We report rigorous quantum calculations of the translation-rotation (TR) eigenstates of para- and ortho-H2O@C60. They provide a comprehensive description of the dynamical behavior of H2O inside the fullerene having icosahedral (Ih) symmetry. The TR eigenstates are assigned in terms of the irreducible representations of the proper symmetry group of H2O@C60, as well as the appropriate translational and rotational quantum numbers. The coupling between the orbital and the rotational angular momenta of the caged H2O 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 H2O@C60 that have not been assigned previously.

In 2011, Murata and co-workers accomplished the feat of encapsulating a single H2O molecule inside the C60 fullerene.1 The resulting endohedral complex H2O@C60 constitutes an extraordinary nanoscale laboratory for detailed experimental and theoretical investigations of the dynamical behavior of an isolated polar molecule, H2O, in a highly symmetrical, nonpolar, and homogeneous environment. The confinement of H2O inside the C60 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 H2O is highly quantum mechanical. It has already been investigated by means of inelastic neutron scattering (INS),2,3 far-infrared,2 and NMR2,4 spectroscopy. The INS spectra recorded at low temperatures in the more recent in-depth study of H2O@C60 exhibit numerous peaks corresponding to the transitions out the ground states of para- and ortho-H2O, 000 and 101 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-H2O 101 ground state is split into two substates, the lowest energy 101a with degeneracy g = 2 and the higher energy 101b with degeneracy g = 1, about 0.48 meV apart.3 This splitting implies a breaking of the icosahedral (Ih) symmetry of the C60 environment, whose origin is not well understood. One possibility is that it arises from the dipolar interactions between the H2O molecules occupying the neighboring C60 cages.3 The low-lying transitions in the measured INS spectra, particularly those originating in the ground para-H2O (000) state, could be assigned to certain rotational excitations based on the close correspondence with the rotational levels of gaseous H2O, revealing that the rotations of the caged H2O 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 H2O molecule. Thus, no information was extracted from the INS spectra about the H2O 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 H2O@C60 can be achieved only with the quantum calculations of the TR eigenstates of the encapsulated H2O, 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 H2@C60.5–7 But the theoretical treatments of H2O@C60 to date have been limited to classical molecular dynamics (MD) simulations of the rotational motions of the trapped H2O 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-H2O inside C60; both H2O and C60 are taken to be rigid. Since we consider H2O in an isolated C60 cage with Ih symmetry, our computed results do not show any splitting of the 101 ground state of ortho-H2O that is observed experimentally.3 Nevertheless, it is essential to have a quantitative description of the dynamical behavior of the entrapped H2O for the well defined, limiting case of isolated C60, 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 H2O in an icosahedral (Ih) environment and reveals its key features. In addition, through comparison with the calculated TR levels, several previously unassigned INS transitions3 are assigned to excitations combining one and two quanta in the translational mode with the H2O rotation.

The methodology used in the quantum 6D calculations of the TR eigenstates of H2O@C60 is described in the supplementary material.10 No ab initio-calculated 6D intermolecular potential energy surface (PES) for H2O@C60 is presently available. Therefore, as was done previously for H2@C60,5 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 H2O with each atom of C60. The LJ parameters employed are the carbon-water oxygen parameters σC−OW = 3.372 Å and ϵC−OW = 0.1039 kcal/mol, and the carbon-water hydrogen parameters σC−HW = 2.640 Å and ϵC−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 Ih(M) and S2. The former comes from the geometry of the C60 cage and the latter is due to the H2O geometry. The I h ( 12 ) = I h 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 Ih, but doubled. Each such pair consists of one irrep that is symmetric under the interchange of the H nuclei (even 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 = Ag, T1g, T2g, etc., is an irrep of Ih. Every TR eigenstate of H2O@C60 must transform according to one of these irreps.

Select TR energy levels from the quantum 6D calculations are shown for para-H2O@C60 in Table I and for ortho-H2O@C60 in Table II. Tables S1 and S2 of the supplementary material10 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-H2O. Also shown there are the energy levels calculated for the free-rotor (rigid) H2O; 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 jkakc, dominant contribution comes from the rotational basis functions having a single j value. It is evident from Tables III, S1 and S210 that the energies of the rotational levels of the caged para- and ortho-H2O are close to those of gaseous, freely rotating H2O, evidence for the weakly hindered rotation of the guest molecule, also seen experimentally.3 For j ≥ 3, the (2j + 1)-fold degeneracy (7 or greater) exceeds the largest degeneracy that the “crystal field” of Ih symmetry can accommodate, which is five-fold. Consequently, every 3kakc rotational level, with g = 7, splits into one g = 3 level (T-type irrep) and one g = 4 level (G-type irrep). Similarly, each 4kakc 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.

TABLE I.

Select translation-rotation (TR) energy levels of para-H2O inside C60 from the quantum 6D calculations. The excitation energies ΔE (in cm−1) are relative to the ground-state energy E0 = − 1987.57 cm−1, and g denotes their degeneracy. λ is the total angular momentum quantum number, l is the angular momentum quantum number of the dominant 3D HO function, j is the quantum number of the dominant Wigner function |jmk〉, and p(j) is its weight in a given TR eigenstate. ΔR, the standard deviation of R, is in bohrs. The irreps are those of the I h ( 12 ) group. The assignments are in terms of the rotational quantum numbers jkakc, and the quantum number n of the 3D isotropic harmonic oscillator. For additional explanations, see the text.

ΔE g λ l j p(j) ΔR Irrep Assignment
0.00  0.969  0.255  A g ( s )   000 
162.08  0.931  0.323  T 1 u ( s )   n = 1 
197.54  0.892  0.326  A g ( s )   n = 1 + 111 
198.33  0.969  0.324  T 1 g ( s )   n = 1 + 111 
200.13  0.851  0.321  H g ( s )   n = 1 + 111 
229.05  0.981  0.325  H u ( s )   n = 1 + 202 
233.26  3  3  0.918  0.322  T 2 u ( s )   [n = 1 + 202](a
233.54  4  3  0.919  0.322  G u ( s )   [n = 1 + 202](b
234.78  0.959  0.323  T 1 u ( s )   n = 1 + 202 
325.97  0.893  0.378  H g ( s )   n = 2 
328.21  0.893  0.377  A g ( s )   n = 2 
ΔE g λ l j p(j) ΔR Irrep Assignment
0.00  0.969  0.255  A g ( s )   000 
162.08  0.931  0.323  T 1 u ( s )   n = 1 
197.54  0.892  0.326  A g ( s )   n = 1 + 111 
198.33  0.969  0.324  T 1 g ( s )   n = 1 + 111 
200.13  0.851  0.321  H g ( s )   n = 1 + 111 
229.05  0.981  0.325  H u ( s )   n = 1 + 202 
233.26  3  3  0.918  0.322  T 2 u ( s )   [n = 1 + 202](a
233.54  4  3  0.919  0.322  G u ( s )   [n = 1 + 202](b
234.78  0.959  0.323  T 1 u ( s )   n = 1 + 202 
325.97  0.893  0.378  H g ( s )   n = 2 
328.21  0.893  0.377  A g ( s )   n = 2 
TABLE II.

Select translation-rotation (TR) energy levels of ortho-H2O inside C60 from the quantum 6D calculations. The excitation energies ΔE (in cm−1) are relative to the ground-state energy E0 = − 1987.57 cm−1. Other symbols have the same meaning as in Table I.

ΔE g λ l j p(j) ΔR Irrep Assignment
23.74  0.987  0.256  T 1 u ( a )   101 
183.12  0.979  0.325  T 1 g ( a )   n = 1 + 101 
186.71  0.952  0.323  H g ( a )   n = 1 + 101 
191.02  0.975  0.322  A g ( a )   n = 1 + 101 
238.34  0.966  0.325  H u ( a )   n = 1 + 212 
243.08  0.926  0.324  T 1 u ( a )   n = 1 + 212 
244.41  4  3  0.768  0.316  G u ( a )   [n = 1 + 212](a
244.64  3  3  0.773  0.316  T 2 u ( a )   [n = 1 + 212](b
362.06  0.946  0.380  T 1 g ( a )   n = 2 + 110 
370.16  3  3  0.740  0.369  T 2 g ( a )   [n = 2 + 110](a
370.20  4  3  0.803  0.371  G g ( a )   [n = 2 + 110](b
371.31  0.884  0.375  H g ( a )   n = 2 + 110 
372.77  0.925  0.376  T 1 g ( a )   n = 2 + 110 
ΔE g λ l j p(j) ΔR Irrep Assignment
23.74  0.987  0.256  T 1 u ( a )   101 
183.12  0.979  0.325  T 1 g ( a )   n = 1 + 101 
186.71  0.952  0.323  H g ( a )   n = 1 + 101 
191.02  0.975  0.322  A g ( a )   n = 1 + 101 
238.34  0.966  0.325  H u ( a )   n = 1 + 212 
243.08  0.926  0.324  T 1 u ( a )   n = 1 + 212 
244.41  4  3  0.768  0.316  G u ( a )   [n = 1 + 212](a
244.64  3  3  0.773  0.316  T 2 u ( a )   [n = 1 + 212](b
362.06  0.946  0.380  T 1 g ( a )   n = 2 + 110 
370.16  3  3  0.740  0.369  T 2 g ( a )   [n = 2 + 110](a
370.20  4  3  0.803  0.371  G g ( a )   [n = 2 + 110](b
371.31  0.884  0.375  H g ( a )   n = 2 + 110 
372.77  0.925  0.376  T 1 g ( a )   n = 2 + 110 
TABLE III.

Experimental (Table 1 of Ref. 3) and calculated energies (in meV) of the transitions in the INS spectra of para-H2O@C60. The assignments of the transitions to the rotational levels of ortho-H2O@C60 are from Ref. 3. The transitions in boldface, left unassigned in Ref. 3, are assigned to TR multiplets shown in Table II, as well as in Table S2 of the supplementary material,10 from the quantum 6D calculations in this work, whose components span the energy ranges shown. Also shown are the energy levels (in meV) calculated for the gaseous, freely rotating rigid H2O. For additional explanations, see the text.

Experiment Theory Free-rotor H2O Assignment
2.50 ± 0.05      0 00 1 01 a  
  2.94  2.95  000 → 101 
3.02 ± 0.05      0 00 1 01 b  
9.20 ± 0.05  9.74  9.86  000 → 212 
16.6 ± 0.1  16.37/16.86  16.78/16.97  000 → 221 303 
22.8±0.1  22.70 − 23.68    000n = 1 + 101 
26.1 ± 0.2  25.78  26.35  000 → 321 
30.8±0.2  29.55 − 30.30    000n = 1 + 212 
39.4±0.2  33.36 − 38.04    000330/n = 1 + 221 
40.7 ± 0.2  39.91  40.41  000 → 505 
45.1±0.2  44.89 − 46.22    000n = 2 + 110 
47.3 ± 0.2  48.40  47.67  000 → 432 
Experiment Theory Free-rotor H2O Assignment
2.50 ± 0.05      0 00 1 01 a  
  2.94  2.95  000 → 101 
3.02 ± 0.05      0 00 1 01 b  
9.20 ± 0.05  9.74  9.86  000 → 212 
16.6 ± 0.1  16.37/16.86  16.78/16.97  000 → 221 303 
22.8±0.1  22.70 − 23.68    000n = 1 + 101 
26.1 ± 0.2  25.78  26.35  000 → 321 
30.8±0.2  29.55 − 30.30    000n = 1 + 212 
39.4±0.2  33.36 − 38.04    000330/n = 1 + 221 
40.7 ± 0.2  39.91  40.41  000 → 505 
45.1±0.2  44.89 − 46.22    000n = 2 + 110 
47.3 ± 0.2  48.40  47.67  000 → 432 

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 H2@C60.5 The first translationally excited state of para-H2O (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 RR), and the dominant j = 0 rotational contribution, and it represents the fundamental translational excitation. Two closely spaced levels of para-H2O with two quanta of translational excitation (n = 2) (Tables I and S110), 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 S210 combine both translational and rotational excitations; they are assigned in terms of the quantum numbers n, l and jkakc.

The I h ( 12 ) irreps uniquely label the TR energy levels of H2O@C60 and account for their degeneracies, while the (n, l, jkakc) 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 H2@C60.5 Adapted to H2O, the model states that the orbital angular momentum l of the c.m. of H2O and the rotational angular momentum j of H2O couple to give the total angular momentum λ = l + j, with λ = l + j, l + j − 1, …, |lj|. 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 jkakc are split into as many distinct levels as there are different values of λ, each with the degeneracy 2λ + 1. It is easy to see that this model applies to all (n, l, jkakc) 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-H2O, 233.26 (g = 3) and 233.54 cm−1 (g = 4) (n = 1 + 202); for ortho-H2O, 244.41 (g = 4) and 244.64 cm−1 (g = 3) (n = 1 + 212), 370.16 (g = 3) and 370.20 cm−1 (g = 4) (n = 2 + 110). 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 Ih “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 Ih corrugation, or deviation from spherical symmetry. We note that the same was observed for H2@C60.5 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 H2O@C60 are dominated by paraortho, as well as orthoortho, transitions.3 Therefore, the transitions out of the ground para-H2O 000 state directly give the energies of the TR levels of ortho-H2O, 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-H2O@C60. 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 S210 (converted to meV). The INS spectra3 show the splitting of the ortho-H2O 101 ground state into two substates, 101a at 2.50 meV and 101b at 3.02 meV. The calculated 101 state is of course not split, and lies at 2.94 meV. The other calculated and measured rotational energy levels of ortho-H2O in Table III differ by no more than 1 meV.

The symmetry-breaking process responsible for the splitting of the ground state of ortho-H2O 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 H2O. Consequently, their important study left a gap in our quantitative understanding of the translational dynamics of H2O and its coupling to the H2O rotations, which our results begin to fill. In Table 1 of Ref. 3 four INS transitions out of 000 are unassigned, since their energies deviate significantly from the rotational levels of gaseous H2O. 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-H2O@C60 in Tables II and S2.

Thus, the transition measured at 22.8 meV is assigned as n = 1 + 101, the first translationally excited state of ortho-H2O. 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 H2O@C60 have been rigorously calculated, as was done previously for H2@C60,6,7 it is not possible to determine the contribution of each λ sublevel to this band. Subtracting from 22.8 meV the mean energy3 of the states 101a and 101b, 2.85 meV, gives 20 meV, what should be a good experimental estimate of the frequency of the translational fundamental of H2O@C60. This happens to be nearly identical to the calculated translational fundamental (n = 1) of para-H2O, 20.1 meV (Table I). It is also surprisingly close to the translational fundamental of H2@C60, calculated6 and observed12 around 22.5 meV, given the large difference in the masses of H2 and H2O.

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-H2O@C60. In particular, the transition measured at 45.1 meV is assigned as n = 2 + 110, i.e., a two-quanta translational excitation combined with 110 rotation. If the calculated energy of 110, 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-H2O, ∼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-H2O@C60. They characterize the quantum dynamical behavior of H2O molecule inside the C60 fullerene with the icosahedral (Ih) symmetry. In particular, this study sheds light on the translational excitations of the entrapped H2O 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 jkakc. 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 H2O 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 Ih symmetry. Several transitions in the measured INS spectra that were left unassigned3 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 H2O@C60 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 H2@C606,7,11,16 exist in H2O@C60 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.

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