] and solid hydrogen [78].

An overview of collision-induced dipoles in gases and gas mixtures is given in a book by Frommhold [79]. The confinement of the endohedral H_{2} introduces two differences when compared with H_{2} in the gas. *First*, the translational energy of H_{2} is quantized. In the gas phase, it is a continuum starting from zero energy. *Second*, the variation of the distance between H_{2} and the carbon atom is limited to the translational amplitude of H_{2} in the confining potential. In the gas phase, the distance varies from infinity to the minimal distance given by the collision radius. The selection rule *ΔN*=±1 for the endohedral H_{2} follows from these two conditions, as shown below.

Quantum mechanical calculations of induced dipoles are available for simple binary systems such as H_{2}–He, H_{2}–Ar and H_{2}–H_{2}. An extensive set of theoretical results for the H_{2}–He system associated with the roto-translation electric dipole transitions, both in the vibrational ground state *v*=0 and accompanying the transition of the H_{2} molecule, can be found in [80–84]. Related to the fullerene studies are calculations of the dipole moment of CO@C_{60} [63] and exohedral H_{2} in solid C_{60} [62].

We express the induced part of the dipole moment as an interaction between a hydrogen molecule and C_{60}. Another approach was used by Ge *et al.* [33], who carried out a summation over 60 pair-wise induced dipole moments between H_{2} and carbon atoms. The relation between two sets of parameters was given [34].

We write the expansion of the dipole moment from the vibrational state *v* to *v*′ in bipolar spherical harmonics and in power series of *R* as
2.18This is similar to the expansion of the potential discussed above, except the dipole moment is a polar vector whereas the potential is a scalar. The dipole moment transforms according to the irreducible representation *T*_{1u} of the symmetry group *I*_{h}. The spherical harmonics of the order *λ*=1,5,7,… transform according to *T*_{1u} of the symmetry group *I*_{h} [67]. We use *λ*=1 and are interested in transitions. In spherical symmetry, it is sufficient to calculate one component of the dipole moment vector, *m*_{λ}=0, and if we drop the explicit dependence of on *v*,*v*′ and of on *λ*,*m*_{λ}, the *m*_{λ}=0 component of the dipole moment reads
2.19

As *λ*=|*l*−*j*|,|*l*−*j*|+1,…,*l*+*j* and *λ*=1 it must be that *l*=|*j*±1|. The possible combinations are (*lj*)∈{(01),(10),(12),(21),(23),…}. If we restrict the expansion up to *n*=*l*=1, we get

*A*^{010} is zero for homonuclear diatomic molecules H_{2} and D_{2}. It is the sum of the HD permanent rotational dipole moment in the gas phase and the induced rotational dipole moment inside C_{60}. The selection rule is *ΔN*=0,*ΔJ*=0,±1, but is forbidden.

The expansion coefficients *A*^{101} and *A*^{121} are allowed by symmetry for both homo- and hetero-nuclear diatomic molecules. The expansion coefficient *A*^{101} describes the induced dipole moment that is independent of the orientation of the diatomic molecular axis **s**, selection rule *ΔN*=0,±1,*ΔJ*=0, but is forbidden. *A*^{121} describes the induced dipole moment that depends on the orientation of **s**, *ΔN*=0,±1, *ΔJ*=0,±2, but and are both forbidden. All terms in equation (2.20) satisfy the selection rule *ΔΛ*=0,±1, but is forbidden.

Although HD has a permanent rotational dipole moment, the induced dipole moment dominates inside C_{60} [34].

IR absorption line intensity [33] is proportional to the thermal population of the initial state. If the thermal relaxation between *para*-H_{2} and *ortho*-H_{2} is very slow we can define temperature-independent fractional *ortho* and *para* populations *n*_{k} of the total number of molecules , where *k*=*o*,*p* selects *ortho*- or *para*-H_{2}. Then the probability that the initial state |*v*_{i}*J*_{i}*N*_{i}*L*_{i}*Λ*_{i}*M*_{Λi}〉 is populated is
2.20where *E*_{i} is the energy of the initial state measured from the ground state *v*=*N*=0 and *j* runs over all *para*- (or *ortho*-) H_{2} states in the basis used. *g*_{j}=2*Λ*_{j}+1 is the degeneracy of the energy level *E*_{j}. Please note that *g*_{i} does not appear in the numerator because *p*_{i} is the population of an individual state |*v*_{i}*J*_{i}*N*_{i}*L*_{i}*Λ*_{i}*M*_{Λi}〉, although the reduced basis |*vJNLΛ*〉 is used. Hetero-nuclear HD has no *para* or *ortho* species and the coefficient *n*_{k} in equation (2.20) must be set to one, *n*_{k}=1.

### (d) Infrared absorption by H_{2}@C_{70}

The infrared absorption in H_{2}@C_{70} is given by electric dipole operators *d*_{0} and *d*_{±1}, which transform as *A*_{2}′′ and *E*_{1}′ irreducible representations of *D*_{5h}. At low temperature the symmetry-allowed transitions from the ground *para* state are and . Two *ortho* states, *A*_{2}′′ and *E*_{1}′, are populated at low temperature (figure 1). The transitions are and . The symmetry-allowed transitions from the *E*_{1}′ state are and .

The IR absorption line area is proportional to the Boltzmann population of the initial state (equation (2.20)). The degeneracy *g*_{j} of the energy level *E*_{j} is one or two in the case of H_{2}@C_{70}. Figure 1 shows the Boltzmann population of the few first *para* and *ortho* levels in the *v*=0 vibrational state. The *para* state (000) population (solid line) starts to drop above 20 K as the first excited state (001) (dotted line) at 54 cm^{−1} above the ground state becomes populated and drains the population from the (000) state. The abrupt change of the population of *ortho* ground states above 2 K is because of the transfer of population from the non-degenerate *A*_{2}′′ ground level (dashed line) to the doubly degenerate *E*_{1}′ (dashed-dotted line) 7 cm^{−1} higher. The populations of the *A*_{2}′′ and *E*_{1}′ states decrease above 30 K as the first excited translational (001) *ortho* state becomes more populated.

## 3. Experiment

The endohedral complexes were prepared by ‘molecular surgery’ as described in [11,12]; H_{2}@C_{60}, D_{2}@C_{60} and H_{2}@C_{70} at Kyoto University, Kyoto, Japan, and HD@C_{60} at Kyoto University and Columbia University, New York, NY. The HD@C_{60} sample was a mixture of the hydrogen isotopologues H_{2}:HD:D_{2} with the ratio 45 : 45 : 10. Since all C_{60} cages are filled, the filling factor for HD is *ρ*=0.45. The content of the C_{70} sample was empty : H_{2} : (H_{2})_{2}=28:70:2 and the filling factor *ρ*=0.7. Experimental absorption spectra were corrected for the filling factor.

The *para*-enriched sample was made at Columbia University using molecular oxygen as a spin catalyst for the *ortho*–*para* conversion [26]. Briefly, the H_{2}@C_{60} adsorbed on the external surface of NaY zeolite was immersed in liquid oxygen at 77 K for 30 min, thereby converting the incarcerated H_{2} spin isomers to the equilibrium distribution at 77 K, *n*_{o}/*n*_{p}=1.0. The liquid oxygen was pumped away and the endofullerene–NaY complex was brought back rapidly to room temperature. The *para*-enriched H_{2}@C_{60} was extracted from the zeolite with CS_{2} and the solvent was evaporated by argon. The powder sample under an argon atmosphere and on dry ice arrived at the National Institute of Chemical Physics and Biophysics, Tallinn, Estonia, 4 days after the preparation.

Powdered samples were pressed under vacuum into 3 mm diameter pellets for IR transmission measurements. Typical sample thickness was 0.3 mm. Two identical vacuum tight chambers with Mylar windows were employed in the IR measurements. The chambers were put inside an optical cold finger-type cryostat with KBr windows. During the measurements, the chamber containing the pellet for analysis was filled with He exchange gas while the empty chamber served as a reference. Transmission spectra were obtained using a Bruker interferometer Vertex 80 v with a halogen lamp and a HgCdTe or an InSb detector. The apodized resolution was typically 0.3 cm^{−1} or better.

Transmission *T*_{r}(*ω*) was measured as the light intensity transmitted by the sample divided by the light intensity transmitted by the reference. The absorption coefficient *α*(*ω*) was calculated from the transmission *T*_{r}(*ω*) through , with the reflection coefficient *R*=[(*η*−1)/(*η*+1)]^{2} calculated assuming a frequency-independent index of refraction [85], *η*=2. Absorption spectra were cut into shorter pieces around groups of H_{2} lines and a baseline correction was performed before fitting the H_{2} lines with Gaussians.

## 4. Results and discussion

### (a) H_{2}, D_{2} and HD in C_{60}

The IR absorption spectra of H_{2}@C_{60} at 6 K together with the simulated spectra are shown in figure 2 and the diagram of the energy levels involved is shown in figure 3*a*. The simulated spectra are calculated using Hamiltonian, dipole moment and the *ortho*–*para* ratio parameters (table 3) obtained from the fit of 200 K spectra [33]. Temperature does not affect these parameters and the line intensities follow the Boltzmann population of initial states [33]. The intensity of three lines, *Q*(0)^{0} and *Q*(1)^{1} in figure 2*a* and *S*(1)^{3} in figure 2*c*, cannot be simulated because the induced dipole moment theory does not describe *ΔN*=0 transitions. However, the position of these three lines was used to fit the Hamiltonian parameters [33]. The intensity of the *ortho* state *N*=*J*=1 into *Λ*=0,1 and 2 by translation–rotation coupling is seen in figure 2*b*. The *para* *N*=1,*J*=2 and *ortho* *N*=1,*J*=3 states are split into three sublevels as well. However, because of the selection rule *ΔΛ*=±1 only one *para* and one *ortho* *S*-transition is IR active.

The intensity of other lines in figure 2*b*–*d* is described accurately by two dipole moment parameters, *A*^{101} and *A*^{121}, and the *ortho*–*para* ratio *n*_{o}/*n*_{p}=2.89±0.045, which is very close to the statistical value three (table 3). We confirmed the assignment of spectral lines to *para*- and *ortho*-H_{2} by measuring the spectrum of a *para*-enriched H_{2}@C_{60} sample [33]. Low-temperature spectra in the region of *Q* lines are shown in figure 4. The time delay between *para* enrichment and the first IR measurement was 4 days. The 4255 cm^{−1} *para* line is stronger and the other three *ortho* lines are weaker in the *para*-enriched sample than in the non-enriched sample.

Figure 4 deserves some attention. Lines *Q*(0)^{1} and *Q*(1)^{0} have a clear multi-component structure and it is different for the two samples. This is not possible in the approximation of spherical symmetry—not even by considering the true icosahedral symmetry of C_{60} because the lowest *Λ* value for the state which is split by the icosahedral symmetry is three while the initial and final states of the optical transitions under discussion only have *Λ*=0,1. Later measurements on the relaxed but initially *para* converted sample showed that the difference between the line shapes of the normal and *para* converted sample are not due to the different *ortho*–*para* ratio. Thus, it is likely that the difference in the line splitting is due to a different impurity content of the two samples. However, it is not completely excluded that the crystal field or the distortion of the C_{60} cage is responsible for part of this splitting. Note that the *Q*(1)^{0} line in D_{2}@C_{60} (figure 5) bears a similar splitting pattern to that in H_{2}@C_{60}.

The D_{2}@C_{60} spectrum (figure 5) is shifted to a lower frequency than that of H_{2}@C_{60} because of the heavier mass of D_{2}. The spectrum has fewer lines than the H_{2}@C_{60} spectrum. The missing transitions in D_{2}@C_{60} (table 4) belong to the *J*-odd species, which are in the minority for D_{2}.

The splitting of *N*=*J*=1 into *Λ*=0,1 and 2 states is similar in D_{2} and H_{2}. The magnitude of the splitting, (table 1), is less in D_{2} because is smaller, although and are similar for the two isotopologues (table 3). Line *Q*(1)^{2} overlaps partially with the *Q*(0)^{1} line (figure 5). However, this is not because the translation–rotation splitting is different for the two isotopologues but because of a smaller anharmonic correction *α*_{e} of the rotational constant for D_{2} (table 3).

The spectrum of HD@C_{60} is more simple (figure 6) because there are no *para* and *ortho* species and therefore only one state is populated at low temperature. There is one spectral line, not present in homonuclear dihydrogen, labelled *R*(0) in figures 3*b* and 6. HD has no inversion symmetry and the ban on *ΔJ*=1 transitions is lifted. The classification of this transition as *ΔJ*=1 is arbitrary, first, because the weight of *JNLΛ*=1201 is only 0.5 in this state [34] and, second, the change of translational state is *ΔN*=2, which is not allowed by any of the dipole operator components in our model. The next component in the final state is *JNLΛ*=2111 with weight 0.29, which makes this transition *A*^{121}-active, , and therefore it would be more appropriate to classify it as an *S* line.

Another interesting feature of the HD spectrum is the absence of the *ΔN*=0,*ΔJ*=1 () transition, although the induced dipole moment coefficient *A*^{010} gives a two orders of magnitude larger dipole moment than the permanent dipole moment of free HD, as was discussed in [34]. This *ΔJ*=1 transition in HD@C_{60} is suppressed because there is an interference of two dipole terms, *A*^{010} and *A*^{101}, which have opposite signs. The final state consists of 80% of the pure rotational state *J*=1,*N*=0 and 18% of the pure translational state *N*=1,*J*=0. This mixed final state has matrix elements from the ground state for both *A* coefficients, *A*^{010} and *A*^{101}, which nearly cancel each other.

The observed low-temperature IR transitions of H_{2}, D_{2} and HD@C_{60} are collected in table 4. The content of the unperturbed |*JNLΛ*〉 state in the final state is above |*ξ*^{1}|^{2}=0.9 in homonuclear species in most cases, while for HD it varies and could be as low as 0.5. The mixing of states *ξ*^{v} is proportional to the energy separation of mixed states, *E*_{j}−*E*_{i} in the first-order perturbation theory. couples states where *L*_{i}=*L*_{j} and *J*_{i}=*J*_{j}. These states are far from each other as *L*_{i}=*L*_{j} only if *N*_{j}=*N*_{i}±2. The other term mixes *L*_{j}=*L*_{i}±2 and *J*_{j}=*J*_{i}±2, which are even further from each other for small *N*. It was found by Xu *et al.* [41] that *J* is almost a good quantum number in H_{2} and D_{2}@C_{60} but not in HD@C_{60}. Indeed, the distance between mixed states decreases for HD as mixes *L*_{j}=*L*_{i}±1 and *J*_{j}=*J*_{i}±1 and for this *N*_{j}=*N*_{i}±1.

The vibrational frequency *ω*_{0} is redshifted from its gas-phase value for H_{2} and D_{2} (table 3). The relative change in the frequency [*ω*_{0}(gas)−*ω*_{0}(*C*_{60})]*ω*_{0}(gas) depends on the cage and is independent of the hydrogen isotopologue [88]. Based on our fit results (table 3), for H_{2} *ω*_{0}(*C*_{60})/*ω*_{0}(gas)=0.9763 and for HD *ω*_{0}(*C*_{60})/*ω*_{0}(gas)=0.9773. We used the average of these two ratios to calculate the D_{2}@C_{60} fundamental vibrational frequency, *ω*_{0}=2924 cm^{−1}, that was necessary to fit the IR spectra to a model Hamiltonian [34]. At this point we cannot say how much of the redshift is caused by a change in the zero-point vibrational energy and how much is caused by the change of anharmonic corrections to the vibrational levels in the C_{60} as our dataset is limited to energy differences of *v*=0 and 1 levels only. Note that the vibrational frequency of H_{2}@C_{60} is *ω*_{0}=4062.4 cm^{−1} (table 3), while the *para* vibrational transition is shifted up by 9 cm^{−1}, to *ω*_{0}=4071.4 cm^{−1} (table 4). This is because the zero-point translational energies are different in the *v*=0 and 1 vibrational states [33].

The rotational constant *B*_{e}, the vibrational correction *α*_{e}, and the centrifugal correction *D*_{e} of the hydrogen inside C_{60} and in free space are compared in table 3. The smaller than the gas-phase value of *B*_{e} may be interpreted as 0.8% (0.9%) stretching of the nucleus–nucleus distance *s* in H_{2}@C_{60} (D_{2}@C_{60}), as *B*_{e}∼〈1/*s*^{2}〉. An attractive interaction between hydrogen atoms and the cage causes *s* to be longer. The elongation of the equilibrium proton–proton distance is consistent with the redshift of *ω*_{0} [89]. However, the anharmonic vibrational correction to the rotational constant, *α*_{e}, is smaller inside the cage than in the gas phase. Here, the cage has the opposite, repulsive, effect and reduces the elongation of the proton–proton distance in the excited *v* states when compared with H_{2} in the free space. This is supported by the fact that within the error bars *α*_{e} of D_{2}@C_{60} is equal to *α*_{e} of D_{2}. The vibrational amplitude of D_{2} is less than that of the H_{2} and therefore the repulsive effect of the cage becomes important at *v*>1 for D_{2}.

Among the rotational and vibrational constants of HD@C_{60} the centrifugal correction *D*_{e} to the rotational constant is anomalously different from its gas-phase value compared with the other two isotopologues (table 3). Positive *D*_{e} means that the faster the molecule rotates, the longer is the bond. We speculate that, since the rotation centre of HD inside the cage is further away from the deuteron, the centrifugal force on the deuteron increases and the bond is stretched more than in the free HD molecule.

A similar system to the one studied here is exohedral H_{2} in C_{60}. There, H_{2} occupies the octahedral interstitial site in the C_{60} crystal. The prominent features in the exohedral H_{2} IR spectra [56] are the translational, *ΔN*=±1, sidebands to the fundamental transitions, *Δv*=1 and *ΔJ*=0,2. The redshift of the fundamental vibrational frequency is about 60 cm^{−1}, which is less than in H_{2}@C_{60}, where it is 98.8 cm^{−1}. Also, the separation of the translational *N*=0 and 1 levels, approximately 120 cm^{−1}, is less than that in H_{2}@C_{60}, 184.4 cm^{−1}. It is likely that the main contribution to the latter difference comes from the larger van der Waals volume available for H_{2} in the octahedral site than in the C_{60} cage.

### (b) H_{2}@C_{70}

IR absorption spectra of H_{2}@C_{70} measured at 5 K are shown in figure 7, and the scheme of energy levels with the low-temperature transitions indicated with arrows is shown in figure 8. The transitions where *ΔJ*=0 and the translational state changes by *Δn*=1 or *Δn*_{z}=1 are labelled as *Q*_{xy}(*J*) and *Q*_{z}(*J*), where *J* is the rotational quantum number of the initial state. The transitions where *ΔJ*=2 are labelled by *S*_{xy}(*J*) and *S*_{z}(*J*). The *z* and *xy* translational modes have distinctly different energies. For the *Q* lines they are shown in different panels (figure 7*a*,*b*). The *para* and *ortho* *S* lines are well separated because of different rotational energies. The *para* *S* lines are shown in figure 7*c*, whereas *ortho* *S* lines are shown in figure 7*d*. The *Q* lines cannot be sorted out into *para* and *ortho* that easily. An exception is the *Q*(1) transition, inset to figure 7*a*. This is a pure vibrational transition, , without the change of translational or rotational states. The corresponding (fundamental) *para* transition *Q*(0) would be at 4069.3 cm^{−1} according to the model (table 5). We expect it to be much weaker than the *ortho* *Q*(1) transition, as in H_{2}@C_{60}, and for this reason it is not observed in the experiment.

In figure 9, we plot the normalized line area and normalized Boltzmann population. Figure 9*a* shows the temperature dependence of *para* lines. The assignment of the 4125.6 cm^{−1} line to *Q*_{z}(0) and the 4219.9 cm^{−1} line to *Q*_{xy}(0) is supported by our model, which is summarized in table 5 and will be discussed below.

The uniaxial symmetry of C_{70} splits the *ortho* ground state *J*=1. This splitting is about 7 cm^{−1}≈10 K and creates a sharp temperature dependence of the *ortho* line intensity at low temperature as is shown in figure 1. This feature could be used to determine which of the *Q* lines belong to the *ortho* species. Two lines, 4110.3 cm^{−1} and 4124.2 cm^{−1}, have a temperature dependence consistent with the *ortho* transitions starting from the ground translational (000) state of *E*_{1}′ symmetry (figure 9*c*). However, according to our model, there is no transition close to 4124.2 cm^{−1} from the thermally excited *ortho* *E*_{1}′ state (table 5). This might be due to our simplified model, where the rotational anisotropy parameter ^{v}*κ* is assumed equal in the *v*=0 and 1 vibrational states. The rest of the *ortho* lines are shown in figure 9*b*. Several lines follow the same temperature dependence, the fundamental transition at 4063.2 cm^{−1}, *Q*_{xy}(1) lines 4213.4, 4224.2 and 4227.2 cm^{−1} and the *S*_{z}(1) line at 4678.8 cm^{−1}. Lines *Q*_{z}(1) at 4115.7 and *S*_{xy}(1) at 4761.3 cm^{−1} have distinctly different temperature dependence from the rest of the lines. Such deviation could be due to the overlap of transitions starting from *A*_{2}′′*and* *E*_{1}′ symmetry states. However, none of the *ortho* lines behaves like the theoretical prediction (black line) so the possibility that the ground *ortho* state is doubly degenerate *E*_{1}′ and not the *A*_{2}′′, i.e. ^{0}*κ*<0, is not ruled out by our data. The dashed black line shows the ground-state population if the ground state is *E*_{1}′ and the *A*_{2}′′ state is 7 cm^{−1} above it. This temperature dependence is closer to the experimental situation than the solid black line, where ^{0}*κ*>0. The change of sign of ^{0}*κ* affects the temperature dependence of intensities only a little if the transitions start from the thermally excited state, as shown in figure 9*c*, where the red dashed line is the normalized Boltzmann population of the *A*_{2}′′ state *if* this state is 7 cm^{−1} above the ground state *E*_{1}′, i.e. if ^{0}*κ*<0.

It is not unreasonable that ^{0}*κ*<0 and the *E*_{1}′ (*J*_{z}=±1) state is the ground rotational state of *ortho*-H_{2}. It requires that there is an attraction instead of repulsion between H and C when H_{2} is in the centre of the C_{70} cage. The attraction is stronger for the H_{2} if its molecular axis is in the *xy* plane, *J*_{z}=±1, where the distance between H and C is less than if the axis is along the *z* plane, *J*_{z}=0, where the H–C distance is longer. Indeed, the attraction between C and H in H_{2}@C_{60} was deduced from the redshift of vibrational energy and reduced rotational constant compared with H_{2} in the gas phase [33]. However, it was found by five-dimensional quantum mechanical calculation that *A*_{2}′′ is the ground state [41]. A more elaborate model with dipole moment parameters that describe the IR line intensities together with the Boltzmann population could resolve the issue.

Table 5 summarizes the assignment of transitions observed in the experiment. Since our model described in §2*b* did not include translation–rotation coupling we used the transitions where both *L* and *J* are zero or one of them is zero in the initial and final states. The fundamental frequency *ω*_{0}=4069.1 cm^{−1} was chosen to match the experimental fundamental *ortho* transition *Q*(1); the rotational constant *B*_{e}=59.865 cm^{−1} and its corrections *α*_{e}=2.974 cm^{−1}, *D*_{e}=0.04832 cm^{−1} for H_{2}@C_{60} were used. Next the two *para* transitions were matched, *Q*_{z}(0) and *Q*_{xy}(0), to obtain cm^{−1}, cm^{−1}. The 4220 cm^{−1} line was chosen as *Q*_{xy}(0) because its temperature dependence (figure 9*a*) is similar to the *S*_{z}(0) and *S*_{xy}(0) line temperature dependence. The *ortho* ground-state splitting is 9.4 cm^{−1} (7.4 cm^{−1} from the five-dimensional calculation [41]) with ^{v}*κ*=3.1 cm^{−1}, which was obtained by assuming that the *Q*_{z}(1)^{a} transition, in figure 8, is centred at 4110.3 cm^{−1}. It was assumed that the anharmonic corrections to translational energy and to ^{v}*κ* are same for the *v*=0 and *v*=1 vibrational states.

The variation in the experimental frequencies compared with the model frequencies is within ±10 cm^{−1} (table 5). This is reasonable since our model did not include translation–rotation coupling, and the translation–rotation coupling in H_{2}@C_{60} creates splittings of this magnitude. The vibrational frequency *ω*_{0} is larger in H_{2}@C_{70} than in H_{2}@C_{60}. This difference should not be overinterpreted, because the correction owing to the difference in zero-point translational energies in two vibrational states was not taken into account for H_{2}@C_{70}, as was done for H_{2}@C_{60} [33]. Again, a more elaborate model is needed for C_{70} than the two-oscillator model described in §2*b*.

## 5. Conclusions

IR absorption spectra of endohedral hydrogen isotopologues H_{2} [32,33], D_{2} [34] and HD [34] in C_{60} are informative, involving excitations of vibrations, rotations and translational motion of dihydrogen. The translational motion of the encapsulated molecule is quantized and coupled to its rotations because of the surrounding C_{60} cage. The vibrational frequency of dihydrogen is redshifted compared with the gas-phase value. Together with the smaller rotational constant it shows that the hydrogen bond is stretched inside the cage and there is an attraction between H (or D) and C. The heteronuclear HD does not rotate about its centre of mass because of the surrounding cage. Different rotational and translational states are mixed and rotational quantum number *J* is not a good quantum number for HD@C_{60}.

Our study shows that the vibrations and rotations of C_{60} and the crystal field effects of solid C_{60} are not important on the energy scale of IR measurements. If these effects are important their contribution to the IR spectra is of the order of one wavenumber splitting of a few absorption lines. Such small splitting is consistent with the NMR [28,29] and heat capacity [30] results.

C_{70} has an ellipsoidal shape and this splits the translation–rotation states of H_{2}. The translational frequency is about 180 cm^{−1} in H_{2}@C_{60}. In C_{70}, this three-dimensional mode is split into a two-dimensional mode at 151 cm^{−1} and a one-dimensional mode at 56 cm^{−1}. The five-dimensional quantum mechanical calculation [41] is in very good agreement with this experimental result.

## Funding statement

This research was supported by the Estonian Ministry of Education and Research, grant SF0690029s09, and by the Estonian Science Foundation, grant nos. ETF8170, ETF8703 and JD187.

## 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.