General Selection Rule in the Inelastic Neutron ... - ACS Publications

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General Selection Rule in the Inelastic Neutron Scattering Spectroscopy of a Diatomic Molecule Confined Inside a NearSpherical Nanocavity Minzhong Xu,*,† Shufeng Ye,† and Zlatko Bačić*,†,‡ †

Department of Chemistry, New York University, New York, New York 10003, United States NYU-ECNU Center for Computational Chemistry at NYU Shanghai, Shanghai, 200062, China

‡

S Supporting Information *

ABSTRACT: Knowledge of the relevant selection rules is crucial for the accurate interpretation of experimental spectra in general. There has been a consensus for a long time that the incoherent inelastic neutron scattering (INS) spectroscopy of the vibrations of discrete molecular compunds is free from any selection rules. We contradict this widely held view by presenting an analytical derivation of the general selection rule for the INS spectroscopy of a hydrogen molecule inside a near-spherical nanocavity. It defines all forbidden transitions, originating in a range of initial translation-rotation (TR) states, ground and excited, of the caged para- and ortho-H2, as well as HD, that are unobservable in the INS spectra. These predictions are amenable to experimental verification. In addition, we demonstrate that the general selection rule applies to the INS spectroscopy of any diatomic molecule in a nanocavity with nearspherical symmetry, which exhibits strong TR coupling. Its existence strongly suggests that similar selection rules apply to the INS spectra of other molecular and supramolecular systems, and need to be identified.

A

experimental and theoretical INS investigation of H2 inside C60, which has icosahedral (Ih) symmetry.18 While this selection rule,16−18 which hereafter we refer to as restricted, set a conceptual precedent, it deals only with the INS transitions originating in the ground TR state of p-H2. It is not applicable to the transitions from either the TR states of orthoH2 (o-H2) or the excited TR states of para-H2 (p-H2). This leaves the following key questions open: (1) Are there other forbidden INS transitions, which originate in the TR states of p-H2, o-H2, and HD, that are not covered by the restricted selection rule? (2) Does a more general version of the selection rule exist, which would define all forbidden INS transitions for a hydrogen molecule in a near-spherical nanocavity? (3) If it does, could it be shown to apply to other diatomic molecules, besides H2 and HD, in such confinement? Here, we answer with a definitive yes to all three questions, by deriving the general selection rule, which has none of the limitations of its restricted counterpart, and identifies all possible forbidden INS transitions. According to the general selection rule, certain INS transtions from a variety of TR states, ground and excited, of both p-H2 and o-H2 molecules, as well as HD, inside a near-spherical nanocavity, such as that of C60, are forbidden. Furthermore, we demonstrate that this

light molecule such as H2 entrapped inside a nanosize cavity exhibits quantized translational center-of-mass (c.m.) motions, which are coupled by the confining potential to its also quantized rotational degrees of freedom. Inelastic neutron scattering (INS) spectroscopy has emerged as the main, highly selective tool for probing the coupled quantum translation-rotation (TR) dynamics of molecules with one or more hydrogen atoms (H2, HD, H2O, CH4) encapsulated in the nanoscale cavities of host materials.1−11 Highly selectivity is due to the unusually large cross section for the incoherent neutron scattering from the 1H nucleus,12 ∼ 15 times greater than for any other nucleus. In addition, INS can induce nuclear spin transitions, thus allowing the observation of the rotational Δj = 1 transitions of diatomic molecules, forbidden in the optical spectroscopy. Selection rules are of fundamental importance in spectroscopy, specifying the pairs of energy levels between which transitions are either allowed or forbidden. They are essential for the analysis of the experimental spectra. It has long been taken for granted that the incoherent INS spectroscopy of the vibrations of discrete (aperiodic) molecular systems is not subject to any selection rules,1,2,13,14 in contrast to optical spectroscopies.15 Recently, we demonstrated that this widespread belief is not entirely correct, by establishing the first ever selection rule in the incoherent INS spectroscopy of the vibrations of molecular compounds,16 specifically a H 2 molecule (and HD17) in a near-spherical nanoconfinement. Its predictions were soon confirmed by the combined © 2015 American Chemical Society

Received: July 14, 2015 Accepted: September 2, 2015 Published: September 2, 2015 3721

DOI: 10.1021/acs.jpclett.5b01505 J. Phys. Chem. Lett. 2015, 6, 3721−3725

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The Journal of Physical Chemistry Letters

Figure 1. Lower-lying TR energy levels of p-H2 and o-H2 molecules inside C60 from the quantum 5D calculations. They are labeled by the quantum numbers (n,j,λ, l) defined in the text, and are arranged in columns according to their j values. The transitions from the ground TR state (0,0,0,0) of p-H2 marked with dashed green arrows are forbidden according to the restricted selection rule (and general as well). Solid blue arrows mark the additional transtions forbidden only by the general selection rule.

latter, denoted Iif , which for homonuclear diatomics is given by26,27

selection rule applies to any diatomic molecule, homo- and heteronuclear, entrapped in a nanocavity with near-spherical symmetry and exhibiting strong TR coupling. The intricate quantum TR dynamics of H2@C6019 was elucidated by us in a series of theoretical studies.20−22 The purely translational eigenstates of H2 can be assigned with the principal quantum number n = 0,1,2,..., of the 3D isotropic harmonic oscillator (HO) and its orbital angular momentum quantum number l = n,n−2,...,1 or 0, for odd and even n, respectively. The quantum number j = 0,1,2,..., of the rigid rotor can be used for assigning the rotational energy levels of the caged H2. The most prominent feature of the “rattling” dynamics of the endohedral hydrogen manifests for the TR states of H2@C60 which are excited both translationally and rotationally: the orbital angular momentum l and the rotational angular momentum j couple vectorially to give the total angular momentum λ = l + j, having the values λ = l + j,l + j−1,..., |l−j|, with the degeneracy of 2λ + 1.20 Due to the TR coupling, the eigenstates labeled by the quantum numbers (n,j,λ,l) and with the same n and j (both nonzero) are split into closely spaced energy levels, each associated with one of the possible values of λ, and having the degeneracy of 2λ + 1.20 The predicted splittings were later observed in the infrared23 and INS spectra6 of H2@C60. In terms of the quantum numbers {n,j,λ,l }, the restricted selection rule16,18 can be stated as follows: for p-H2 in a nearspherical nanocavity, INS transitions involving the ground TR state (0,0,0,0) and the excited states (n,j,λ, l) are forbidden for λ = j + l − 1. The general selection rule that we prove here states the following: in a near-spherical nanocavity, INS transitions betweeen the pairs of TR states (n′,j′,λ′,l′) and (n,j,λ,l) of the entrapped p-H2, o-H2 or HD molecule are forbidden if λ′ = 0 and λ= j + l− o, where o is a positive odd integer (o = 1,3,...) . Note that the quantum numbers n′ and n do not appear in this selection rule. The restricted selection rule16,18 is just a special case of the general selection rule, when n′ = j′ = λ′ = l′ = 0 and o = 1. The INS cross section factorizes into the sum of products of nuclear-spin matrix elements and spatial matrix elements.24−28 The derivation of the general selection rule involves only the

⎛ κ ⃗·ρ ⃗ ⎞ 5D ⃗ ⎟|Ψ ⟩ I if = ⟨Ψ5D f | exp(iκ ⃗· R ) exp⎜i ⎝ 2 ⎠ i

(1)

5D In eq 1, |Ψ5D i ⟩ and |Ψf ⟩ are two different 5D eigenstates of the TR Hamiltonian (see eq 14 in ref 27), constituting the spatial parts of the |i ⟩ and |f ⟩ states of the INS transition, respectively,26,27 R⃗ is the position vector of the c.m. of H2, ρ⃗ the vector connecting the two atoms of the guest molecule, and κ⃗ is the wave vector of the neutron momentum transfer.26,27 |Ψ5D i ⟩ and |Ψf5D⟩ fully include the TR coupling of the guest molecule.26−28 Assuming the l−j (or TR) coupling above, the TR 5D eigenstates |Ψ5D τ ⟩ (τ = i,f) in eq 1 can be written as |ψτ ⟩ = |njλ l,mλ⟩ = |nl⟩ | jλ l, mλ⟩ (−λ ≤ mλ ≤ λ): |nl⟩ denotes the radial (R) component of the eigenstate. For the states with λ = 0 it must be that j = l and mλ = 0. Therefore, if we identify |Ψ5D i ⟩ as the (primed) TR eigenstate having λ′ = 0, then |Ψ5D i ⟩ = |n′ l′0l′,0⟩ = |n′l′⟩ | l′0 l′, 0⟩ and |ψ5D f ⟩ = |njλ l,mλ⟩ = |nl⟩ | jλ l, mλ⟩. Since the nanocavity is presumed to be near-spherical, e.g., that of C60 with Ih symmetry, for the purposes of the proof |nl⟩ can be approximated with high accuracy by nl (R ), the radial part of the wave function of the 3D isotropic HO in spherical polar coordinates.16 Then, for λ′ = 0, and |Ψi5D⟩ = |n′l′⟩|l′0l′, 0⟩ = n ′ l ′(R )|l′0l′, 0⟩, |ψ f5D⟩ = |nl⟩|jλl , mλ⟩ = nl (R )|jλl , mλ⟩. After these preliminary steps, the derivation presented in Sections S1 and S2 of the Supporting Information yields the following expression for Iif in eq 1:

I if =

∑ i L+J LJ

⎛ κρ ⎞ 4π  nL′ l ′ , nljJ ⎜ ⎟( −1) J + j + l ′+ mλ ⎝2 ⎠

× (2L + 1)(2J + 1) ×

(2l + 1)(2j + 1)(2l′ + 1)

⎛ l L l′⎞⎛ j J l′⎞⎛ J L λ ⎞⎧ J L λ ⎫ ⎬ ⎟⎜ ×⎜ ⎟⎜ ⎟⎨ ⎝ 0 0 0 ⎠⎝ 0 0 0 ⎠⎝ 0 0 0 ⎠⎩ l j l′ ⎭ × Yλ − mλ(θ κ ⃗ , ϕ κ ⃗) 3722

⎪

⎪

⎪

⎪

(2) DOI: 10.1021/acs.jpclett.5b01505 J. Phys. Chem. Lett. 2015, 6, 3721−3725

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inspection and analysis of the INS spectra of H2@C60 taken at higher temperatures, such as those of Horsewill et al.6 at 120 and 240 K. The confirmation would be much easier, and more unambiguous, if the INS spectra of pure p-H2@C60 would be measured, since this would reduce the likelihood of the forbidden transtions being obscured by some nearby intense lines. The general selection rule is defined in terms of j,l and λ, which are assumed to be good quantum numbers for the system considered. This is the case for H2 (and D2) in C60.20−22 But for the heteronuclear isotopologue HD inside C60, at higher excitation energies j is generally no longer a good quantum number.17,21 Its TR eigenstates exhibit strong mixing of two or more rotational basis functons with different j values, and the contribution c(j) of the dominant basis function is often only slightly greater than 0.5. Nevertheless, our rigorous calculations of the INS spectra of HD@C60 demonstrated that the transitions originating from the ground TR state of HD are subject to the restricted selection rule,17 although their final states exhibit strong j-mixing. The INS transition listed in Table 1, calculated to have zero intensity are those predicted to be

where  nL′ l ′ , nl is given by eq S9 (Supporting Information), Yλ − mλ is a spherical harmonic, while (···) and {···} stand for 3j and 6j symbols, respectively. Based on the properties of 3j symbols (eq A7 of the Supporting Information and ref 29), for Ifi to be nonzero, the three 3j symbols in eq 2 must simultaneously satisfy the following conditions: ⎧ l + L + l′ = even integer ⎪ ⎨ j + J + l′ = even integer ⎪ ⎩ J + L + λ = even integer

(3)

Adding the first two conditions in eq 3 gives J + L + l + j + 2l′= even integer, hence J + L + l + j = even integer. Subtracting this from the third condition above, one obtains λ − l − j = even integer

(4)

Iif

Iif

as the condition for nonzero in eq 2. It follows that must be zero if λ − l − j = odd integer, or equivalently, λ = l + j + o, where o stands for “odd integer”. Since λ cannot be greater than l + j, o must be negative, i.e., λ = l + j − o, o being a positive odd integer. This concludes the proof of the general selection rule. Like the restricted selection rule,16,18 the general selection rule will manifest only in the INS spectra of those systems where l−j coupling is strong enough to give rise to λ as a good quantum number, e.g., H2 in C60. Table S1 in the Supporting Information gives the calculated intensities of the INS transitions out of the first excited translational state (1,1,0,1) of o-H2, for the incident neutron scattering wavelength λn = 1.1 Å used to record the INS spectra of [email protected] The bold-faced transitions are those forbidden by the general selection rule [but not the restricted selection rule, except for the transtion to the ground TR state (0,0,0,0) ]; all of them have zero intensity, providing a computational validation of the general selection rule. The considerably greater scope of the general selection rule, and the much larger number of the INS transtions it predicts to be forbiden relative to the restricted selection rule, are conveyed by Figure 1. The TR energy levels of p-H2 and o-H2 in C60 shown in Figure 1 were calculated using the wellestablished 5D TR Hamiltonian for a caged diatomic molecule,18,27 and the efficient computational methodology developed in our group.26,27 For nonzero n and j quantum numbers, the levels are split by the TR coupling into closely spaced multiplets, each componet of which is associated with one value of the good quantum number λ. In the energy range covered by Figure 1 there are four INS transitions, indicated with the dashed green arrows, that are forbidden by the restricted selection rule16 (and general as well); they are all out of the ground TR state (0,0,0,0) of p-H2, and there is no trace of them in the experimental INS spectra.18 Besides these four transitions, the general selection rule predicts eight additional forbidden INS transtions, marked by the solid blue arrows, in the same energy range; the restricted selection rule does not cover these transitions. They originate in the sublevels (1,1,1,1) and (1,1,0,1) of the first excited translational state of o-H2, and the second translationally excited TR state (2,0,0,0) of p-H2; none of them is from the ground TR state (0,1,1,0) of o-H2. The excited TR states in question are not populated at the temperatures of several kelvins, at which the low-temperature INS spectra are recorded.6,18 Consequently, confirming the zero intensities of the transitions in Figure 1 forbidden by the general selection rule will require careful

Table 1. Forbidden INS Transitions of HD@C60 out of the Ground TR State (0,0,0,0) of HD to the Final States (n,j,λ,l)a i

ΔE (cm−1)

ΔE (meV)

g

(n,j,λ,l)

1 2 3 4 5 6

228.02 363.51 417.69 506.65 506.88 523.05

28.270 45.069 51.787 62.816 62.845 64.849

3 5 5 3 4 3

(1,1,1,1) (2,1,2,2) (1,2,2,1) (3,1,3,3) (3,1,3,3) (3,1,1,1)

c(j) (1) (1) (1) (1) (1)

(1) 0.973 0.726 (2) 0.271 0.236 (2) 0.751 0.578 (2) 0.405 0.576 (2) 0.407 0.550 (2) 0.439

The degeneracy of the final state is denoted with g, while c(j) is the contribution of the rotational basis function j to the final state; if its value is greater than 0.95, then it is the only one shown. The numbers in the parentheses next to the c(j) values are those of the corresponding j. The dominant j is in boldface, and is used in the (n,j,λ,l) assignments.

a

forbidden, provided their quantum number j, and therefore λ, is assigned according to the largest c(j) value. In the following, we analyze why the INS selection rule, restricted and general, applies to HD. Calculating the intensity of the INS transitions for HD@C60 requires the evaluation of the transition matrix element Iif very similar to that for H2@C60 in eq 1. For a general heteronuclear molecule AB, the slightly modified Iif is given by28 5D ⃗ I if = ⟨Ψ5D f | exp(iκ ⃗· R ) exp(iηnκ ⃗· ρ ⃗ )|Ψi ⟩ n = A, B

(5)

For the case of HD, ηH = 2/3 and ηD = −1/3, while R⃗ and ρ⃗ are defined as in eq 1. The initial state is the ground TR state of HD, |Ψ5D i ⟩ = |0000⟩. The final states are represented as ∑njλlm A nj5Dλl , mλ |njλl , mλ⟩, with |njλ l,mλ⟩ defined in eq |Ψ5D ⟩ = f λ S7 of the Supporting Information. For the (triply degenerate) TR level (3,1,1,1), denoted i = 6 in Table 1, the dominant |njλ l,mλ⟩ basis states and their respective expansion coefficients A5D njλ l,mλ are listed in Table S2, Supporting Information. It is evident that the only basis states with significant weight are |3111,mλ⟩ and |2212,mλ⟩ (mλ = 0, ± 1), and each of them separately satisfies the condition λ = j + l − o (o = 1 or 3) of the INS selection rule, restricted or general. Tables S3−S7 in the 3723

DOI: 10.1021/acs.jpclett.5b01505 J. Phys. Chem. Lett. 2015, 6, 3721−3725

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Supporting Information give the dominant |njλ l,mλ⟩ basis states for the TR levels i = 1−5 in Table 1, and they all satisfy the condition λ= j + l − o. This explains why the general selection rule applies to the INS spectra of HD@C60, despite the fact that for its higher-lying TR states j ceases to be a good quantum number. The detailed expression for Iif in eq 2, which is at the heart of the general selection rule, is valid for all diatomics molecules. The nuclear-spin components of the INS cross section differ from one molecule to another,24−28 but they have no bearing on the general selection rule. Therefore, the general selection rule is valid in the INS spectroscopy of any diatomic molecule, homo- or heteronuclear, inside a nanocavity with a nearspherical symmetry, which stongly couples its translational and rotational motions. INS measurements of HD or HF in C60 would provide the experimental validation of the general selection rule for systems where j is not a good qantum number. In summary, we have presented an analytical derivation of the general selection rule for the INS spectroscopy of p-H2 and o-H2, as well as HD, in a near-spherical nanosize cavity. This selection rule defines all forbidden INS transitions, originating in the TR states, ground and excited, of these guest molecules, and other caged diatomics, which are therefore absent from the INS spectra. The general rule derived in the present work, together with the restricted version established earlier,16,18 should alert experimentalists to the possibility that hitherto unsuspected selection rules operate in the INS spectroscopy of other dicrete molecular and supramolecular systems whose quantum dynamics involve coupling of angular momenta. This possibility will need to be kept in mind whenever the INS spectra of such systems are analyzed and interpreted in the future. Promising candidates are symmetric light polyatomic molecules, e.g., H2O and CH4, in high-symmetry confining nanospaces, like C60 and C70, carbon nanotubes, and metal− organic frameworks. In fact, the introduction of the restricted selection rule16 has already prompted the suggestion that certain transitions apparently missing from the INS spectra recorded for H2O inside C60 may be forbidden by a selection rule,11 which for this type of system remains to be formulated.

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REFERENCES

(1) Ramirez-Cuesta, A. J.; Jones, M. J.; David, W. I. F. Neutron scattering and hydrogen storage. Mater. Today 2009, 12 (11), 54. (2) Kearley, G. J.; Johnson, M. R. Vibrational spectroscopy with neutrons - Where are we now? Vib. Spectrosc. 2010, 53, 54. (3) Yildirim, T.; Harris, A. B. Rotational and vibrational dynamics of interstitial molecular hydrogen. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 66, 214301. (4) Horsewill, A. J.; Panesar, K. S.; Rols, S.; Johnson, M. R.; Murata, Y.; Komatsu, K.; Mamone, S.; Danquigny, A.; Cuda, F.; Maltsev, S.; Grossel, M. C.; Carravetta, M.; Levitt, M. H. Quantum -rotator: Inelastic neutron scattering of dihydrogen molecules trapped inside anisotropic fullerene cages. Phys. Rev. Lett. 2009, 102, 013001. (5) Horsewill, A. J.; Rols, S.; Johnson, M. R.; Murata, Y.; Murata, M.; Komatsu, K.; Carravetta, M.; Mamone, S.; Levitt, M. H.; Chen, J. Y.C.; Johnson, J. A.; Lei, X.; Turro, N. J. Inelastic neutron scattering of a quantum translator-rotator encapsulated in a closed fullerene cage: Isotope effects and translation-rotation coupling in H2@C60 and HD@ C60. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 82, 081410(R). (6) Horsewill, A. J.; Panesar, K. S.; Rols, S.; Ollivier, J.; Johnson, M. R.; Carravetta, M.; Mamone, S.; Levitt, M. H.; Murata, Y.; Komatsu, K.; Chen, J. Y.-C.; Johnson, J. A.; Lei, X.; Turro, N. J. Inelastic neutron scattering investigations of the quantum molecular dynamics of a H2 molecule entrapped inside a fullerene cage. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 205440. (7) Horsewill, A. J.; Goh, K.; Rols, S.; Ollivier, J.; Johnson, M. R.; Levitt, M. H.; Carravetta, M.; Mamone, S.; Murata, Y.; Chen, J. Y.-C.; Johnson, J. A.; Lei, X.; Turro, N. J. Quantum rotation and translation of hydrogen molecules encapsulated inside C60: temperature dependence of inelastic neutron scattering spectra. Philos. Trans. R. Soc., A 2013, 371, 20110627. (8) Ulivi, L.; Celli, M.; Giannasi, A.; Ramirez-Cuesta, A. J.; Bull, D. J.; Zoppi, M. Quantum rattling of molecular hydrogen in clathrate hydrate nanocavities. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 161401(R). (9) Ulivi, L.; Celli, M.; Giannasi, A.; Ramirez-Cuesta, A. J.; Zoppi, M. Inelastic neutron scattering from hydrogen clathrate hydrates. J. Phys.: Condens. Matter 2008, 20, 104242. (10) Beduz, C.; Carravetta, M.; Chen, J. Y.-C.; Concistre, M.; Denning, M.; Frunzi, M.; Horsewill, A. J.; Johannessen, O. G.; Lawler, R.; Lei, X.; Levitt, M. H.; Li, Y.; Mamone, S.; Murata, Y.; Nagel, U.; Nishida, T.; Ollivier, J.; Rols, S.; Rõõm, T.; Sarkar, R.; Turro, N. J.; Yang, Y. Quantum rotation of ortho and para-water encapsulated in a fullerene cage. Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 12894. (11) Goh, K. S. K.; Jiménez-Ruiz, M.; Johnson, M. R.; Rols, S.; Ollivier, J.; Denning, M. S.; Mamone, S.; Levitt, M. H.; Lei, X.; Li, Y.; Turro, N. J.; Murata, Y.; Horsewill, A. J. Symmetry-breaking in the endofullerene H2O@C60 revealed in the quantum dynamics of ortho and para-water: a neutron scattering investigation. Phys. Chem. Chem. Phys. 2014, 16, 21330. (12) Squires, G. L. Introduction to the Theory of Thermal Neutron Scattering; Dover Press: New York, 1996. (13) Hudson, B. S. Vibrational spectroscopy using inelastic neutron scattering: Overview and outlook. Vib. Spectrosc. 2006, 42, 25. (14) Mitchell, P. C. H., Parker, S. F., Ramirez-Cuesta, A. J., Tomkinson, J. Vibrational Spectroscopy with Neutrons; World Scientific: Singapore, 2005. (15) Bunker, P. R.; Jensen, P. Molecular Symmetry and Spectroscopy, E-book ed.; NRC Research Press: Ottawa, Ontario, Canada, 2006. (16) Xu, M.; Ye, S.; Powers, A.; Lawler, R.; Turro, N. J.; Bačić, Z. Inelastic neutron scattering spectrum of H2@C60 and its temperature dependence decoded using rigorous quantum calculations and a new selection rule. J. Chem. Phys. 2013, 139, 064309. (17) Xu, M.; Ye, S.; Lawler, R.; Turro, N. J.; Bačić, Z. HD in C60: theoretical prediction of the inelastic neutron scattering spectrum and its temperature dependence. Philos. Trans. R. Soc., A 2013, 371, 20110630. (18) Xu, M.; Jiménez-Ruiz, M.; Johnson, M. R.; Rols, S.; Ye, S.; Carravetta, M.; Denning, M. S.; Lei, X.; Bačić, Z.; Horsewill, A. J.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b01505. Deatiled derivation of the key INS transition matrix element in eq 2, seven tables referenced in the main text: the first giving the calculated intensities of the INS transitions out of the TR state (1,1,0,1) of o-H2@C60, the other six giving the dominant basis functions for the TR levels in Table 1. (PDF)

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Letter

AUTHOR INFORMATION

Corresponding Authors

*Electronic mail: [email protected]. *Electronic mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Z.B. thanks the National Science Foundation for financial support of this research through the Grant CHE-1112292. 3724

DOI: 10.1021/acs.jpclett.5b01505 J. Phys. Chem. Lett. 2015, 6, 3721−3725

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The Journal of Physical Chemistry Letters Confirming a predicted selection rule in inelastic neutron scattering spectroscopy: The quantum translator-rotator H2 entrapped inside C60. Phys. Rev. Lett. 2014, 113, 123001. (19) Mamone, S.; Chen, J. Y.-C.; Bhattacharyya, R.; Levitt, M. H.; Lawler, R. G.; Horsewill, A. J.; Rõõm, T.; Bačić, Z.; Turro, N. J. Theory and spectroscopy of an incarcerated quantum rotor. The infrared spectroscopy, inelastic neutron scattering and nuclear magnetic resonance of H2@C60 at cryogenic temperatures. Coord. Chem. Rev. 2011, 255, 938. (20) Xu, M.; Sebastianelli, F.; Bačić, Z.; Lawler, R.; Turro, N. J. Quantum dynamics of coupled translational and rotational motions of H2 inside C60. J. Chem. Phys. 2008, 128, 011101. (21) Xu, M.; Sebastianelli, F.; Bačić, Z.; Lawler, R.; Turro, N. J. H2, HD, and D2 inside C60: Coupled translation-rotation eigenstates of the endohedral molecules from quantum five-dimensional calculations. J. Chem. Phys. 2008, 129, 064313. (22) Xu, M.; Sebastianelli, F.; Gibbons, B. R.; Bačić, Z.; Lawler, R.; Turro, N. J. Coupled translation-rotation eigenstates of H2 in C60 and C70 on the spectroscopically optimized interaction potential: Effects of cage anisotropy on the energy level structure and assignments. J. Chem. Phys. 2009, 130, 224306. (23) Mamone, S.; Ge, M.; Hüvonen, D.; Nagel, U.; Danquigny, A.; Cuda, F.; Grossel, M. C.; Murata, Y.; Komatsu, K.; Levitt, M. H.; Rõõm, T.; Carravetta, M. Rotor in a cage: Infrared spectroscopy of an endohedral hydrogen-fullerene complex. J. Chem. Phys. 2009, 130, 081103. (24) Sears, V. F. Theory of cold neutron scattering by homonuclear diatomic liquids. Can. J. Phys. 1966, 44, 1279. (25) Stein, H.; Stiller, H.; Stockmeyer, R. Phonons, librons, and rotational state J = 1 in HCP and FCC solid hydrogen by neutron spectroscopy. J. Chem. Phys. 1972, 57, 1726. (26) Xu, M.; Ulivi, L.; Celli, M.; Colognesi, D.; Bačić, Z. Quantum calculation of inelastic neutron-scattering spectra of a hydrogen molecule inside a nanoscale cavity based on rigorous treatment of the coupled translation-rotation dynamics. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 241403(R). (27) Xu, M.; Bačić, Z. Inelastic neutron scattering spectra of a hydrogen molecule in a nanocavity: Methodology for quantum calculations incorporating the coupled five-dimensional translationrotation eigenstates. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 195445. (28) Xu, M.; Ulivi, L.; Celli, M.; Colognesi, D.; Bačić, Z. Rigorous quantum treatment of inelastic neutron scattering spectra of a heteronuclear diatomic molecule in a nanocavity: HD in the small cage of structure II clathrate hydrate. Chem. Phys. Lett. 2013, 563, 1. (29) Brink, D. M.; Satchler, G. R. Angular Momentum; Clarendon Press: Oxford, 1968.

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