H2 in solid C60: Coupled translation-rotation eigenstates in the octahedral interstitialsite from quantum five-dimensional calculationsShufeng Ye, Minzhong Xu, Stephen FitzGerald, Kirill Tchernyshyov, and Zlatko BaiCitation: The Journal of Chemical Physics 138, 244707 (2013); doi: 10.1063/1.4811220View online: Table of Contents: 38/24?ver pdfcovPublished by the AIP PublishingThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

THE JOURNAL OF CHEMICAL PHYSICS 138, 244707 (2013)H2 in solid C60 : Coupled translation-rotation eigenstates in the octahedralinterstitial site from quantum five-dimensional calculationsShufeng Ye,1 Minzhong Xu,1 Stephen FitzGerald,2 Kirill Tchernyshyov,2and Zlatko Bačić1,a)12Department of Chemistry, New York University, New York, New York 10003, USADepartment of Physics and Astronomy, Oberlin College, Oberlin, Ohio 44074, USA(Received 30 April 2013; accepted 3 June 2013; published online 27 June 2013)We report rigorous quantum five-dimensional (5D) calculations of the coupled translation-rotation(TR) energy levels and wave functions of an H2 molecule, in the ground (ν 0) and vibrationallyexcited (ν 1) states, confined inside the octahedral interstitial site of solid C60 with S6 symmetry.Translational and rotational excitations of H2 in this nanocavity have been measured by the inelasticneutron scattering (INS) and infrared (IR) spectroscopy, enabling direct comparison between theoryand experiment. A pairwise additive 5D intermolecular potential energy surface (PES) was employedin the calculations. The quantum calculations cover the range of energies and types of translationaland rotational excitations of the guest molecule which go substantially beyond those considered inthe earlier theoretical investigations of this system, revealing new information about the TR energylevel structure. The computed j 1 and j 2 rotational levels and their splittings, as well as thetranslational fundamental, are in semi-quantitative agreement with the available INS and IR data,indicating the need for a more accurate intermolecular PES. Our calculations reveal a strong dependence of the TR energy levels, in particular their splittings, on the setting angle which defines theorientation of the C60 molecules relative to their local threefold axes. 2013 AIP Publishing LLC.[]I. INTRODUCTIONEntrapment of hydrogen molecules inside nanoscale cavities of host materials has received considerable attention inrecent years, from experimentalists and theorists alike. Its relevance for hydrogen storage applications has been the maindriving force behind much of the research aimed at molecularhydrogen in clathrate hydrates1–5 and metal-organic frameworks (MOFs).6–9 Endohedral fullerene complexes encapsulating the H2 molecule, because of their low weight percentage of hydrogen, are unlikely candidates for hydrogenstorage applications. Nevertheless, a great deal of experimental research has been directed at H2 inside the C60 molecule(H2 @C60 ) and aza-thia-open-cage fullerene (ATOCF),10–19 togain understanding of the dynamics of the nanoconfined H2molecule. These spectroscopic investigations have been complemented by the rigorous theoretical treatments of the quantum dynamics of H2 in C60 ,20–22 C70 ,22, 23 and ATOCF.24Confinement of the H2 molecule results in the quantization of the three translational degrees of freedom of its centerof mass (cm). The discrete translational eigenstates are wellseparated in energy because of the small mass of the H2 andthe tightness of the confining cavity. The same holds for thequantized rotational levels of H2 owing to its exceptionallylarge rotational constant. The resulting coupled translationrotation (TR) energy level structure is sparse. It is even sparserbecause of the symmetry constraints on the total wave function of H2 (and D2 ), which lead to the existence of two distinct species, para-H2 (p-H2 ) which has only even-j rotationala) Electronic mail: 12/ 30.00states ( j 0, 2, . . . ), and ortho-H2 (o-H2 ) with odd-j rotational states only ( j 1, 3, . . . ). Consequently, the TR dynamics of the encapsulated H2 is highly quantum mechanical,especially at the low temperatures at which most of the spectroscopic measurements are performed.The quantum TR dynamics of the trapped H2 moleculeis strongly influenced by the symmetry of the confiningnanocage, which leaves clear fingerprints in the patterns ofthe degeneracies of the TR energy levels and their splittings,the types of quantum numbers appropriate for the assignmentof the translational excitations, and the nature of couplingbetween the angular momenta associated with the translational and rotational motions, respectively. This was broughtto light and analyzed in our systematic studies of the quantum five-dimensional (5D) TR dynamics and eigenstates ofH2 inside fullerenes of decreasing symmetry, C60 (Ih ),20–22C70 (D5h ),22, 23 and ATOCF (C1 ).24This line of investigation has lead us to consider the quantum TR dynamics of H2 confined inside the interstitial cavities of solid C60 . The centers of the C60 molecules forman fcc lattice. For temperatures above about 260 K, the C60molecules are orientationally disordered, rotate rather freely.At 260 K a phase transition occurs to the low-temperaturestructure which is orientationally ordered and has the P a 3̄crystal symmetry.25–27 It has been well established that H2singly occupies the octahedral interstitial sites of solid C60 .In the orientationally disordered phase, above 260 K, theoctahedral site has Oh symmetry. Below this temperature,the orientational ordering lowers the local symmetry of theoctahedral interstitial site to that of the point group S6 .11, 28Since the neighboring octahedral sites are well separated, the138, 244707-1 2013 AIP Publishing LLCThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

244707-2Ye et al.interaction between the H2 molecules occupying them is negligible. Consequently, the system can be treated as an isolatedH2 molecule inside the octahedral interstitial cavity.The quantum TR dynamics of interstitial H2 in solid C60has been probed experimentally using inelastic neutron scattering (INS),28 NMR,11, 29 infrared (IR) spectroscopy,30, 31 andRaman spectroscopy.32 It has also been the subject of several theoretical studies.28, 33, 34 Moreover, the diffusion of H2in solid C60 has been investigated both theoretically35 andexperimentally.36 Despite considerable experimental and theoretical efforts, quantitative understanding of the quantum TRdynamics of H2 in the octahedral site is still lacking. The experimental data regarding the TR excitations are rather limited and insufficiently resolved. The theoretical studies33, 34have provided valuable insights into the TR dynamics, butthe definitive interpretation of the measured spectra has notbeen achieved. This is in part due to the deficiencies of theexisting potentials for the interaction of H2 with the C60molecules forming the octahedral site. In addition, these theoretical treatments have employed various decoupling approximations aimed at reducing the dimensionality of the quantumdynamics treatment, which introduce uncertainties in the results and the conclusions based on them. There has been nofully coupled quantum 5D calculation of the TR eigenstatesof H2 in the octahedral cavity.This has motivated us to undertake the theoretical studywhose results are reported in this paper. We have performedrigorous quantum 5D calculations of the TR energy levels and wave functions of an H2 molecule in the octahedral interstitial site of solid C60 , for both the ground (ν 0) and the vibrationally excited (ν 1) states of the guestmolecule. The translational and rotational degrees of freedom of H2 are treated as fully coupled, without invoking anyreduced-dimensionality approximations. A pairwise additiveintermolecular potential energy surface (PES) was employed,which was used in several earlier studies of this system.11, 28, 33Our results are numerically exact for the PES employed,and constitute a benchmark with which other theoretical approaches can be compared. The TR eigenstates characterizedin the present work span the range of energies and types oftranslational and rotational excitations which go well beyondthose probed in the previous theoretical investigations, leading to new and interesting insights. Extensive comparison ismade with the existing spectroscopic data and the results ofearlier theoretical studies. In the next stage of the investigations, already under way in our group, the 5D coupled TRwave functions obtained in the present study will serve as aninput for the quantum simulation of the INS spectra of H2 insolid C60 , using the methodology recently developed by us,which allows rigorous calculations of the INS spectra of a hydrogen molecule inside a nanoscale cavity.37–39II. THEORETICAL METHODOLOGYA. Potential energy surface for H2 in the octahedralinterstitial siteWe consider solid C60 having the orientationally ordered low-temperature structure with the overall P a 3̄ crys-J. Chem. Phys. 138, 244707 (2013)Z(Bohr)Z20YXY(Bohr)20- 2020X(Bohr)- 20- 20FIG. 1. A schematic depiction of the octahedral interstitial site of solid C60 ,inside which an H2 molecule is trapped. Black circles represent the C60molecules.tal symmetry, and C60 molecules in the “major” (or p),orientation.11, 28 The local symmetry of the octahedral interstitial site is that of the point group S6 . The octahedral site isshown schematically in Fig. 1. It is formed by the six nearestneighbor C60 molecules, whose centers lie on the three mutually orthogonal axes, at the distance of 13.27 bohrs from thecenter of the octahedral cavity. The C60 molecules are takento be static (i.e., their centers and orientations fixed) and internally rigid.The bond length of H2 is held fixed. This is justifiedby the fact that the intramolecular stretch frequency of H2 , 4100 cm 1 , is much higher than those of the intermolecularTR modes. Therefore, the H2 stretch vibration is coupled veryweakly to the TR motions, and can be treated as frozen. Theposition of the H2 molecule within the site is defined completely by the five coordinates q (x, y, z, θ , φ); x, y, andz are the Cartesian coordinates of the center of mass (cm) ofH2 , while the two polar angles θ and φ specify the orientationof the molecule. As in our previous investigations of H2 infullerenes,20, 22–24 the intermolecular 5D PES of H2 in the octahedral site, V (q), is constructed by summing over the pairwise interactions of each of the two H atoms with every oneof the 360 C atoms of the six C60 molecules forming the site: V (q) VH C (rij ),(1)i H2 j C60where VH C is the H–C atom-atom pair potential to be specified shortly, and rij is the distance between the ith H atom andthe jth C atom. For VH C , we employ the potential:VH C (r) Be Cr A/r 6 ,(2)This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

244707-3Ye et al.J. Chem. Phys. 138, 244707 (2013)from that on the concave inner surface, and so does the interaction of H2 with the exterior and the interior of the fullerene.Consequently, H2 –C pair potential optimized for H2 insideC60 should not be expected to provide an accurate descriptionof the H2 -host interaction in the octahedral interstitial site.B. Calculation of the coupled translation-rotationeigenstatesThe methodology for accurate and efficient calculationof the coupled 5D TR energy levels and wave functions ofa hydrogen molecule inside nanoscale cavities, employed inthis work, has been described in detail in Ref. 38. This approach has evolved in our group over a number of years, inthe course of the theoretical investigations of the quantum TRdynamics of H2 and its isotopologues entrapped in the cagesof clathrate hydrates,40–42 fullerenes C60 and C70 ,20, 22 and anopen-cage derivative of C60 (ATOCF).24For H2 in the interstitial site formed by the six static C60molecules, using the coordinates q (x, y, z, θ , φ), the 5DHamiltonian for its coupled TR motions can be written as40 2H 2mFIG. 2. The 3D isosurfaces of the 5D PES for H2 inside the octahedral interstitial site. They are obtained by minimizing the PES with respect to thetwo angular coordinates of the H2 molecule, at every position of its center ofmass. In the bottom panel, the isosurfaces extend to much higher energies, inorder to reveal features of the PES which are not visible in the top panel.with A 5.941 eV Å6 , B 678.2 eV, and C 3.67 Å 1 . ThisH–C potential has been used in several earlier studies of H2 insolid C60 .11, 28, 33 The 3D isosurface representation of the PESV (q) is displayed in Fig. 2.An issue which merits a comment here is that of the suitability of the three-site H2 –C pair potential,22 developed byus earlier for H2 @C60 , for the present system. This H2 –Cpair potential was optimized to reproduce quantitatively theIR spectra of H2 @C60 . In the early stages of this work, weutilized it also to construct another 5D PES of H2 in the octahedral interstitial site. However, the results of the preliminary quantum 5D calculations on this PES were in significantly worse agreement with the experimental data than thoseobtained using the PES defined by Eqs. (1) and (2), and itsuse was discontinued. In the hindsight, this is not surprising.In the octahedral interstitial site, H2 interacts with the exterior surfaces of the six C60 molecules forming the site. Theπ -electron density on the convex outer surface of C60 differs 2 2 2 x 2 y 2 z2 Bj2 V (q).(3)In Eq. (3), m is the mass of H2 (2.016 amu), B and j2 denotethe rotational constant and the angular momentum operator ofthe diatomic, respectively, and V (q) is the 5D PES defined inEqs. (1) and (2).Comparison with the spectroscopic measurements on thissystem requires the computation of the TR eigenstates of thetrapped H2 in both the ground vibrational state ν 0 and the ν 1 vibrationally excited state. Using Be 60.853 cm 1 andα e 3.062 cm 1 for the gas-phase H2 , and Bν Be α e (ν 1/2), one obtains B0 59.322 cm 1 and B1 56.260 cm 1 ,for the ν 0 and ν 1 states, respectively. However, our calculations of the TR eigenstates of H2 (ν 1) in the octahedralsite of solid C60 utilized B1 55.6 cm 1 , the value estimatedfrom the IR spectroscopic measurements of this system.31This value is close to that for H2 (ν 1) inside C60 , B1 55.404 cm 1 , extracted from the IR spectra of H2 @C60 .18Both rotational constants are smaller than the gas-phase value;this is caused by the softening of the H2 intramolecular potential due to the interaction with the host environment.In the bound-state calculations,38 the dimension of thesinc-discrete variable representation (DVR) basis was 20 foreach of the three Cartesian coordinates x, y, and z, spanningthe range 3.78 bohr λ 3.78 bohrs (λ x, y, z). For pH2 , the angular basis included even-j rotational functions upto jmax 6, while the angular basis for o-H2 included odd-j rotational functions up to jmax 7. The cutoff parameter for thesize of the intermediate 3D eigenvector basis was set to 400lowest energy eigenvectors, resulting in the final 5D Hamiltonian matrices of dimension 11 200 for p-H2 and 14 400 foro-H2 . These basis set parameters were chosen following extensive testing, assuring that the TR energy levels reported inthis paper are converged to five significant figures or better.Diagonalization of the final 5D Hamiltonian matrices yieldsThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

Ye et al.244707-4J. Chem. Phys. 138, 244707 (2013)the fully coupled TR energy levels and wave functions whichare numerically exact for the 5D PES employed.III. RESULTS AND DISCUSSIONA. Translation-rotation energy levels of p-H2The first 39 excited TR energy levels of p-H2 (ν 0) inthe octahedral site of solid C60 , from our quantum 5D calculations, are given in Table I, together with their degeneracies. They encompass all j 0 TR levels with up to fourquanta of translational excitation; in this energy range arealso the j 2 TR levels with zero, one, and two quanta inTABLE I. TR energy levels of p-H2 (ν 0) molecule inside the octahedral cavity of solid C60 . The excitation energies E5D (in cm 1 ) from thequantum 5D calculations in this work are relative to the TR ground state E0 769.48 cm 1 ; g denotes the degeneracy of the levels. The quantum numbers n and l are those of the 3D isotropic harmonic oscillator. The quantum3D translational ( j 0) energy levels E3D (in cm 1 ) are from Ref. 33.iE5DgnljE3D (Ref. AuEuAuAgAgEgEgAgEgEgEgAgS6the translational modes. The global minimum of the PES is at 926.93 cm 1 while the TR ground-state energy E0 is equalto 769.48 cm 1 for p-H2 (ν 0); therefore, the zero-pointenergy (ZPE) of the TR motions is 157.45 cm 1 . To put this inperspective, the ZPE of the TR motions of p-H2 inside C60 is241.55 cm 1 .22 This value was calculated for H2 in theν 1 state, but the excitation of the H2 stretch mode has avery small effect on the TR ZPE. The fact that the TR ZPE forH2 in the octahedral site of solid C60 is substantially smallerthan that of H2 inside the C60 molecule implies that H2 is confined more tightly in the latter.Useful information regarding the nature of the translational excitations of the guest H2 molecule is provided by the3D reduced probability density (RPD) ρ i (x, y, z) in the translational (Cartesian) coordinates:40 ρi (x, y, z) ψi (x, y, z, θ, φ)ψi (x, y, z, θ, φ) sin θ dθ dφ,(4)where ψ i (x, y, z, θ , φ) is the ith T-R eigenfunction of the encapsulated p-H2 or o-H2 . Fig. 3 shows the RPDs of the threej 0 states (i 1, 2 in Table I) having one quantum of translational excitation, while Fig. 4 displays the RPDs of the sixj 0 states (i 3–6 in Table I) with two quanta of translational excitation. Fig. 4 in particular suggests strongly thatthe quantum numbers most appropriate for the assignment ofthe translationally excited TR eigenstates in the octahedralsite are those of the 3D isotropic harmonic oscillator (HO),the principal quantum number n and the orbital angular momentum quantum number l, whose allowed values are n,n 2, . . . 1 or 0, for odd or even n, respectively.43 When thepossible values of m, l m l, are taken into account, thedegree of degeneracy of the energy levels of the isotropic 3DHO is 12 (n 1)(n 2), e.g., 3 for n 1, 6 for n 2, 10 for n 3, and 15 for n 4.The five n 2 TR levels i 3, 4, 6, whose RPDs inFig. 4 resemble closely the d (l 2) orbitals of the hydrogenatom, are clearly the members of the n 2, l 2 quintuplet,and the sixth level i 5, with an almost spherical RPD, is thesingle n 2, l 0 state. Along the same lines, within the j 0, n 3 manifold, seven states, i 7, 8, 12, 14, and 16,can be assigned as n 3, l 3, while three states, i 9 and15, are assigned as n 3, l 1. The fact that energies of theTR levels depend not only on n, as they do in the 3D isotopicHO,43 but also on l, is evidence of the anharmonicity of thePES, which we observed also in our previous studies of H2inside C60 20–22 and the large cage of the structure II clathratehydrate.42However, the five n 2, l 2 states are not degenerate,and neither are the seven states with n 3, l 3 nor the threewith n 3, l 1. In fact, Table I shows that the TR levelsexhibit at most double degeneracy, starting with the translational fundamental whose n 1, l 1 triplet is split into anondegenerate (i 1) and a doubly degenerate level (i 2).This large reduction of degeneracy must be due to somethingother than the anharmonicity of the potential. The most obvious assumption is that it is caused by the crystal field of theinterstitial site, i.e., the symmetry of the potential felt by theH2 , which is S6 in this case. Our earlier theoretical studiesThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

244707-5Ye et al.FIG. 3. The 3D isosurfaces of the reduced probability densities in the translational (Cartesian) coordinates of the three n 1, l 1 states of p-H2 (ν 0) in the octahedral interstitial site. These j 0 states have one quantum ofexcitation in the translational modes. The excitation energies E are relativeto the ground state.have identified crystal-field induced splittings of degeneratetranslationally excited states of H2 in C60 (Ih ) for n 3,20C70 (D5h ),22 and the large cage of the sII clathrate hydrate (Td ,framework O atoms only).42J. Chem. Phys. 138, 244707 (2013)Yildirim and Harris have provided a beautiful in-depthanalysis of the effects of the S6 symmetry on the degeneraciesof the TR energy levels of H2 in the octahedral site of solidC60 .33 For the j 0 manifold, they investigated by means ofthe perturbation theory what happens to the n 1–3 energylevels of the 3D isotropic HO as its symmetry is graduallylowered, first by the anharmonicity, and then by the external potentials having (a) Oh symmetry appropriate for orientationally disordered solid C60 , and finally (b) S6 symmetrywhich applies to the orientationally ordered (P a 3̄) phase ofthe solid C60 ; see Fig. 1 of Ref. 33. Their perturbative model,despite its simplicity, incorporates the essential features of thefull problem. The pattern of degeneracies that it yields for thepotential of S6 symmetry, with the nondegenerate levels belonging to the 1D Au and Ag irreducible representations (irreps), and the doubly degenerate levels associated with the2D Eu and Eg irreps, matches that from our quantum 5D calculations in Table I, allowing a straightforward symmetry assignments of the latter. In the j 0, n 3 manifold, the energy ordering of our nondegenerate and degenerate quantum5D levels is generally reversed relative to that from the modelcalculations33 showing that, not surprisingly, the model is notsufficiently quantitative to account for such fine details.In addition to the perturbation treatment above, Yildirimand Harris have also performed a quantum 3D calculation ofthe purely translational ( j 0) energy levels of H2 with n 1–3 in the octahedral site, by treating H2 as a sphericalparticle on a radially anisotropic 3D PES with S6 symmetry,which depends only on the position of the cm of H2 .33 TheirPES was obtained by averaging the 5D PES employed in thiswork, specified by Eqs. (1) and (2), over the angular coordinates of H2 . The ( j 0) energy levels from their 3D calculation, for n 1–3, are also shown in Table I. In most cases,they differ by less than 1 cm 1 from the corresponding energy levels obtained in the 5D calculations. The differencesare larger in several instances. According to the 3D calculations, the levels i 8 and i 9 are nearly degenerate, separated by only 0.16 cm 1 , while in the 5D calculations theirenergies differ by 2.45 cm 1 . In addition, the energies of thelevels i 5, 15, and 16 from the 5D calculations are 2.21,3.92, and 2.65 cm 1 , respectively, higher than those obtainedin the 3D calculations of Yildirim and Harris.33 Very goodoverall agreement between the 3D and 5D quantum calculations implies that treating H2 as a spherical particle is a goodapproximation for calculating its translational ( j 0) energylevels in the octahedral site of solid C60 .Also shown in Table I are the quantum 5D TR eigenstatesbelonging to the j 2 manifold. The five j 2, n 0 statesare interspersed among the n 3 states with j 0. All 15j 2, n 1 states lie entirely in the energy range spannedby the n 4 states in the ground rotational state j 0, together with five (out of 30) j 2, n 2 TR eigenstates. The j 2 manifold, with or without translational excitations, wasnot treated at all by Yildirim and Harris.33 It is not clear howaccurate would be their computational approach, in which every n, j manifold is treated separately, as decoupled from others, in the case when several manifolds overlap.The five j 2 energy levels in the ground translationalstate n 0, i 10, 11, and 13, are split in a 1:2:2 degeneracyThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

244707-6Ye et al.J. Chem. Phys. 138, 244707 (2013)FIG. 4. The 3D isosurfaces of the reduced probability densities (RPDs) in the translational (Cartesian) coordinates of the six n 2 states of p-H2 (ν 0) inthe octahedral interstitial site. These j 0 states have two quanta of excitation in the translational modes. The panels (a)–(c), (e), and (f) show the RPDs of thefive n 2, l 2 states, while panel (d) shows the RPD of the n 2, l 0 state. The excitation energies E are relative to the ground state.pattern by the angular anisotropy (crystal-field effects) of theoctahedral site. The splittings between the three componentsof the quintuplet are highly uneven. The nondegenerate andthe closest doubly degenerate j 2 levels, i 10 and 11, aresplit by only 1.93 cm 1 , while the second degenerate j 2level, i 13, lies 5.16 cm 1 above i 11.Table II compares the energies of j 2 (and j 1) rotational levels from the quantum 5D calculations (for the groundtranslational state) with those from the purely rotational quantum 2D bound-state calculation on the same PES in which thecm of H2 (ν 0) is fixed at the center of the octahedral site.Also shown are the overall splittings , which for j 2 aredefined as the energy difference between the highest energydoubly degenerate component and the nondegenerate component of the quintuplet. The overall splitting of the quantum 5D j 2 levels (the energy difference between the levelsi 10 and i 13 in Table I), 7.09 cm 1 , is more than a factorof two greater than that from the quantum 2D calculations,3.27 cm 1 . This demonstrates the importance of includingthe vibrational averaging over ground-state wave function forThis article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP: On: Fri, 14 Mar 2014 14:52:32

244707-7Ye et al.J. Chem. Phys. 138, 244707 (2013)TABLE II. Energies (in cm 1 ) of the j 1 and j 2 rotational levels,together with their respective splittings (in cm 1 ), from the quantum 5Dcalculations (for the ground translational state) and purely rotational quantum2D calculations, for H2 (ν 0) molecule inside the octahedral cavity of solidC60 in this work. g denotes the degeneracy of the level. For j 1, is theenergy difference between the doubly degenerate and nondegenerate components of the triplet. For j 2, is the energy difference between the highestenergy doubly degenerate component and the nondegenerate component ofthe quintuplet.j 1j 2g5D2D12115.61119.26 3.65116.81119.29 2.48122351.79353.72358.88 7.09352.55353.52355.82 3.27obtaining the accurate value of the splitting of the j 2 manifold. The larger value of the splitting obtained from the 5Dcalculations can be readily explained: the ground-state wavefunction of H2 is significantly delocalized, and samples theregions of the PES closer to the site walls, where the angularanisotropy is stronger than in the vicinity of the center of thesite, at which the H2 is fixed in the quantum 2D calculations.B. Translation-rotation energy levels of o-H2When both the translational and rotational degrees offreedom of a nanoconfined H2 molecule are excited, as in thej 1, n 1 manifold of the caged o-H2 , the TR coupling liftsin part the degeneracy of the manifold. The remaining degeneracies are reduced further by the crystal-field effects of theenvironment. Therefore, the final pattern of the level degeneracies is the result of the interplay between the TR couplingand the symmetry of the nanocavity in which H2 is entrapped.For H2 inside the highly symmetric (icosahedral)fullerene C60 , our quantum 5D calculations of the TR eigenstates and their analysis20–22 have shown that the orbital angular momentum l and the rotational angular momentum j couple vectorially to give the total angular momentum λ l jhaving the values λ l j, l j 1, . . . , l j , and thedegeneracy of 2λ 1. The eigenstates with the same quantum numbers n and j (both nonzero) split into as many distinct levels as there are different values of λ, e

X(Bohr) Y(Bohr) Z(Bohr) 20 20-20-20-20 20 FIG. 1. A schematic depiction of the octahedral interstitial site of solid C60, inside which an H2 molecule is trapped. Black circles represent the C60 molecules. tal symmetry, and C 60 molecules in the “major” (or p), orientation.11,28 The local