Translation-rotation Eigenstates Research Articles

Flexible water molecule in C60: Intramolecular vibrational frequencies and translation-rotation eigenstates from fully coupled nine-dimensional quantum calculations with small basis sets.

We present a method for efficient calculation of intramolecular vibrational excitations of H2O inside C60, together with the low-energy intermolecular translation-rotation states within each intramolecular vibrational manifold. Apart from assuming rigid C60, this nine-dimensional (9D) quantum treatment is fully coupled. Following the recently introduced approach [P. M. Felker and Z. Bačić, J. Chem. Phys. 151, 024305 (2019)], the full 9D vibrational Hamiltonian of H2O@C60 is partitioned into two reduced-dimension Hamiltonians, a 6D one for the intermolecular vibrations and another in 3D for the intramolecular degrees of freedom, and a 9D remainder term. The two reduced-dimension Hamiltonians are diagonalized, and their eigenvectors are used to build up a product contracted basis in which the full vibrational Hamiltonian is diagonalized. The efficiency of this methodology derives from the insight of our earlier study referenced above that converged high-energy intramolecular vibrational excitations of weakly bound molecular complexes can be obtained from fully coupled quantum calculations where the full-dimensional product contracted basis includes only a small number of intermolecular vibrational eigenstates spanning the range of energies much below those of the intramolecular vibrational states of interest. In this study, the eigenstates included in the 6D intermolecular contacted basis extend to only 410 cm-1 above the ground state, which is much less than the H2O stretch and bend fundamentals, at ≈3700 and ≈1600 cm-1, respectively. The 9D calculations predict that the fundamentals of all three intramolecular modes, as well as the bend overtone, of the caged H2O are blueshifted relative to those of the gas-phase H2O, the two stretch modes much more so than the bend. Excitation of the bend mode affects the energies of the low-lying H2O rotational states significantly more than exciting either of the stretching modes. The center-of-mass translational fundamental is virtually unaffected by the excitation of any of the intramolecular vibrational modes. Further progress hinges on the experimental measurement of the vibrational frequency shifts in H2O@C60 and ab initio calculation of a high-quality 9D potential energy surface for this endohedral complex, neither of which is presently available.

The Endofullerene HF@C60: Inelastic Neutron Scattering Spectra from Quantum Simulations and Experiment, Validity of the Selection Rule, and Symmetry Breaking.

Accurate quantum simulations of the low-temperature inelastic neutron scattering (INS) spectra of HF@C60 are reported for two incident neutron wavelengths. They are distinguished by the rigorous inclusion of symmetry-breaking effects in the treatment and having the spectra computed with HF as the guest, rather than H2 or HD, as in the past work. The results demonstrate that the precedent-setting INS selection rule, originally derived for H2 and HD in near-spherical nanocavities, applies also to HF@C60, despite the large mass asymmetry of HF and the strongly mixed character of its translation-rotation eigenstates. This lends crucial support to the theoretical prediction made earlier that the INS selection rule is valid for any diatomic molecule in near-spherical nanoconfinement. The selection rule remains valid in the presence of symmetry breaking but is modified slightly in an interesting way. Comparison is made with the recently published experimental INS spectrum of HF@C60. The agreement is very good, apart from one peak for which our calculations suggest a reassignment. This reassignment is consistent with the measured INS spectrum presented in this work, which covers an extended energy range.

Perspective: Accurate treatment of the quantum dynamics of light molecules inside fullerene cages: Translation-rotation states, spectroscopy, and symmetry breaking

In this perspective, I review the current status of the theoretical investigations of the quantum translation-rotation (TR) dynamics and spectroscopy of light molecules encapsulated inside fullerenes, mostly C60 and C70. The methodologies developed in the past decade allow accurate quantum calculations of the TR eigenstates of one and two nanoconfined molecules and have led to deep insights into the nature of the underlying dynamics. Combining these bound-state methodologies with the formalism of inelastic neutron scattering (INS) has resulted in the novel and powerful approach for the quantum calculation of the INS spectra of a diatomic molecule in a nanocavity with an arbitrary geometry. These simulations have not only become indispensable for the interpretation and assignment of the experimental spectra but are also behind the surprising discovery of the INS selection rule for diatomics in near-spherical nanocavities. Promising directions for future research are discussed.

The effect of the condensed-phase environment on the vibrational frequency shift of a hydrogen molecule inside clathrate hydrates.

We report a theoretical study of the frequency shift (redshift) of the stretching fundamental transition of an H2 molecule confined inside the small dodecahedral cage of the structure II clathrate hydrate and its dependence on the condensed-phase environment. In order to determine how much the hydrate water molecules beyond the confining small cage contribute to the vibrational frequency shift, quantum five-dimensional (5D) calculations of the coupled translation-rotation eigenstates are performed for H2 in the v=0 and v=1 vibrational states inside spherical clathrate hydrate domains of increasing radius and a growing number of water molecules, ranging from 20 for the isolated small cage to over 1900. In these calculations, both H2 and the water domains are treated as rigid. The 5D intermolecular potential energy surface (PES) of H2 inside a hydrate domain is assumed to be pairwise additive. The H2-H2O pair interaction, represented by the 5D (rigid monomer) PES that depends on the vibrational state of H2, v=0 or v=1, is derived from the high-quality ab initio full-dimensional (9D) PES of the H2-H2O complex [P. Valiron et al., J. Chem. Phys. 129, 134306 (2008)]. The H2 vibrational frequency shift calculated for the largest clathrate domain considered, which mimics the condensed-phase environment, is about 10% larger in magnitude than that obtained by taking into account only the small cage. The calculated splittings of the translational fundamental of H2 change very little with the domain size, unlike the H2 j = 1 rotational splittings that decrease significantly as the domain size increases. The changes in both the vibrational frequency shift and the j = 1 rotational splitting due to the condensed-phase effects arise predominantly from the H2O molecules in the first three complete hydration shells around H2.

Effects of symmetry breaking on the translation-rotation eigenstates of H2, HF, and H2O inside the fullerene C60.

Splittings of the translation-rotation (TR) eigenstates of the solid light-molecule endofullerenes M@C60 (M = H2, H2O, HF) attributed to the symmetry breaking have been observed in the infrared (IR) and inelastic neutron scattering spectra of these species in the past couple of years. In a recent paper [Felker et al., Phys. Chem. Chem. Phys., 2017, 19, 31274], we established that the electrostatic, quadrupolar interaction between the guest molecule M and the twelve nearest-neighbor C60 cages of the solid is the main source of the symmetry breaking. The splittings of the three-fold degenerate ground states of the endohedral ortho-H2, ortho-H2O and the j = 1 level of HF calculated using this model were found to be in excellent agreement with the experimental results. Utilizing the same electrostatic model, this theoretical study investigates the effects of the symmetry breaking on the excited TR eigenstates of the three species, and how they manifest in their simulated low-temperature (5-6 K) near-IR (NIR) and far-IR (FIR) spectra. The TR eigenstates are calculated variationally for both the major P and minor H crystal orientations. For the H orientation, the calculated splittings of all of the TR levels of these species are less than 0.1 cm-1. For the dominant P orientation, the splittings vary strongly depending on the character of the excitations involved. In all of the species, the splittings of the higher rotationally excited levels are comparable in magnitude to those for the j = 1 levels. For the levels corresponding to purely translational excitations, the calculated splittings are about an order of magnitude smaller than those of the purely rotational eigenstates. Based on the computed TR eigenstates, the low-temperature NIR (for M = H2) and FIR (for M = HF and H2O) spectra are simulated for both the P and H orientations, and also combined as their weighted sum (0.15H + 0.85P). The weighted sum spectra computed for M = H2 and HF match quantitatively the corresponding measured spectra, while for M = H2O, the weighted sum FIR spectrum predicts features that can potentially be observed experimentally.

Electric-dipole-coupled H2O@C60 dimer: Translation-rotation eigenstates from twelve-dimensional quantum calculations.

We report on variational solutions to the twelve-dimensional (12D) Schrödinger equation appertaining to the translation-rotation (TR) eigenstates of H2O@C60 dimer, associated with the quantized "rattling" motions of the two encapsulated H2O molecules. Both H2O and C60 moieties are treated as rigid and the cage-cage geometry is taken to be fixed. We consider the TR eigenstates of H2O@C60 monomers in the dimer to be coupled by the electric dipole-dipole interaction between water moieties and develop expressions for computing the matrix elements of that interaction in a dimer basis composed of products of monomer 6D TR eigenstates reported by us recently [P. M. Felker and Z. Bačić, J. Chem. Phys. 144, 201101 (2016)]. We use these expressions to compute TR Hamiltonian matrices of H2O@C60 dimer for two values of the water dipole moment and for various dimer geometries. 12D TR eigenstates of the dimer are then obtained by filter diagonalization. The results reveal two classes of eigenstates, distinguished by the leading order (first or second) at which dipole-dipole coupling contributes to them. The two types of eigenstates differ in the general magnitude of their dipole-induced energy shifts and in the dependence of those shifts on the value of the water dipole moment and on the distance between the H2O@C60 monomers. The dimer results are also found to be markedly insensitive to any change in the orientations of the C60 cages. Finally, the results lend some support for the interpretation that electric dipole-dipole coupling is at least partially responsible for the apparent reduced-symmetry environment experienced by H2O in the powder samples of H2O@C60 [K. S. K. Goh et al., Phys. Chem. Chem. Phys. 16, 21330 (2014)], but only if the water dipole is taken to have a magnitude close to that of free water. The methodology developed in the paper is transferable directly to the calculation of TR eigenstates of larger H2O@C60 assemblies, that will be required for more extensive modeling of crystalline H2O@C60.

Accurate quantum calculations of translation-rotation eigenstates in electric-dipole-coupled H2O@C60 assemblies

We present methodology for variational calculation of the 6n-dimensional translation-rotation (TR) eigenstates of assemblies of n H2O@C60 moieties coupled by dipole-dipole interactions. We show that the TR Hamiltonian matrix for any n can be constructed from dipole-dipole matrix elements computed for n=2. We present results for linear H2O@C60 assemblies. Two classes of eigenstates are revealed. One class comprises excitations of the 111 rotational level of H2O. The lowest-energy 111-derived eigenstate for each assembly exhibits significant dipole ordering and shifts down in energy with the assembly size.

Communication: Quantum six-dimensional calculations of the coupled translation-rotation eigenstates of H2O@C60.

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.

Inelastic neutron scattering spectrum of H2@C60 and its temperature dependence decoded using rigorous quantum calculations and a new selection rule

In the supramolecular complex H2@C60, the lightest of molecules, H2, is encapsulated inside the most highly symmetric molecule C60. The elegance and apparent simplicity of H2@C60 conceal highly intricate quantum dynamics of the coupled translational and rotational motions of the guest molecule in a nearly spherical nanoscale cavity, which embodies some of the most fundamental concepts of quantum mechanics. Here we present the first rigorous and highly accurate quantum calculations of the inelastic neutron scattering (INS) spectra of this prototypical endohedral fullerene complex and their temperature dependence. The calculations enable complete assignment of the recently reported experimental INS spectra of H2@C60 measured at several temperatures. We also derive a new and unexpected selection rule for the INS spectroscopy of H2 in a near-spherical confinement, which explains why the INS transitions between certain translation-rotation eigenstates of H2 in C60 have zero intensity and do not appear in the spectra.

H2 in solid C60: Coupled translation-rotation eigenstates in the octahedral interstitial site from quantum five-dimensional calculations

We report rigorous quantum five-dimensional (5D) calculations of the coupled translation-rotation (TR) energy levels and wave functions of an H2 molecule, in the ground (ν = 0) and vibrationally excited (ν = 1) states, confined inside the octahedral interstitial site of solid C60 with S6 symmetry. Translational and rotational excitations of H2 in this nanocavity have been measured by the inelastic neutron scattering (INS) and infrared (IR) spectroscopy, enabling direct comparison between theory and experiment. A pairwise additive 5D intermolecular potential energy surface (PES) was employed in the calculations. The quantum calculations cover the range of energies and types of translational and rotational excitations of the guest molecule which go substantially beyond those considered in the earlier theoretical investigations of this system, revealing new information about the TR energy level structure. The computed j = 1 and j = 2 rotational levels and their splittings, as well as the translational fundamental, are in semi-quantitative agreement with the available INS and IR data, indicating the need for a more accurate intermolecular PES. Our calculations reveal a strong dependence of the TR energy levels, in particular their splittings, on the setting angle which defines the orientation of the C60 molecules relative to their local threefold axes.

Hydrogen adsorbed in a metal organic framework-5: Coupled translation-rotation eigenstates from quantum five-dimensional calculations

We report rigorous quantum five-dimensional (5D) calculations of the coupled translation-rotation (T-R) eigenstates of a H(2) molecule adsorbed in metal organic framework-5 (MOF-5), a prototypical nanoporous material, which was treated as rigid. The anisotropic interactions between H(2) and MOF-5 were represented by the analytical 5D intermolecular potential energy surface (PES) used previously in the simulations of the thermodynamics of hydrogen sorption in this system [Belof et al., J. Phys. Chem. C 113, 9316 (2009)]. The global and local minima on this 5D PES correspond to all of the known binding sites of H(2) in MOF-5, three of which, α-, β-, and γ-sites are located on the inorganic cluster node of the framework, while two of them, the δ- and ε-sites, are on the phenylene link. In addition, 2D rotational PESs were calculated ab initio for each of these binding sites, keeping the center of mass of H(2) fixed at the respective equilibrium geometries; purely rotational energy levels of H(2) on these 2D PESs were computed by means of quantum 2D calculations. On the 5D PES, the three adjacent γ-sites lie just 1.1 meV above the minimum-energy α-site, and are separated from it by a very low barrier. These features allow extensive wave function delocalization of even the lowest translationally excited T-R eigenstates over the α- and γ-sites, presenting significant challenges for both the quantum bound-state calculations and the analysis of the results. Detailed comparison is made with the available experimental data.

Inelastic neutron scattering spectra of a hydrogen molecule in a nanocavity: Methodology for quantum calculations incorporating the coupled five-dimensional translation-rotation eigenstates

We present an in-depth description of the methodology for accurate quantum calculation of the inelastic neutron scattering (INS) spectra of an H${}_{2}$ molecule confined inside a nanosize cavity of an arbitrary shape. This methodology was introduced in a recent work [M. Xu, L. Ulivi, M. Celli, D. Colognesi, and Z. Ba\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{}, Phys. Rev. B 83, 241403(R) (2011)], where the INS spectra of para- and ortho-H${}_{2}$ in the small cage of the structure II clathrate hydrate were simulated and compared with the measured spectra. The key distinctive feature of our approach, and its main strength and advantage, is the use of the coupled quantum 5D translation-rotation (TR) energy levels and wave functions of the entrapped H${}_{2}$ molecule, rigorously calculated on the 5D intermolecular potential energy surface (PES), as the initial and the final states of the INS transitions. In this work, we describe the implementation of the 5D TR wave functions within the quantum INS formalism, and obtain the working expressions for the matrix elements required to compute the INS spectra of the nanoconfined H${}_{2}$ molecule. The computational approach devised for efficient calculation of the 5D TR eigenstates in the compact contracted basis, indispensable for our quantum simulation of the INS spectra, is presented as well. Since the TR coupling is fully taken into account, the computed INS spectra exhibit a uniquely high degree of realism and faithfully reflect the quantum dynamics of H${}_{2}$ on the PES of the host environment.

Methane molecule confined in the small and large cages of structure I clathrate hydrate: Quantum six-dimensional calculations of the coupled translation-rotation eigenstates

We report fully coupled quantum six-dimensional (6D) calculations of the translation-rotation (T-R) energy levels of CH(4) molecule inside the small dodecahedral (5(12)) and large tetracaidecahedral (5(12)6(2)) cages of the structure I clathrate hydrate. The quantum dynamics of the three translational and three rotational degrees of freedom of CH(4) are treated rigorously, while the guest molecule and the host cavities are taken to be rigid. The matrix of the full 6D T-R Hamiltonian is diagonalized in the product basis of contracted translational and angular basis functions, generated by solving two reduced-dimension (3D) eigenvalue problems. A pairwise additive CH(4)-cage 6D potential energy surface (PES) is employed, constructed using the anisotropic CH(4)-H(2)O pair potential which was utilized previously in the molecular dynamics simulations of methane hydrate. Our calculations elucidate the key features of the T-R energy level structure of the nanoconfined CH(4). The rotational levels of methane exhibit an elaborate pattern of splittings caused by the angular anisotropy of the environment; the splitting patterns are identical for both types of cages. Translationally excited T-R states in the small cage are assigned in terms of the quantum numbers n and l of the 3D isotropic harmonic oscillator and those in the large cage using the Cartesian quantum numbers. Extensive comparison is made with the data from the inelastic neutron scattering studies of methane hydrate, allowing an assessment of the accuracy of the 6D PES employed.

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

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).

H 2 , 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.