We report rigorous quantum calculations of the translation-rotation (TR) eigenstates of *para*- and *ortho*-H_{2}O@C_{60}. They provide a comprehensive description of the dynamical behavior of H_{2}O inside the fullerene having icosahedral (*I _{h}*) symmetry. The TR eigenstates are assigned in terms of the irreducible representations of the proper symmetry group of H

_{2}O@C

_{60}, as well as the appropriate translational and rotational quantum numbers. The coupling between the orbital and the rotational angular momenta of the caged H

_{2}O gives rise to the total angular momentum

*λ*, which additionally labels each TR level. The calculated TR levels allow tentative assignments of a number of transitions in the recent experimental INS spectra of H

_{2}O@C

_{60}that have not been assigned previously.

In 2011, Murata and co-workers accomplished the feat of encapsulating a single H_{2}O molecule inside the C_{60} fullerene.^{1} The resulting endohedral complex H_{2}O@C_{60} constitutes an extraordinary nanoscale laboratory for detailed experimental and theoretical investigations of the dynamical behavior of an isolated polar molecule, H_{2}O, in a highly symmetrical, nonpolar, and homogeneous environment. The confinement of H_{2}O inside the C_{60} cage results in the quantization of the translational motions associated with the center-of-mass (c.m.) displacements of the guest molecule and their coupling to its quantized rotational degrees of freedom. As a result, the coupled translation-rotation (TR) dynamics of the caged H_{2}O is highly quantum mechanical. It has already been investigated by means of inelastic neutron scattering (INS),^{2,3} far-infrared,^{2} and NMR^{2,4} spectroscopy. The INS spectra recorded at low temperatures in the more recent in-depth study of H_{2}O@C_{60} exhibit numerous peaks corresponding to the transitions out the ground states of *para*- and *ortho*-H_{2}O, 0_{00} and 1_{01} respectively, to a broad range of excited TR states of these two nuclear spin-isomers.^{3} A surprising finding of this study was that the three-fold degenerate *ortho*-H_{2}O 1_{01} ground state is split into two substates, the lowest energy 1_{01}^{a} with degeneracy *g* = 2 and the higher energy 1_{01}^{b} with degeneracy *g* = 1, about 0.48 meV apart.^{3} This splitting implies a breaking of the icosahedral (*I _{h}*) symmetry of the C

_{60}environment, whose origin is not well understood. One possibility is that it arises from the dipolar interactions between the H

_{2}O molecules occupying the neighboring C

_{60}cages.

^{3}The low-lying transitions in the measured INS spectra, particularly those originating in the ground

*para*-H

_{2}O (0

_{00}) state, could be assigned to certain rotational excitations based on the close correspondence with the rotational levels of gaseous H

_{2}O, revealing that the rotations of the caged H

_{2}O are weakly hindered. However, as the resolution of the spectra decreases with increasing energy transfer, assignments relying solely on comparison with gaseous water become increasingly ambiguous, leaving many higher-energy peaks unassigned. Moreover, Horsewill

*et al.*

^{3}were unable to assign any of the measured INS peaks to translational excitations of the entrapped H

_{2}O molecule. Thus, no information was extracted from the INS spectra about the H

_{2}O translational dynamics and eigenstates, and their coupling to the quantized rotation of the guest molecule.

Quantitative analysis and reliable assignment of the measured INS spectra of H_{2}O@C_{60} can be achieved only with the quantum calculations of the TR eigenstates of the encapsulated H_{2}O, followed by rigorous quantum simulations of the INS spectra and their comparison with experiment. This was done very successfully in the case of the endofullerene H_{2}@C_{60}.^{5–7} But the theoretical treatments of H_{2}O@C_{60} to date have been limited to classical molecular dynamics (MD) simulations of the rotational motions of the trapped H_{2}O molecule.^{8,9}

In this Communication, we report the results of the first fully coupled quantum six-dimensional (6D) calculations of the TR eigenstates of *para*- and *ortho*-H_{2}O inside C_{60}; both H_{2}O and C_{60} are taken to be rigid. Since we consider H_{2}O in an isolated C_{60} cage with *I _{h}* symmetry, our computed results do not show any splitting of the 1

_{01}ground state of

*ortho*-H

_{2}O that is observed experimentally.

^{3}Nevertheless, it is essential to have a quantitative description of the dynamical behavior of the entrapped H

_{2}O for the well defined, limiting case of isolated C

_{60}, as an indispensable starting point for future efforts to model and understand the symmetry-breaking effects. The TR energy level structure computed in this study provides a detailed picture of the quantum dynamics of the endohedral H

_{2}O in an icosahedral (

*I*) environment and reveals its key features. In addition, through comparison with the calculated TR levels, several previously unassigned INS transitions

_{h}^{3}are assigned to excitations combining one and two quanta in the translational mode with the H

_{2}O rotation.

The methodology used in the quantum 6D calculations of the TR eigenstates of H_{2}O@C_{60} is described in the supplementary material.^{10} No *ab initio*-calculated 6D intermolecular potential energy surface (PES) for H_{2}O@C_{60} is presently available. Therefore, as was done previously for H_{2}@C_{60},^{5} this intermolecular PES is constructed as a sum over the pairwise interactions, modeled with the Lennard-Jones (LJ) 12-6 potentials, of each atom of H_{2}O with each atom of C_{60}. The LJ parameters employed are the carbon-water oxygen parameters *σ*_{C−OW} = 3.372 Å and *ϵ*_{C−OW} = 0.1039 kcal/mol, and the carbon-water hydrogen parameters *σ*_{C−HW} = 2.640 Å and *ϵ*_{C−HW} = 0.0256 kcal/mol. They correspond to the parameter set referred to as MD1 in the work of Aluru *et al.*,^{9} obtained by fitting to DFT-SAPT *ab initio* graphene-water interaction curve.

The TR Hamiltonian is invariant to all the operations of the group comprised of the direct product of *I _{h}*(

*M*) and

*S*

_{2}. The former comes from the geometry of the C

_{60}cage and the latter is due to the H

_{2}O geometry. The $ I h ( 12 ) = I h \u2297 S 2 $ group has been described recently by Poirier.

^{11}It has order 240 and 20 irreducible representations (irreps). The irreps are just those of

*I*, but doubled. Each such pair consists of one irrep that is symmetric under the interchange of the H nuclei (even

_{h}*k*or

*para*) and another that is antisymmetric under such interchange (odd

*k*or

*ortho*). These are denoted as “Γ

^{(s)}” and “Γ

^{(a)},” respectively, where Γ =

*A*,

_{g}*T*

_{1g},

*T*

_{2g}, etc., is an irrep of

*I*. Every TR eigenstate of H

_{h}_{2}O@C

_{60}must transform according to one of these irreps.

Select TR energy levels from the quantum 6D calculations are shown for *para*-H_{2}O@C_{60} in Table I and for *ortho*-H_{2}O@C_{60} in Table II. Tables S1 and S2 of the supplementary material^{10} present all calculated TR energy levels of these two endohedral complexes up to about 400 cm^{−1} above the ground state of the entrapped *para*-H_{2}O. Also shown there are the energy levels calculated for the free-rotor (rigid) H_{2}O; a few of these appear in Table III as well. Each TR level has a symmetry assignment in terms of one of the irreps of $ I h ( 12 ) $. These assignments were made based (a) on level degeneracies (*A* irreps are 1D, *T* are 3D, *G* are 4D, and *H* are 5D) and (b) on the results of direct computation of the transformation properties of the eigenstates under the operations of $ I h ( 12 ) $. For purely rotational states, labeled as *j*_{kakc}, dominant contribution comes from the rotational basis functions having a single *j* value. It is evident from Tables III, S1 and S2^{10} that the energies of the rotational levels of the caged *para*- and *ortho*-H_{2}O are close to those of gaseous, freely rotating H_{2}O, evidence for the weakly hindered rotation of the guest molecule, also seen experimentally.^{3} For *j* ≥ 3, the (2*j* + 1)-fold degeneracy (7 or greater) exceeds the largest degeneracy that the “crystal field” of *I _{h}* symmetry can accommodate, which is five-fold. Consequently, every 3

_{kakc}rotational level, with

*g*= 7, splits into one

*g*= 3 level (

*T*-type irrep) and one

*g*= 4 level (

*G*-type irrep). Similarly, each 4

_{kakc}level, with

*g*= 9, splits into a

*g*= 4 level (

*G*-type irrep) and a

*g*= 5 level (

*H*-type irrep). In Tables I, II, S1 and S2,

^{10}such pairs, typically split by less than 1 cm

^{−1}, are denoted as (

*a*) and (

*b*) in the assignments.

ΔE
. | g
. | λ
. | l
. | j
. | p(j)
. | ΔR
. | Irrep . | Assignment . |
---|---|---|---|---|---|---|---|---|

0.00 | 1 | 0 | 0 | 0 | 0.969 | 0.255 | $ A g ( s ) $ | 0_{00} |

162.08 | 3 | 1 | 1 | 0 | 0.931 | 0.323 | $ T 1 u ( s ) $ | n = 1 |

197.54 | 1 | 0 | 1 | 1 | 0.892 | 0.326 | $ A g ( s ) $ | n = 1 + 1_{11} |

198.33 | 3 | 1 | 1 | 1 | 0.969 | 0.324 | $ T 1 g ( s ) $ | n = 1 + 1_{11} |

200.13 | 5 | 2 | 1 | 1 | 0.851 | 0.321 | $ H g ( s ) $ | n = 1 + 1_{11} |

229.05 | 5 | 2 | 1 | 2 | 0.981 | 0.325 | $ H u ( s ) $ | n = 1 + 2_{02} |

233.26 | 3 | 3 | 1 | 2 | 0.918 | 0.322 | $ T 2 u ( s ) $ | [n = 1 + 2_{02}](a) |

233.54 | 4 | 3 | 1 | 2 | 0.919 | 0.322 | $ G u ( s ) $ | [n = 1 + 2_{02}](b) |

234.78 | 3 | 1 | 1 | 2 | 0.959 | 0.323 | $ T 1 u ( s ) $ | n = 1 + 2_{02} |

325.97 | 5 | 2 | 2 | 0 | 0.893 | 0.378 | $ H g ( s ) $ | n = 2 |

328.21 | 1 | 0 | 0 | 0 | 0.893 | 0.377 | $ A g ( s ) $ | n = 2 |

ΔE
. | g
. | λ
. | l
. | j
. | p(j)
. | ΔR
. | Irrep . | Assignment . |
---|---|---|---|---|---|---|---|---|

0.00 | 1 | 0 | 0 | 0 | 0.969 | 0.255 | $ A g ( s ) $ | 0_{00} |

162.08 | 3 | 1 | 1 | 0 | 0.931 | 0.323 | $ T 1 u ( s ) $ | n = 1 |

197.54 | 1 | 0 | 1 | 1 | 0.892 | 0.326 | $ A g ( s ) $ | n = 1 + 1_{11} |

198.33 | 3 | 1 | 1 | 1 | 0.969 | 0.324 | $ T 1 g ( s ) $ | n = 1 + 1_{11} |

200.13 | 5 | 2 | 1 | 1 | 0.851 | 0.321 | $ H g ( s ) $ | n = 1 + 1_{11} |

229.05 | 5 | 2 | 1 | 2 | 0.981 | 0.325 | $ H u ( s ) $ | n = 1 + 2_{02} |

233.26 | 3 | 3 | 1 | 2 | 0.918 | 0.322 | $ T 2 u ( s ) $ | [n = 1 + 2_{02}](a) |

233.54 | 4 | 3 | 1 | 2 | 0.919 | 0.322 | $ G u ( s ) $ | [n = 1 + 2_{02}](b) |

234.78 | 3 | 1 | 1 | 2 | 0.959 | 0.323 | $ T 1 u ( s ) $ | n = 1 + 2_{02} |

325.97 | 5 | 2 | 2 | 0 | 0.893 | 0.378 | $ H g ( s ) $ | n = 2 |

328.21 | 1 | 0 | 0 | 0 | 0.893 | 0.377 | $ A g ( s ) $ | n = 2 |

ΔE
. | g
. | λ
. | l
. | j
. | p(j)
. | ΔR
. | Irrep . | Assignment . |
---|---|---|---|---|---|---|---|---|

23.74 | 3 | 1 | 0 | 1 | 0.987 | 0.256 | $ T 1 u ( a ) $ | 1_{01} |

183.12 | 3 | 1 | 1 | 1 | 0.979 | 0.325 | $ T 1 g ( a ) $ | n = 1 + 1_{01} |

186.71 | 5 | 2 | 1 | 1 | 0.952 | 0.323 | $ H g ( a ) $ | n = 1 + 1_{01} |

191.02 | 1 | 0 | 1 | 1 | 0.975 | 0.322 | $ A g ( a ) $ | n = 1 + 1_{01} |

238.34 | 5 | 2 | 1 | 2 | 0.966 | 0.325 | $ H u ( a ) $ | n = 1 + 2_{12} |

243.08 | 3 | 1 | 1 | 2 | 0.926 | 0.324 | $ T 1 u ( a ) $ | n = 1 + 2_{12} |

244.41 | 4 | 3 | 1 | 2 | 0.768 | 0.316 | $ G u ( a ) $ | [n = 1 + 2_{12}](a) |

244.64 | 3 | 3 | 1 | 2 | 0.773 | 0.316 | $ T 2 u ( a ) $ | [n = 1 + 2_{12}](b) |

362.06 | 3 | 1 | 2 | 1 | 0.946 | 0.380 | $ T 1 g ( a ) $ | n = 2 + 1_{10} |

370.16 | 3 | 3 | 2 | 1 | 0.740 | 0.369 | $ T 2 g ( a ) $ | [n = 2 + 1_{10}](a) |

370.20 | 4 | 3 | 2 | 1 | 0.803 | 0.371 | $ G g ( a ) $ | [n = 2 + 1_{10}](b) |

371.31 | 5 | 2 | 2 | 1 | 0.884 | 0.375 | $ H g ( a ) $ | n = 2 + 1_{10} |

372.77 | 3 | 1 | 0 | 1 | 0.925 | 0.376 | $ T 1 g ( a ) $ | n = 2 + 1_{10} |

ΔE
. | g
. | λ
. | l
. | j
. | p(j)
. | ΔR
. | Irrep . | Assignment . |
---|---|---|---|---|---|---|---|---|

23.74 | 3 | 1 | 0 | 1 | 0.987 | 0.256 | $ T 1 u ( a ) $ | 1_{01} |

183.12 | 3 | 1 | 1 | 1 | 0.979 | 0.325 | $ T 1 g ( a ) $ | n = 1 + 1_{01} |

186.71 | 5 | 2 | 1 | 1 | 0.952 | 0.323 | $ H g ( a ) $ | n = 1 + 1_{01} |

191.02 | 1 | 0 | 1 | 1 | 0.975 | 0.322 | $ A g ( a ) $ | n = 1 + 1_{01} |

238.34 | 5 | 2 | 1 | 2 | 0.966 | 0.325 | $ H u ( a ) $ | n = 1 + 2_{12} |

243.08 | 3 | 1 | 1 | 2 | 0.926 | 0.324 | $ T 1 u ( a ) $ | n = 1 + 2_{12} |

244.41 | 4 | 3 | 1 | 2 | 0.768 | 0.316 | $ G u ( a ) $ | [n = 1 + 2_{12}](a) |

244.64 | 3 | 3 | 1 | 2 | 0.773 | 0.316 | $ T 2 u ( a ) $ | [n = 1 + 2_{12}](b) |

362.06 | 3 | 1 | 2 | 1 | 0.946 | 0.380 | $ T 1 g ( a ) $ | n = 2 + 1_{10} |

370.16 | 3 | 3 | 2 | 1 | 0.740 | 0.369 | $ T 2 g ( a ) $ | [n = 2 + 1_{10}](a) |

370.20 | 4 | 3 | 2 | 1 | 0.803 | 0.371 | $ G g ( a ) $ | [n = 2 + 1_{10}](b) |

371.31 | 5 | 2 | 2 | 1 | 0.884 | 0.375 | $ H g ( a ) $ | n = 2 + 1_{10} |

372.77 | 3 | 1 | 0 | 1 | 0.925 | 0.376 | $ T 1 g ( a ) $ | n = 2 + 1_{10} |

Experiment . | Theory . | Free-rotor H_{2}O
. | Assignment . |
---|---|---|---|

2.50 ± 0.05 | $ 0 00 \u2192 1 01 a $ | ||

2.94 | 2.95 | 0_{00} → 1_{01} | |

3.02 ± 0.05 | $ 0 00 \u2192 1 01 b $ | ||

9.20 ± 0.05 | 9.74 | 9.86 | 0_{00} → 2_{12} |

16.6 ± 0.1 | 16.37/16.86 | 16.78/16.97 | 0_{00} → 2_{21} 3_{03} |

22.8±0.1 | 22.70 − 23.68 | 0 → _{00}n = 1 + 1 _{01} | |

26.1 ± 0.2 | 25.78 | 26.35 | 0_{00} → 3_{21} |

30.8±0.2 | 29.55 − 30.30 | 0 → _{00}n = 1 + 2 _{12} | |

39.4±0.2 | 33.36 − 38.04 | 0 → _{00}3/_{30}n = 1 + 2 _{21} | |

40.7 ± 0.2 | 39.91 | 40.41 | 0_{00} → 5_{05} |

45.1±0.2 | 44.89 − 46.22 | 0 → _{00}n = 2 + 1 _{10} | |

47.3 ± 0.2 | 48.40 | 47.67 | 0_{00} → 4_{32} |

Experiment . | Theory . | Free-rotor H_{2}O
. | Assignment . |
---|---|---|---|

2.50 ± 0.05 | $ 0 00 \u2192 1 01 a $ | ||

2.94 | 2.95 | 0_{00} → 1_{01} | |

3.02 ± 0.05 | $ 0 00 \u2192 1 01 b $ | ||

9.20 ± 0.05 | 9.74 | 9.86 | 0_{00} → 2_{12} |

16.6 ± 0.1 | 16.37/16.86 | 16.78/16.97 | 0_{00} → 2_{21} 3_{03} |

22.8±0.1 | 22.70 − 23.68 | 0 → _{00}n = 1 + 1 _{01} | |

26.1 ± 0.2 | 25.78 | 26.35 | 0_{00} → 3_{21} |

30.8±0.2 | 29.55 − 30.30 | 0 → _{00}n = 1 + 2 _{12} | |

39.4±0.2 | 33.36 − 38.04 | 0 → _{00}3/_{30}n = 1 + 2 _{21} | |

40.7 ± 0.2 | 39.91 | 40.41 | 0_{00} → 5_{05} |

45.1±0.2 | 44.89 − 46.22 | 0 → _{00}n = 2 + 1 _{10} | |

47.3 ± 0.2 | 48.40 | 47.67 | 0_{00} → 4_{32} |

Purely translationally excited states can be assigned using the quantum numbers *n* and *l* of the 3D isotropic harmonic oscillator (HO), as in the case of H_{2}@C_{60}.^{5} The first translationally excited state of *para*-H_{2}O (Tables I and S1) is a three-fold degenerate $ ( g = 3 ) T 1 u ( s ) $ level at 162.08 cm^{−1} (20.10 meV). It is readily assigned as *n* = 1 due to the large value of the standard deviation of *R* (Δ*R*), and the dominant *j* = 0 rotational contribution, and it represents the fundamental translational excitation. Two closely spaced levels of *para*-H_{2}O with two quanta of translational excitation (*n* = 2) (Tables I and S1^{10}), at 325.97 cm^{−1} (*g* = 5) and 328.21 cm^{−1} (*g* = 1), correspond to *l* = 2 and *l* = 0, respectively. Their splitting reflects the (weak) anharmonicity of the translational mode. In addition to these and other states that are predominantly either rotational or translational, many TR states shown in Tables S1 and S2^{10} combine both translational and rotational excitations; they are assigned in terms of the quantum numbers *n*, *l* and *j*_{kakc}.

The $ I h ( 12 ) $ irreps uniquely label the TR energy levels of H_{2}O@C_{60} and account for their degeneracies, while the (*n*, *l*, *j*_{kakc}) assignments specify the nature of their excitation. There is also a complementary labeling scheme which, as shown below, provides valuable physical insight. It follows from the model previously developed to account for the TR level splittings and degeneracies in H_{2}@C_{60}.^{5} Adapted to H_{2}O, the model states that the orbital angular momentum **l** of the c.m. of H_{2}O and the rotational angular momentum **j** of H_{2}O couple to give the total angular momentum *λ* = **l** + **j**, with *λ* = *l* + *j*, *l* + *j* − 1, …, |*l* − *j*|. The values of *l* are those allowed for the quantum number *n* of the 3D isotropic HO. The TR states having the same quantum numbers *n* and *j*_{kakc} are split into as many distinct levels as there are different values of *λ*, each with the degeneracy 2*λ* + 1. It is easy to see that this model applies to all (*n*, *l*, *j*_{kakc}) multiplets in Tables I and II, correctly accounting for the number of levels and their degeneracies.

The calculated TR energy level structure exhibits a conspicuous feature for which the above angular-momentum-coupling model provides a straightforward and natural explanation. In Tables I and II, there are pairs of TR levels (marked in italics), belonging to different irreps, that are nearly degenerate, separated by 1 cm^{−1} or less: for *para*-H_{2}O, 233.26 (*g* = 3) and 233.54 cm^{−1} (*g* = 4) (*n* = 1 + 2_{02}); for *ortho*-H_{2}O, 244.41 (*g* = 4) and 244.64 cm^{−1} (*g* = 3) (*n* = 1 + 2_{12}), 370.16 (*g* = 3) and 370.20 cm^{−1} (*g* = 4) (*n* = 2 + 1_{10}). Many more examples are readily found in Tables S1 and S2.^{10}

We note that all closely spaced pairs of TR levels (in italics) in Tables I and II have *λ* = 3. In a spherically symmetric environment each pair would appear as a single *λ* = 3 seven-fold degenerate level. But in the *I _{h}* “crystal field,” the largest degeneracy possible is five-fold. This leads to splitting of the seven-fold degeneracy into two

*λ*= 3 levels, one that is three-fold degenerate and belongs to a

*T*-type irrep and another that is four-fold degenerate and belongs to a

*G*-type irrep—precisely the splitting pattern of the near-degenerate pairs in Tables I and II. The very small magnitudes of the splittings reflect and are a measure of the weak

*I*corrugation, or deviation from spherical symmetry. We note that the same was observed for H

_{h}_{2}@C

_{60}.

^{5}This illustrates the physical significance of the quantum number

*λ*, and can be viewed as its distinct fingerprint in the TR level structure.

The INS spectra of H_{2}O@C_{60} are dominated by *para*↔*ortho*, as well as *ortho*↔*ortho*, transitions.^{3} Therefore, the transitions out of the ground *para*-H_{2}O 0_{00} state directly give the energies of the TR levels of *ortho*-H_{2}O, which can be compared with our calculations. Table III presents the comparison between the experimental and calculated energies of the transitions in the INS spectra of *para*-H_{2}O@C_{60}. The experimental values, and the assignments of the purely rotational transitions, are taken from Table 1 of Ref. 3, while the theoretical results are from Table S2^{10} (converted to meV). The INS spectra^{3} show the splitting of the *ortho*-H_{2}O 1_{01} ground state into two substates, 1_{01}^{a} at 2.50 meV and 1_{01}^{b} at 3.02 meV. The calculated 1_{01} state is of course not split, and lies at 2.94 meV. The other calculated and measured rotational energy levels of *ortho*-H_{2}O in Table III differ by no more than 1 meV.

The symmetry-breaking process responsible for the splitting of the ground state of *ortho*-H_{2}O is expected to split all excited rotational (and translation-rotation) states of the guest molecule. However, the resulting fine structure is not resolved in the experimental spectra, but its presence is indicated by the asymmetry of a number of peaks observed in the spectra.^{3}

As mentioned earlier, Horsewill *et al.*^{3} could not assign any of the peaks in their measured INS spectra to translational excitations of the caged H_{2}O. Consequently, their important study left a gap in our quantitative understanding of the translational dynamics of H_{2}O and its coupling to the H_{2}O rotations, which our results begin to fill. In Table 1 of Ref. 3 four INS transitions out of 0_{00} are unassigned, since their energies deviate significantly from the rotational levels of gaseous H_{2}O. In Table III, these transitions are boldfaced, and we can assign them tentatively, with reasonable confidence, by matching them to the calculated TR levels of *ortho*-H_{2}O@C_{60} in Tables II and S2.

Thus, the transition measured at 22.8 meV is assigned as *n* = 1 + 1_{01}, the first translationally excited state of *ortho*-H_{2}O. It is a triplet of levels with *λ* = 1, 2, 0 in the energy range 22.70–23.68 meV. Until the INS spectra of H_{2}O@C_{60} have been rigorously calculated, as was done previously for H_{2}@C_{60},^{6,7} it is not possible to determine the contribution of each *λ* sublevel to this band. Subtracting from 22.8 meV the mean energy^{3} of the states 1_{01}^{a} and 1_{01}^{b}, 2.85 meV, gives 20 meV, what should be a good experimental estimate of the frequency of the translational fundamental of H_{2}O@C_{60}. This happens to be nearly identical to the calculated translational fundamental (*n* = 1) of *para*-H_{2}O, 20.1 meV (Table I). It is also surprisingly close to the translational fundamental of H_{2}@C_{60}, calculated^{6} and observed^{12} around 22.5 meV, given the large difference in the masses of H_{2} and H_{2}O.

The assignments of the other three previously unassigned transitions shown in Table III are also made by comparison with the calculated TR levels of *ortho*-H_{2}O@C_{60}. In particular, the transition measured at 45.1 meV is assigned as *n* = 2 + 1_{10}, i.e., a two-quanta translational excitation combined with 1_{10} rotation. If the calculated energy of 1_{10}, 5.19 meV, is subtracted from 45.1 meV, one obtains ∼40 meV. That is close to the calculated energy of the *n* = 2 level of *para*-H_{2}O, ∼40.5 meV, and twice the translational fundamental at 20 meV.

In conclusion, we have presented the results of the first rigorous quantum 6D calculations of the coupled TR eigenstates of *para*- and *ortho*-H_{2}O@C_{60}. They characterize the quantum dynamical behavior of H_{2}O molecule inside the C_{60} fullerene with the icosahedral (*I _{h}*) symmetry. In particular, this study sheds light on the translational excitations of the entrapped H

_{2}O that have not been identified experimentally.

^{3}The computed TR eigenstates have been symmetry-labeled by the irreps of the $ I h ( 12 ) $ group and assigned with the translational quantum numbers

*n*,

*l*of the 3D isotropic HO and the rotational quantum numbers

*j*

_{kakc}. In addition, every TR level can be labeled by the total angular momentum quantum number

*λ*, arising from the coupling of the orbital angular momentum of the c.m. of H

_{2}O and the rotational angular momentum of the guest molecule.

^{5}TR levels with

*λ*≥ 3 are split into closely spaced doublets due to the “crystal field” of

*I*symmetry. Several transitions in the measured INS spectra that were left unassigned

_{h}^{3}have been tentatively assigned as combined translation-rotation excitations. From this, the translational fundamental is deduced to be at ∼20 meV. More complete analysis and definitive assignment of the experimental INS spectra of H

_{2}O@C

_{60}requires generalization of the recently developed quantum methodology for accurate simulation of the INS spectra of nanoconfined diatomics.

^{13–15}This will also allow us to investigate whether INS selection rules analogous to those discovered for H

_{2}@C

_{60}

^{6,7,11,16}exist in H

_{2}O@C

_{60}as well.

Partial support of this research by the NSF through the Grant No. CHE-1112292 to Z.B. is gratefully acknowledged. P.M.F. thanks Professor Daniel Neuhauser for generously allowing access to his computational resources.

## REFERENCES

_{2}O and C

_{60}employed in the calculations.