Hydrogen molecules inside fullerene C70: quantum dynamics, energetics, maximum occupancy, and comparison with C60.

Recent synthesis of the endohedral complexes of C(70) and its open-cage derivative with one and two H(2) molecules has opened the path for experimental and theoretical investigations of the unique dynamic, spectroscopic, and other properties of systems with multiple hydrogen molecules confined inside a nanoscale cavity. Here we report a rigorous theoretical study of the dynamics of the coupled translational and rotational motions of H(2) molecules in C(70) and C(60), which are highly quantum mechanical. Diffusion Monte Carlo (DMC) calculations were performed for up to three para-H(2) (p-H(2)) molecules encapsulated in C(70) and for one and two p-H(2) molecules inside C(60). These calculations provide a quantitative description of the ground-state properties, energetics, and the translation-rotation (T-R) zero-point energies (ZPEs) of the nanoconfined p-H(2) molecules and of the spatial distribution of two p-H(2) molecules in the cavity of C(70). The energy of the global minimum on the intermolecular potential energy surface (PES) is negative for one and two H(2) molecules in C(70) but has a high positive value when the third H(2) is added, implying that at most two H(2) molecules can be stabilized inside C(70). By the same criterion, in the case of C(60), only the endohedral complex with one H(2) molecule is energetically stable. Our results are consistent with the fact that recently both (H(2))(n)@C(70) (n = 1, 2) and H(2)@C(60) were prepared, but not (H(2))(3)@C(70) or (H(2))(2)@C(60). The ZPE of the coupled T-R motions, from the DMC calculations, grows rapidly with the number of caged p-H(2) molecules and is a significant fraction of the well depth of the intermolecular PES, 11% in the case of p-H(2)@C(70) and 52% for (p-H(2))(2)@C(70). Consequently, the T-R ZPE represents a major component of the energetics of the encapsulated H(2) molecules. The inclusion of the ZPE nearly doubles the energy by which (p-H(2))(3)@C(70) is destabilized and increases by 66% the energetic destabilization of (p-H(2))(2)@C(60). For these reasons, the T-R ZPE has to be calculated accurately and taken into account for reliable theoretical predictions regarding the stability of the endohedral fullerene complexes with hydrogen molecules and their maximum H(2) content.

Coupled translation-rotation eigenstates of H2, HD, and D2 in the large cage of structure II clathrate hydrate: comparison with the small cage and rotational Raman spectroscopy.

We report fully coupled quantum five-dimensional calculations of the translation-rotation (T-R) energy levels of one H(2), HD, and D(2) molecule confined inside the large hexakaidecahedral (5(12)6(4)) cage of the structure II clathrate hydrate. Highly converged T-R eigenstates have been obtained for excitation energies beyond the j = 2 rotational levels of the guest molecules, in order to allow comparison with the recent Raman spectroscopic measurements. The translationally excited T-R states are assigned with the quantum numbers n and l of the 3D isotropic harmonic oscillator. However, the translational excitations are not harmonic, since the level energies depend not only on n but also on l. For l > 1, the T-R levels having the same n,l values are split into groups of almost degenerate levels. The splitting patterns follow the predictions of group theory for the environment of T(d) symmetry, which is created by the configuration of the oxygen atoms of the large cage. The 2j + 1 degeneracy of the j = 1 and 2 rotational levels of the encapsulated hydrogen molecule is lifted entirely by the angular anisotropy of the H(2)-cage interaction potential. The patterns and magnitudes of the j = 1, 2 rotational level splittings, and the energies of the sublevels, in the large cage are virtually identical with those calculated for the small cage. This is in agreement with, and sheds light on, the observation that the S(0)(0) (j = 0-->2) bands in the rotational Raman spectra measured for simple H(2) hydrate and the binary hydrate of H(2) with tetrahydrofuran are remarkably similar with respect to their frequencies, widths, shapes, and internal structure, when the H(2) occupancy of the large cage of simple H(2) hydrate is low.

Coupled translation-rotation eigenstates of H(2) in C(60) and C(70) on the spectroscopically optimized interaction potential: Effects of cage anisotropy on the energy level structure and assignments.

We have developed a quantitatively accurate pairwise additive five-dimensional (5D) potential energy surface (PES) for H(2) in C(60) through fitting to the recently published infrared (IR) spectroscopic measurements of this system for H(2) in the vibrationally excited nu=1 state. The PES is based on the three-site H(2)-C pair potential introduced in this work, which in addition to the usual Lennard-Jones (LJ) interaction sites on each H atom of H(2) has the third LJ interaction site located at the midpoint of the H-H bond. For the optimal values of the three adjustable parameters of the potential model, the fully coupled quantum 5D calculations on this additive PES reproduce the six translation-rotation (T-R) energy levels observed so far in the IR spectra of H(2)@C(60) to within 0.6%. This is due in large part to the greatly improved description of the angular anisotropy of the H(2)-fullerene interaction afforded by the three-site H(2)-C pair potential. The same H(2)-C pair potential spectroscopically optimized for H(2)@C(60) was also used to construct the pairwise additive 5D PES of H(2) (nu=1) in C(70). This PES, because of the lower symmetry of C(70) (D(5h)) relative to that of C(60) (I(h)), exhibits pronounced anisotropy with respect to the direction of the translational motion of H(2) away from the cage center, unlike that of H(2) in C(60). As a result, the T-R energy level structure of H(2) in C(70) from the quantum 5D calculations on the optimized PES, the quantum numbers required for its assignment, and the degeneracy patterns which arise from the T-R coupling for translationally excited H(2) are all qualitatively different from those determined previously for H(2)@C(60) [M. Xu et al., J. Chem. Phys. 128, 011101 (2008).

H2, HD, and D2 inside C60: coupled translation-rotation eigenstates of the endohedral molecules from quantum five-dimensional calculations.

We have performed rigorous quantum five-dimensional (5D) calculations of the translation-rotation (T-R) energy levels and wave functions of H(2), HD, and D(2) inside C(60). This work is an extension of our earlier investigation of the quantum T-R dynamics of H(2)@C(60) [M. Xu et al., J. Chem. Phys. 128, 011101 (2008)] and uses the same computational methodology. Two 5D intermolecular potential energy surfaces (PESs) were employed, differing considerably in their well depths and the degree of confinement of the hydrogen molecule. Our calculations revealed pronounced sensitivity of the endohedral T-R dynamics to the differences in the interaction potentials, and to the large variations in the masses and the rotational constants of H(2), HD, and D(2). The T-R levels vary significantly in their energies and ordering on the two PESs, as well as from one isotopomer to another. Nevertheless, they all display the same distinctive patterns of degeneracies, which can be qualitatively understood and assigned in terms the model which combines the isotropic three-dimensional harmonic oscillator, the rigid rotor, and the coupling between the orbital and the rotational angular momenta of H(2)/HD/D(2). The quantum number j associated with the rotation of H(2), HD, and D(2) was found to be a good quantum number for H(2) and D(2) on both PESs, while most of the T-R levels of HD exhibit strong mixing of two or more rotational basis functions with different j values.

Quantum dynamics of H2, D2, and HD in the small dodecahedral cage of clathrate hydrate: evaluating H2-water nanocage interaction potentials by comparison of theory with inelastic neutron scattering experiments.

We have performed rigorous quantum five-dimensional (5D) calculations and analysis of the translation-rotation (T-R) energy levels of one H(2), D(2), and HD molecule inside the small dodecahedral (H(2)O)(20) cage of the structure II clathrate hydrate, which was treated as rigid. The H(2)- cage intermolecular potential energy surface (PES) used previously in the molecular dynamics simulations of the hydrogen hydrates [Alavi et al., J. Chem. Phys. 123, 024507 (2005)] was employed. This PES, denoted here as SPC/E, combines an effective, empirical water-water pair potential [Berendsen et al., J. Phys. Chem. 91, 6269 (1987)] and electrostatic interactions between the partial charges placed on H(2)O and H(2). The 5D T-R eigenstates of HD were calculated also on another 5D H(2)-cage PES denoted PA-D, used by us earlier to investigate the quantum T-R dynamics of H(2) and D(2) in the small cage [Xu et al., J. Phys. Chem. B 110, 24806 (2006)]. In the PA-D PES, the hydrogen-water pair potential is described by the ab initio 5D PES of the isolated H(2)-H(2)O dimer. The quality of the SPC/E and the PA-D H(2)-cage PESs was tested by direct comparison of the T-R excitation energies calculated on them to the results of two recent inelastic neutron scattering (INS) studies of H(2) and HD inside the small clathrate cage. The translational fundamental and overtone excitations, as well as the triplet splittings of the j=0-->j=1 rotational transitions, of H(2) and HD in the small cage calculated on the SPC/E PES agree very well with the INS results and represent a significant improvement over the results computed on the PA-D PES. Our calculations on the SPC/E PES also make predictions about several spectroscopic observables for the encapsulated H(2), D(2), and HD, which have not been measured yet.

Quantum dynamics of coupled translational and rotational motions of H2 inside C60.

We report rigorous quantum calculations of the translation-rotation (T-R) eigenstates of the H2 molecule in C60. The resulting level structure can be explained in terms of a few dominant features. These include the coupling between the orbital and the rotational angular momenta of H2 to give the total angular momentum lambda, and the splitting of the sevenfold degeneracy of T-R levels with lambda=3 by the nonsphericity of C60, according to the rules of the icosahedral I h group.

Quantum dynamics of small H2 and D2 clusters in the large cage of structure II clathrate hydrate: energetics, occupancy, and vibrationally averaged cluster structures.

We report diffusion Monte Carlo (DMC) calculations of the quantum translation-rotation (T-R) dynamics of one to five para-H(2) (p-H(2)) and ortho-D(2) (o-D(2)) molecules inside the large hexakaidecahedral (5(12)6(4)) cage of the structure II clathrate hydrate, which was taken to be rigid. These calculations provide a quantitative description of the size evolution of the ground-state properties, energetics, and the vibrationally averaged geometries, of small (p-H(2))(n) and (o-D(2))(n) clusters, n=1-5, in nanoconfinement. The zero-point energy (ZPE) of the T-R motions rises steeply with the cluster size, reaching 74% of the potential well depth for the caged (p-H(2))(4). At low temperatures, the rapid increase of the cluster ZPE as a function of n is the main factor that limits the occupancy of the large cage to at most four H(2) or D(2) molecules, in agreement with experiments. Our DMC results concerning the vibrationally averaged spatial distribution of four D(2) molecules, their mean distance from the cage center, the D(2)-D(2) separation, and the specific orientation and localization of the tetrahedral (D(2))(4) cluster relative to the framework of the large cage, agree very well with the low-temperature neutron diffraction experiments involving the large cage with the quadruple D(2) occupancy.

Hydrogen molecule in the small dodecahedral cage of a clathrate hydrate: quantum translation-rotation dynamics at higher excitation energies.

Higher-lying five-dimensional translation-rotation (T-R) eigenstates of a single p-H2 and o-D2 molecule confined inside the small dodecahedral (512) cage of the structure II clathrate hydrate are calculated rigorously, as fully coupled, with the cage assumed to be rigid. The calculations cover the excitation energies up to and beyond the j=2 rotational level of the free molecule, 356 cm(-1) for H2 and 179 cm(-1) for D2. It is found that j is a good quantum number for all the T-R states of p-H2, j=0 and j=2, considered. The same is not true for o-D2, where a number of T-R states in the neighborhood of the j=2 level show significant mixing of j=0 and j=2 rotational basis functions. The 5-fold degeneracy of the j=2 level of p-H2 is lifted completely due to the anisotropy of the cage environment, as is the 3-fold degeneracy of the j=1 level of o-H2 studied by us previously. Pure translational mode excitations with up to four quanta display negative anharmonicity, which was observed earlier for the translational fundamentals and their first overtones. The issues of assigning the combination states of p-H2 with excitations of two or all three translational modes, and of the strength of the mode coupling as a function of the excitation energy, are studied carefully for a range of quantum numbers. The average T-R energy of the encapsulated p-H2 is calculated as a function of temperature from 0 to 150 K.

One and two hydrogen molecules in the large cage of the structure II clathrate hydrate: quantum translation-rotation dynamics close to the cage wall.

We have performed a rigorous theoretical study of the quantum translation-rotation (T-R) dynamics of one and two H2 and D2 molecules confined inside the large hexakaidecahedral (5(12)6(4)) cage of the sII clathrate hydrate. For a single encapsulated H2 and D2 molecule, accurate quantum five-dimensional calculations of the T-R energy levels and wave functions are performed that include explicitly, as fully coupled, all three translational and the two rotational degrees of freedom of the hydrogen molecule, while the cage is taken to be rigid. In addition, the ground-state properties, energetics, and spatial distribution of one and two p-H2 and o-D2 molecules in the large cage are calculated rigorously using the diffusion Monte Carlo method. These calculations reveal that the low-energy T-R dynamics of hydrogen molecules in the large cage are qualitatively different from that inside the small cage, studied by us recently. This is caused by the following: (i) The large cage has a cavity whose diameter is about twice that of the small cage for the hydrogen molecule. (ii) In the small cage, the potential energy surface (PES) for H2 is essentially flat in the central region, while in the large cage the PES has a prominent maximum at the cage center, whose height exceeds the T-R zero-point energy of H2/D2. As a result, the guest molecule is excluded from the central part of the large cage, its wave function localized around the off-center global minimum. Peculiar quantum dynamics of the hydrogen molecule squeezed between the central maximum and the cage wall manifests in the excited T-R states whose energies and wave functions differ greatly from those for the small cage. Moreover, they are sensitive to the variations in the hydrogen-bonding topology, which modulate the corrugation of the cage wall.

Hydrogen molecule in the small dodecahedral cage of a clathrate hydrate: quantum five-dimensional calculations of the coupled translation-rotation eigenstates.

We report quantum five-dimensional (5D) calculations of the energy levels and wave functions of the hydrogen molecule, para-H2 and ortho-H2, confined inside the small dodecahedral (H2O)20 cage of the sII clathrate hydrate. All three translational and the two rotational degrees of freedom of H2 are included explicitly, as fully coupled, while the cage is treated as rigid. The 5D potential energy surface (PES) of the H2-cage system is pairwise additive, based on the high-quality ab initio 5D (rigid monomer) PES for the H2-H2O complex. The bound state calculations involve no dynamical approximations and provide an accurate picture of the quantum 5D translation-rotation dynamics of H2 inside the cage. The energy levels are assigned with translational (Cartesian) and rotational quantum numbers, based on calculated root-mean-square displacements and probability density plots. The translational modes exhibit negative anharmonicity. It is found that j is a good rotational quantum number, while the threefold degeneracy of the j = 1 level is lifted completely. There is considerable translation-rotation coupling, particularly for excited translational states.

HF in clusters of molecular hydrogen: II. Quantum solvation by H2 isotopomers, cluster rigidity, and comparison with CO-doped parahydrogen clusters.

We present a comprehensive theoretical study of the quantum solvation of the HF molecule by small clusters of the H2 isotopomers, p-H2, HD, and o-D2, with up to 13 hydrogen solvent molecules. This complements our earlier work on the HF-doped parahydrogen clusters [H. Jiang and Z. Bacic, J. Chem. Phys. 122, 244306 (2005)]. The ground-state properties of the clusters are calculated exactly using the diffusion Monte Carlo method. Detailed information is obtained regarding the size and isotopomer dependences of the energetics, vibrationally averaged structures, and their rigidity. The rigidity of these clusters is investigated further by analyzing the distributions of their principal moments of inertia from the diffusion Monte Carlo simulations. The clusters are found to be rather rigid, especially when compared with the pure parahydrogen clusters of the same size. Extensive comparison is made with the quantum Monte Carlo results for the CO-doped parahydrogen clusters and significant differences are observed in the size evolution of certain properties, notably the chemical potential.

Microsolvation of LiH+ in helium clusters: many-body effects and additivity models for the interaction forces.

The ab initio calculation of the interaction forces between the LiH+ molecular ion, at its equilibrium geometry, and several He atoms is carried out in order to isolate and assess the importance of many-body contributions in the search for realistic energy and geometry data. The full potential energy surface (PES) with a single helium partner is obtained first by using an aug-cc-pVQZ basis set for He and higher quality ones for Li and H. The calculations were performed at the CAS-SCF plus MRCI level for the lowest potential energy surface over a total of 480 grid points of the two intermolecular Jacobi coordinates, whereas the excited state surface has also been examined in order to exclude the presence of any significant nonadiabatic interaction between the two PESs. A numerical fit of the lower surface is presented and the general physical changes of the ionic interaction when going from the lower to the upper of the two potentials are described and discussed. The fairly limited importance of many-body effects for such systems is seen from further ab initio calculations including several He atoms: our results suggest that, at least in the present case, no strong charge migration occurs after He attachment, and therefore, one could realistically model larger clusters by implementing a sum-of-potentials approach via the presently computed PES.