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Salvatore
Mamone
*^{a},
Mónica
Jiménez-Ruiz
^{b},
Mark R.
Johnson
^{b},
Stéphane
Rols
^{b} and
Anthony J.
Horsewill
*^{a}
^{a}School of Physics and Astronomy, University of Nottingham, NG7 2RD Nottingham, UK. E-mail: salvatore.mamone@nottingham.ac.uk; a.horsewill@nottingham.ac.uk
^{b}Institut Laue-Langevin, BP 156, 38042 Grenoble, France

Received
1st September 2016
, Accepted 6th October 2016

First published on 6th October 2016

In this paper we report a methodology for calculating the inelastic neutron scattering spectrum of homonuclear diatomic molecules confined within nano-cavities of spherical symmetry. The method is based on the expansion of the confining potential into multipoles of the coupled rotational and translational angular variables. The Hamiltonian and the INS transition probabilities are evaluated analytically. The method affords a fast and computationally inexpensive way to simulate the inelastic neutron scattering spectrum of molecular hydrogen confined in fullerene cages. The potential energy surface is effectively parametrized in terms of few physical parameters comprising an harmonic term, anharmonic corrections and translation–rotation couplings. The parameters are refined by matching the simulations against the experiments and the excitation modes are identified for transfer energies up to 215 meV.

With a hydrogen molecule entrapped inside C_{60}, the endofullerene H_{2}@C_{60} affords an actual realization of a quantum rotor confined in a three dimensional cage. The dynamics of H_{2} encapsulated within a fullerene is fundamentally quantum mechanical for several reasons. Firstly, owing to its low mass, free molecular hydrogen is characterized by a large rotational constant with sparse levels whose separation is in excess of 170 K in temperature units. Secondly once hydrogen is confined inside the C_{60} nano-cavity its translational degrees of freedom becomes quantised. Significantly, since the internuclear bond length in hydrogen (0.74 Å) is in the same scale as the fullerene radius, the quantised rotational and translational degrees of freedom become coupled. Thirdly, since the nuclei comprising H_{2} are indistinguishable fermions, the antisymmetry principle applies in determining the allowable eigenstates and there exists ortho- and para-nuclear spin isomers of H_{2}. At low temperature these remain largely decoupled from each other and the two co-exist as independent and identifiable species in the absence of time-dependent inhomogeneous magnetic fields.

The study of the quantum dynamics of H_{2}@C_{60} is relevant for understanding the interaction between carbon based materials and hydrogen. Molecular endofullerenes are ideal compounds in which the guest molecules are relatively isolated from each other within the homogenous environment provided by the cages. As a result, the spectrum of translation–rotation energy levels is sparse and when probed by various spectroscopic techniques, narrow well-separated peaks are observed.^{5–8} This reveals in fine detail the quantum dynamics of the entrapped translator–rotator and enables the potential energy surface (PES) experienced by the rotor to be probed meticulously along with the various interactions that characterise the Hamiltonian. With its unique ability to induce transitions between ortho- and para-spin-isomers, inelastic neutron scattering (INS) is a prime technique for studying confined hydrogen.^{9}

Pioneering work by the group of Bačić using a rigorous computational approach to rigorously solve the Hamiltonian^{10–14} has been instrumental in developing an understanding of confined H_{2}, including endofullerenes.^{15–17} Their methodology has been extended to determine the INS spectrum of hydrogen confined in nanocavities^{18,19} and in fullerenes.^{20,21} There, the potential energy surface is modelled using modified Lennard-Jones functions to represent the atom–atom potentials, then the Hamiltonian is solved exactly from which the spectrum may be determined.

The scope of this paper is different. Starting from the experimental spectrum, the observed low energy transitions are described in terms of excitations of a rotator angularly coupled to a harmonic oscillator. The quantum dynamics of a homonuclear diatomic molecules in a weakly perturbed isotropic harmonic trap is investigated by expanding the confining potential in symmetry adapted multipoles of the coupled angular variables. The Hamiltonian and the INS transition probabilities are represented analytically in the basis of the coupled rotor and harmonic oscillator. The INS spectrum is simulated by refining few Hamiltonian parameters, including harmonic and anharmonic terms along with translation–rotation coupling constants. In this methodology, the PES is not built from specific microscopic interactions but it is determined from a direct fit of the Hamiltonian to the experimental spectrum with the advantage that the fitted parameters represent physical attributes, such as the harmonic potential, the anharmonicity and translation–rotation coupling constants.

In the context of the spectroscopy of H_{2}@C_{60}, it is well-established that the quantum states of the coupled harmonic translator–rotator can be described in terms of the quantum numbers (n,l,j,λ).^{22} The entrapped rotor possesses both rotational and translational energy. The rotational states are characterised by j, while the translational states, defined by displacements of the centre-of-mass of the particle within the 3D-cage, are characterised by n and l. The former, n, is the principle quantum number while the latter, l, determines the translational angular momentum. Rotational and translational angular momenta couple according to the usual vector addition rules and we identify the quantum number λ to characterise the coupled translation–rotation states. A more complete account of the rotational and translational states will be discussed in Section 4. Finally, applying the antisymmetry principle, the Pauli Exclusion Principle (PEP) demands that the nuclear spin isomer para-H_{2} is a nuclear spin singlet with total spin I_{N} = 0 and characterised by even j, while ortho-H_{2} is a nuclear spin triplet with I_{N} = 1 and odd values of j. In this way the spatial and spin degrees of freedom are entangled. This has consequences for experiments since ortho ↔ para transitions are spin-restricted and forbidden to photon spectroscopy, whereas neutron scattering interactions couple space and spin enabling ortho ↔ para transitions to be directly observed in the INS spectrum with high intensity.

The paper is organised as follows. First the experimental INS spectrum is presented in Sections 2 and 3. The analytical approach used to represent the Hamiltonian and to determine the INS transition probabilities is discussed in Sections 4 and 5, respectively. The method is then applied to simulate the INS spectrum of H_{2}@C_{60} in Section 6. The parameters characterising the cage potential and the spectral assignments are discussed in Section 7.

The sample of H_{2}@C_{60} was prepared according to published methods.^{1} In a final stage of purification, the presence of any intercalated solvent was substantially reduced by sublimation. The powdered sample with mass 101 mg was uniformly loaded inside a rectangular Al foil sachet, folded into a cylindrical annulus of approximate diameter 1 cm and mounted in the Lagrange spectrometer. The sample temperature was maintained at 2.5 K using a closed cycle He refrigerator. To subtract background and scattering from Al and from the C_{60} cage, a “blank” sample of empty C_{60} cages with matching mass was prepared in an identical Al foil sachet.

INS spectra of H_{2}@C_{60} were recorded using the three monochromators and the corresponding energy transfer ranges were selected to provide the best available resolution as follows: Si(331) 12.7 < ΔE ≤ 27 meV; Cu(220) 27 < ΔE ≤ 63 meV and Cu(311) 63 < ΔE ≤ 215 meV. The intensities of the INS peaks recorded with the three monochromators were normalized to the same scale by comparing the integrated intensities of peaks recorded in overlapping energy transfer regions. The systematic error in energy transfer is estimated to be 0.1 meV below approximately 50 meV rising to 0.5 meV at the highest energy transfers recorded. Random errors in energy transfer were small by comparison, typically 0.02 meV for singlet peaks. Unless otherwise stated, these error bars will apply in the following discussions.

Fig. 1 The INS spectrum of H_{2}@C_{60} at 2.5 K as recorded on the IN1-Lagrange spectrometer at ILL, Grenoble (France). The vertical axis represents neutron counts in arbitrary units. |

At low temperature, and particularly in the lower energy transfer region, the spectrum is sparse with sharply resolved peaks revealing a remarkable quantity of detail. H_{2}@C_{60} is clearly an extremely well-characterized and homogenous system, revealing the quantum nature of the H_{2} molecule and its wavelike nature in exquisite detail.

Fig. 2 shows an extract from the experimental spectrum for transfer energy up to 73 meV. The first rotational transition (0,0,0,0) → (0,0,1,1) is clearly observed as the singlet peak with energy transfer 2B = 14.7 meV, interconverting the ground states of para-H_{2} and ortho-H_{2}. Another singlet peak is the second rotational transition (0,0,1,1) → (0,0,2,2) which has energy transfer approximating 4B and is centered on 29.2 meV. A third singlet with width equal to the resolution function appears at 72.2 meV. This energy approximates to 10B so the peak assigned to the ortho–ortho pure rotational transition (0,0,1,1) → (0,0,3,3). These observations indicates that barriers impeding free rotations are small in H_{2}@C_{60}.

Fig. 2 Expansion of the INS spectrum of H_{2}@C_{60} at 2.5 K of Fig. 1 in the range 10–75 meV. Each peak is assigned to an INS transitions for which the labels indicate the value of n, l, j in the final states. Transitions originating from the para-H_{2} ground state are in blue and transitions originating from the ortho-H_{2} ground state are in red, respectively. The vertical axis represents neutron counts in arbitrary units. |

The prominent band centered on 22.5 meV received particular interest in our earlier papers,^{24} since it represents the transition from the ground state of ortho-H_{2} to the first excited translational state of ortho-H_{2}, (0,0,1,1) → (1,1,1,λ). This is a ubiquitous “particle-in-a-box” mode and in the Lagrange spectrum this is observed with similar resolution to the earlier IN4 spectrum (λ_{n} = 1.6 Å). This mode is a partially resolved triplet comprising the λ = 0, 1, 2 components of the excited state. This band was significant since it represented the first observation in an INS spectrum of a splitting arising from TR coupling of rotational and translational angular momentum. Moving to higher energy transfer, a doublet is observed with energies 37.4 and 38.5 meV. Considering the energy level diagram and consistent with the assignments in our earlier papers,^{8,24} we infer this band is the transition (0,0,0,0) → (1,1,1,λ) involving the same final state triplet referred to in the previous paragraph, but this time originating in the ground para-H_{2} state. Within the resolution of the spectrometer only two components of this band can be identified while the third one is apparently missing. We shall return to this point more widely in Section 7 in the context of the discussion of the INS transition probability for diatomic molecules in spherical potentials.

The second excited translational state of ortho-H_{2}, n = 2, j = 1, is revealed in a doublet with energy transfers 45.8 and 47.0 meV and in the small peak at 49.8 meV. These peaks were observed with the Cu(220) monochromator and have the resolution line width. The former doublet has a partner observed at 61.3 meV while the latter peak is closely related to the singlet at 64.2 meV observed with the Cu(331) monochromator, which also has the resolution line width. Allowing for systematic errors that arise from two different monochromators, these two set of peaks differ in energy transfer by 2B, so we are able to make the assignment to the same final state within the manifold n = 2, j = 1; the lower energy of the two originates in ortho-H_{2}, while the higher energy peak originates in para-H_{2}.

The peak in the region 51.5 meV has been observed with the Cu(220) monochromator. It is slightly broader than the resolution line width and slightly asymmetric so we can infer the feature has multiple components. As with the examples above, applying an offset equal to a multiple of 2B provides an energy that coincides with a known splitting in the spectrum. In this case subtracting 4B coincides with the translational splitting so we can infer these two peaks correspond to the ortho–para transition from (0,0,1,1) → (1,1,2,λ).

Finally, two low intensity peaks occur at 32.3 and 34.3 meV. These identify the states n = 2, j = 0, corresponding to the transitions (0,0,1,1) → (2,0,0,0) and (0,0,1,1) → (2,2,0,2).

As the above few examples show, with good instrument resolution and a spectrum which is sufficiently sparse, substantial progress can be made in the assignment of n, l, j quantum numbers based solely on the experimental data. This enables the energy level diagram to be determined for all low lying states up to approximately 73 meV. As both para to ortho and ortho to ortho transitions are allowed, and the same final states are represented by multiple peaks, the assignments may be confirmed by correlating transition energies. However, with increasing energy transfer, the number of transitions significantly increases while anharmonicities lead to non-integer relationships between peak energies. It becomes clear that a definitive assignment of the INS peaks, including the TR multiplet components, increasingly demands the assistance of a more sophisticated theoretical framework. Therefore we shall return to consider more detailed assignments in the spectrum in Section 7 after describing the computational simulation of the H_{2}@C_{60} spectrum.

The molecule–cage interaction is described by the potential V(R,r;Ω_{LC}). The function V can be thought as the PES of an H_{2} molecule confined inside the fullerene cage. It is worth noting that any symmetry operation Ô of the icosahedral point group I_{h} acting on all the position coordinates leave the potential unaffected:

V(R,r;Ω_{LC}) = V(ÔR,Ôr;Ô[Ω_{LC}]) = V(ÔR,Ôr;Ω_{LC}) | (1) |

(2) |

(3) |

The symmetry of the confinement is reflected in the number and types of multipoles present in the potential expansion, eqn (2). In icosahedral symmetry it is possible to prove that the rank of the multipoles that form invariant functions is restricted to Λ = 0, 6, 10, 12, 16.^{28} Since the centre of the potential is a point of inversion symmetry, only multipoles with L + J = even are allowed. For multipoles with rank Λ = 0, only terms with Λ = 0, M_{Λ} = 0 and L = J are allowed. The next lowest non-zero rank terms for an icosahedral potential is:^{29}

(4) |

The vibrationally averaged effective five-dimensional Hamiltonian for a diatomic molecules moving in a spherical potential can be written as

(5) |

It is a general statement of quantum mechanics that the total angular momentum of a system with spherical symmetry commutes with the Hamiltonian, so providing “good” quantum numbers for labelling the energy levels. For a diatomic molecule in a spherical trap, the total angular momentum = + Ĵ is given by the sum of the orbital angular momentum of the centre of mass and of the end to end rotational angular momentum Ĵ. The bipolar spherical harmonics are eigenfunctions of the angular momentum operators ^{2} and _{z} with eigenvalues λ(λ + 1) and m_{λ}, respectively. They provide a complete basis set for the coupled angular motion.

A complete basis set for the radial motion is provided by the set of radial eigenfunctions for the Schrödinger equation of a particle in a central potential. Early ab initio determinations of the potential energy surface of H_{2}@C_{60} suggest to use the eigenfunctions of a three-dimensional isotropic harmonic trap. These are discussed in the appendix, Section A.1. The coupled harmonic translation–rotation wave-function is written in ket notation as

(6) |

E^{0}_{n,j} = ħω(n + 3/2) + Bj(j + 1) | (7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

Interestingly, the derivation of the transition probability via the Wigner–Eckart theorem, as discussed in Section A.4, allows us to recover the selection rule for INS transitions in diatomic molecules confined in potentials of spherical symmetry, originally found by Xu et al.^{20,31} and later extended and explained by Poirier in the context of group theory.^{26} Indeed in the evaluation of the reduced matrix element of eqn (13) the terms with

(14) |

To achieve a more direct visual comparison between experiments and simulations, the transition probability, eqn (10), is integrated over the q-range appropriate to the instrumental apparatus at energy ΔE_{fi} = E_{f} − E_{i}. In an inverse geometry neutron spectrometer the final energy of the scattered neutron is fixed to E_{n}, while the incident neutron energy is scanned through. A detector at angle θ with respect the direction of the incident neutron beam records scattered neutrons at transferred momentum

(15) |

(16) |

In the simulation of the IN1-Lagrange spectrum of H_{2}@C_{60} the number of scattered neutrons was determined by averaging over a uniform distribution of 20 polar angles in the nominal angular ranges 34° ≤ θ ≤ 70° at fixed E_{n} = 4.5 meV and using the instrumental resolution discussed in Section 2.

Hamiltonian parameter | Optimized value |
---|---|

B
_{0}/meV |
7.375 |

D
_{0}/meV |
0.012 |

ρ/Å | 0.311 |

Ṽ
^{00;4}_{00}/meV |
−3.90 |

Ṽ
^{00;6}_{00}/meV |
1.52 |

Ṽ
^{22;2}_{00}/meV |
8.9 |

Fig. 3 shows the comparison between the experimental and calculated spectrum of H_{2}@C_{60} at 2.5 K.

The calculated spectra confirm the peak assignment up to 73 meV given in Section 3, that were mainly based on the approximation of uncoupled rotation and translation, see eqn (7). However even at low energy the presence of couplings has a major impact on the appearance of the spectrum. By providing intensities and transition energies as well, the calculations establish a firm framework for the analysis of the spectrum.

The experimental peaks at 14.7, 29.2 and 72.2 meV are singlet rotational peaks, corresponding to the para–ortho (0,0,0,0) → (0,0,1,1) and ortho–para (0,0,1,1) → (0,0,2,2) transitions, respectively. The presence of anharmonic and translation–rotation coupling has a minor impact on pure rotational transition between states with n = 0, l = 0. However other rotational transitions are more difficult to identify because of the reduced intensity and their use as milestones for spectral assignment is less poignant.

The structure underlying multi-component peaks is clearly resolved in the stick spectrum. The experimental peak at 22 meV is resolved into three main components corresponding to the intra-ortho translational transitions (0,0,1,1) → (1,1,1,λ) with λ = 1, 2, 0 in order of increasing energy. As shown in ref. 37 using group theoretical methods and successively in ref. 25 by using perturbation theory, the order of the λ levels in the (1,1,1,λ) multiplet is dictated by the positive sign of the translational–rotational coupling term Ṽ^{22;2}_{00}. As observed previously the λ = 0 component is much weaker in intensity than the other two and, importantly, these observations are in agreement with the simulation of the H_{2}@C_{60} spectrum by Xu et al.^{15,20} In a similar manner, the experimental peaks at 37.4 and 38.5 meV are assigned to the λ = 2 and λ = 0 components of the para–ortho translational transition (0,0,0,0) → (1,1,1,λ), respectively. There is no evidence of the λ = 1 component of the triplet. This highlights a striking prediction from the earlier papers,^{20,21} namely the existence of a selection rule for INS transitions in diatomic molecules in a spherical confinement.^{26,31}

Transitions towards levels with two quanta of translational excitations have been identified in the experimental discussion. The peaks at 32.3 and 34.3 meV are assigned to the transitions (0,0,1,1) → (2,2,0,2) and (0,0,1,1) → (2,0,0,0), respectively. Transitions towards states with n = 2 and j = 1 reveal a larger degree of mixing between different translational–rotational states. The calculations identifies the transitions towards (2,2,1,2) and (2,2,1,3) with the peaks at 45.8 and 47.0 meV, respectively. Concealed under the latter peak, there is a transition to an energy level comprising a superposition of the (2,0,1,1) and (2,2,1,1) states with a ratio close to 1:2. Similarly the peak at 49.8 meV corresponds to a transition to an energy level comprising a superposition of the (2,0,1,1) and (2,2,1,1) states in a ratio close to 2:1. The partner transitions to these superposition states originating from the para ground state are found at 61.3 and 64.2 meV, respectively, while the para transition towards (2,2,1,2) is forbidden and the para transition towards (2,2,1,3) has negligible intensity.

The peak centered at 51.5 meV is resolved in three transitions starting from the ortho ground state towards (1,2,1,λ) state with λ = 2, 3, 1 in order of increasing energy.

The stick spectrum demonstrates the existence of para–para transitions that are easily overlooked in the experimental spectrum. Owing to the small ratio between the coherent and incoherent cross section for ^{1}H, para–para peaks generate a very low scattered neutron count and are difficult to observe. For example the fundamental translational transition in the para manifold (0,0,0,0) → (1,1,0,1) is shadowed by the ortho transitions (0,0,1,1) → (1,1,1,λ).

At higher energy transfer the absolute resolution becomes worse, the number of accessible transitions increases and the peaks become relatively less sparse and less intense. This is particularly evident in the region above 96 meV. The calculations identify 162 transitions originating from the ground para state and 191 transitions originating ground ortho state in the energy range between 96 and 215 meV. In contrast to the low energy case, where most of the levels are represented by the nominal (n,l,j,λ) state with at least 95% probability except where noted, states with different n, l, j become more and more mixed at high energies, so hampering their use as labels for the energy levels. A semi-empirical approach that tries to fit more prominent features to a minimal number of Gaussian lines may still work where few high intensity transitions are clustered.^{8} However in other regions where transitions are more spread in energy and there is no dominant component, it is practically impossible to produce any meaningful assignment of this type. Simulations remains the only method to make sense of the complexity of the spectrum at high energy transfer. Indeed the simulated spectrum provide a realistic representation of the experimental observations over all the accessed energy range. The simulated transitions follow closely but not exactly the experimental ones. Higher order spherical terms may improve the match at the expense of increasing the parameter space. However improved accuracy appears to be not worthwhile with the current experimental resolution also considering that shifts and splittings of the m_{λ} components in the energy levels of order of fractions of meV are introduced by terms of icosahedral symmetry^{38} as well as intermolecular effects.^{39}

The theoretical and computational model developed in this paper may be useful for investigating the quantum dynamics and the INS spectrum of confined molecular hydrogen in systems such as clathrates hydrates,^{40,41} zeolites^{42,43} and MOFs.^{9,44,45} We anticipate its extension to treat heteronuclear diatomic molecules as well as water molecules confined in C_{60}. In systems with high symmetry or when the effect of non spherical multipoles are small, such as H_{2}@C_{60}, realistic INS spectra can be generated promptly. For systems with lower symmetry the method can be used to provide a starting approximate description useful to identify the main features of the INS spectrum and it can be eventually augmented to include non-spherical terms perturbatively.

(17) |

(18) |

(19) |

(20) |

Similarly 6-j symbols and 9-j symbols appear when combining the angular momentum for three subsystems and four subsystems. 9-j symbols appear in the evaluation of the reduced matrix element of two coupled subsystem as well, see eqn (29). They are defined in terms of 6-j symbols as:

(21) |

3-j symbols and 6-j symbols can be calculated from generating formulae^{27} and are available as built-in objects in computational softwares such as Mathematica.^{34}

(22) |

A.3.1 Radial matrix elements.
The evaluation of radial matrix elements for non-negative power of local operator f(R) = R^{k} over the radial translational wave-functions, |n,l;ρ〉 of eqn (17), is found by integration

where Γ is the gamma function^{47} and

are the coefficients for the Laguerre expansion of L^{(α)}_{n}(x) in power of x^{m}.

For sake of completeness, we note that only matrix elements of spherical Bessel functions j_{β} with |l′ − l| ≤ β ≤ l′ + l and l′ + l + β = even needs to be evaluated. In such case the confluent hypergeometric function _{1}F_{1} reduces to:^{47}

where

is the Pochammer symbol.^{47}

(23) |

(24) |

Similarly the radial matrix elements of the spherical Bessel function^{47}j_{β}(qR) for integer β ≥ 0, which enter in the space part of the INS transition matrix, can be integrated in a closed form

(25) |

(26) |

(a)_{n} = a(a + 1)(a + 2)…(a + n − 1), (a)_{0} = a | (27) |

A.3.2 Angular matrix elements.
The evaluation of the matrix elements of bipolar harmonics over the coupled spherical basis is easily achieved via the use of the Wigner–Eckart theorem^{48}

where the round brackets represent a 3j symbol and the double bars represents the reduced matrix elements for the bipolar spherical harmonics. The reduced matrix elements in the coupled angular momentum scheme^{48}

is given in terms of reduced matrix elements for spherical harmonics^{48}

(28) |

(29) |

(30) |

The angular matrix elements in eqn (28) are diagonal in λ and m_{λ} for bipolar harmonics with Λ = m_{Λ} = 0:

(31) |

(32) |

(33) |

(34) |

When evaluating the neutron cross-section for homonuclear diatomic molecules, only the symmetric combination intervenes in transitions where the total molecular nuclear spin does not change, ΔI_{N} = 0, while the antisymmetric combination intervenes in transitions where the total molecular nuclear spin changes of one unit, |ΔI_{N}| = 1. In experiments with non polarised neutrons and non polarised molecular system, the cross section can be written as product of a spin dependent part s_{fi} and space dependent part S_{fi}:

(35) |

(36) |

The spatial dependent part of the INS cross section involves the evaluation of the matrix elements of the local operator g_{±}(q) where q = k − k′ is the transferred momentum. The plus and minus sign are alternatively required for direct (ortho–ortho and para–para) and cross spin transitions (ortho–para and para–ortho), respectively. For our scopes, this operator can be conveniently written as

(37) |

(38) |

(39) |

(40) |

In systems with spherical symmetry, the space dependent part of the INS cross section for transitions between an initial energy level i = E_{λ} and a final energy level is given by the transition probability of the operator g_{±}(q) summed over the final states m_{λ′} and averaged over the 2λ + 1 initial states m_{λ}:

(41) |

(42) |

(43) |

(44) |

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