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Impact of the Condensed-Phase Environment on the Translation-Rotation Eigenstates and Spectra of a Hydrogen Molecule in Clathrate Hydrates Anna Powers, Ondrej Marsalek, Minzhong Xu, Lorenzo Ulivi, Daniele Colognesi, Mark Edward Tuckerman, and Zlatko Bacic J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02611 • Publication Date (Web): 04 Jan 2016 Downloaded from http://pubs.acs.org on January 7, 2016

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The Journal of Physical Chemistry Letters

Impact of the Condensed-Phase Environment on the Translation-Rotation Eigenstates and Spectra of a Hydrogen Molecule in Clathrate Hydrates Anna Powers,1 Ondrej Marsalek,1 Minzhong Xu,1 Lorenzo Ulivi,2 Daniele Colognesi,2 Mark E. Tuckerman∗ ,3,4 and Zlatko Baˇcić†1,4 1

Department of Chemistry, New York University, New York, NY 10003, USA 2

Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, I-50019, Sesto Fiorentino, Italy

3

Department of Chemistry and Courant Institute of Mathematical Sciences, New York University, New York, NY 10003, USA 4

NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China (Dated: December 24, 2015)

∗ †

Electronic mail: [email protected] Electronic mail: [email protected]

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Abstract We investigate systematically the manifestations of the condensed-phase environment of the structure II clathrate hydrate in the translation-rotation (TR) dynamics and the inelastic neutron scattering (INS) spectra of an H2 molecule confined in the small dodecahedral cage of the hydrate. The aim is to elucidate the extent to which these properties are affected by the clathrate water molecules beyond the confining cage and the proton disorder of the water framework. For this purpose, quantum calculations of the TR eigenstates and INS spectra are performed for H2 inside spherical clathrate domains of gradually increasing radius and the number of water molecules ranging from 20 for the isolated small cage to more than 1800. For each domain size, several hundred distinct hydrogen-bonding topologies are constructed in order to simulate the effects of the proton disorder. Our study reveals that the clathrate-induced splittings of the j = 1 rotational level and the translational fundamental of the guest H2 are influenced by the condensed-phase environment to a dramatically different degree, the former very strongly and the latter only weakly. Keywords: dihydrogen, clathrate hydrates, condensed phase, eigenstates, spectroscopy, neutron scattering


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Hydrogen clathrate hydrates are inclusion compounds where hydrogen molecules are trapped inside closely packed polyhedral cavities, or cages, within the ice-like lattice of hydrogen-bonded water molecules.1–3 Simple hydrogen clathrate hydrates, having hydrogen molecules as the sole guests,4,5 adopt the classical structure II (sII).1,2,5 Its cubic unit cell has sixteen small cages, each comprised of 20 H2 O molecules and denoted 512 for their 12 pentagonal faces, as well as eight large cages, each formed by 28 H2 O molecules and denoted 512 64 for their 12 pentagonal and 4 hexagonal faces. The small cage can accommodate only one H2 molecule, while up to four H2 molecules can be entrapped in the large cage.6 Hydrogen clathrate hydrates have generated a great deal of interest because of the potential they may have as environmentally friendly hydrogen storage materials.1,2,7–10 They also provide an outstanding opportunity for investigating the dynamical consequences and spectroscopic signatures of the nanoscale confinement in the quantum regime. The translational center-of-mass (c.m.) motions of the guest molecule(s) are quantized, and coupled by the confining potential to their quantized rotational degrees of freedom. For a single hydrogen molecule in the cages of the sII clathrate hydrate, our quantum 5D calculations have revealed the salient features of the translation-rotation (TR) eigenstates.11–13 These predictions have been validated by the inelastic neutron scattering (INS)14,15 and the Raman spectra16–18 of the binary tetrahydrofuran (THF) + H2 /HD/D2 sII clathrate hydrate. Recently, we have developed the quantum methodology for rigorous calculation of the INS spectra of a nanoconfined hydrogen molecule with a uniquely high degree of realism, owing to the use of the quantum 5D TR energy levels and wave functions.19–21 This novel approach has been used to compute the INS spectra of H2 and HD in two binary D2 O clathrates, one with the cubic sII structure,19,22,23 and the other having the hexagonal structure (sH).23,24 In both, a large promoter molecule resides in the large cages, while a single H2 /HD occupies the small cages, and in the case of the sH clathrate also the medium cages. Semiquantitative agreement was found between the simulated and experimental spectra, allowing the assignment of the latter in terms of various TR excitations of the guest molecule. From the generally good agreement between theory and experiment, one might be tempted to conclude that the main facets of the TR dynamics of molecular hydrogen in clathrate hydrates are well understood. Nevertheless, the following basic question has not been addressed in a rigorous fashion thus far: Is the dynamical behavior of the caged H2 molecule determined primarily by its interactions with the nearest water molecules of the ACS Paragon 3 Plus Environment


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confining cage, or do the framework water molecules further away make a significant contribution, and if yes, out to what distance from the cage center? To date, the theoretical studies of the quantum dynamics and INS spectra have considered one11–13,19–25 and several H2 molecules26–30 in the isolated clathrate hydrate cages, where H2 interacts only with the H2 O molecules of the individual cages. Therefore, they could not shed light on this problem. In an effort to close this gap in our fundamental understanding, here we explore systematically, for the first time, how the condensed-phase environment of the clathrate hydrate manifests in the TR dynamics and the INS spectra of the guest H2 molecule. A hydrogen molecule inside the small dodecahedral cage of the sII clathrate hydrate is considered. We investigate (i) the degree to which the dynamics and spectroscopy of the entrapped H2 are affected by its interactions with the clathrate water molecules beyond those in the confining nanocage, (ii) the size of the clathrate domain, i.e., the number of water molecules around H2 , for which the spectroscopic observables considered approach their respective bulk condensed-phase values, and (iii) the effects of the proton disorder of the water framework. For a single hydrogen molecule in the small cage of the sII clathrate, the threefold degeneracy of the translational fundamental and the j = 1 rotational level is lifted completely, and these appear as closely spaced triplets of states.11,12,14 Both the translational and rotational splittings are caused by the cage anisotropies, the former with respect to the translational motion of the c.m. of the guest molecule and the latter with respect to its angular orientation within the cage. The splittings reflect the strength of the anisotropy, radial or angular, while the splitting patterns are related to the symmetry of the nanocavity. Therefore, translational and rotational splittings, observable in the experimental INS spectra,14,19,21–24 are sensitive probes of the confining environment. For this reason, our efforts to elucidate the condensed-phase effects focus on the splittings of the j = 1 rotational level and the translational fundamental of the H2 molecule in clathrate hydrate domains of different sizes. In this study, the crystalline sII clathrate hydrate is represented by a 3 × 3 × 3 supercell, having 27 primitive unit cells of the crystal. The O atoms of the water molecules in the supercell are placed at their crystallographic positions.31 However, the H atoms are configurationally disordered. Different random distributions of the framework water protons were then generated, each consistent with the Bernal-Fowler ice rules32 and the periodic boundary conditions on the supercell. One proton configuration with a negligible dipole moment was selected for use in further calculations. In the next step, the energy of the empty supercell ACS Paragon 4 Plus Environment

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with the chosen proton distribution was minimized by allowing the (essentially rigid) H2 O molecules to change their orientation slightly, while keeping the O atoms fixed at their crystallographic positions. This was accomplished using the lattice dynamics program GULP,33 assuming the SPC/E effective pair potential34 for the H2 O-H2 O interactions, and (stiff) harmonic forces for the intramolecular H-O-H interaction. In these calculations, the O-H distance was set to 0.9716 ˚ A and the H-O-H angle to 104.69◦ .11 The sII clathrate supercell constructed in this manner was used in two ways. First, 432 (27×16) small cages were extracted from the supercell, each having a distinct distribution of the water protons, and quantum 5D calculations of the TR energy levels and wave functions for H2 inside each of the cages, taken to be rigid, were performed. The results, which are discussed extensively later in the paper, provide detailed information about the sensitivity of the TR excitations of H2 to different proton configurations for the isolated small cages. Next, we sought to investigate the combined influence of an increasing number of water molecules beyond the small cage and the proton disorder on the TR dynamics and INS spectra of the encapsulated H2 molecule. This was done by defining several spherical clathrate domains of increasing radius, with a growing number of H2 O molecules in them, around each of the 432 small cages, by the following process: (1) The c.m. of a given small cage is computed and the entire system is moved so as to make this c.m. the origin of the coordinate system, by wrapping all the water molecules of the supercell around it into the closest image to the origin. (2) Guided by the radial distribution function of the O atoms, radial distances from the cage center were chosen that include entire well-defined shells of water molecules: (a) Setting the cutoff radius to 5.0 ˚ A results in the sperical domain consisting of the 20 water molecules of the small cage itself. (b) Increasing the radius to 7.5 ˚ A includes additional 20 water molecules, those directly hydrogen-bonded to the small cage, creating a spherical domain with a total of 40 water molecules. (c) The cutoff radius set to 9.0 ˚ A encompasses another shell of 32 water molecules, resulting in a sphere containing 72 water mlecules. (d) The largest spherical domain possible for our supercell has a radius of 25.65 ˚ A, with a total of 1872 water molecules in it. This domain constitutes our condensed-phase environment for the small cage considered. These four domains are shown in Fig. 1, for one particular small cage at the center of the supercell. The domains of a given size (radius) have different water proton distributions for each of the 432 small cages at the center. The results obtained using the largest domain are consistent with the calculations performed ACS Paragon 5 Plus Environment


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with the electrostatic potential evaluated using the 3 × 3 × 3 supercell in periodic boundary conditions, as detailed in the Supporting Information. Quantum 5D TR eigenstate computations were carried out for a single H2 inside the central small cages of the 432 condensed-phase domains with 1872 H2 O molecules each (Fig. 1(d)), treated as rigid. All other cages of the domain are empty. To better understand the size evolution of the condensed-phase effects, such calculations were performed also for 432 intermediate-size domains with the radius of 9 ˚ A and 72 water molecules each [Fig. 1(c)]. Our computational methodolodology employed in the quantum 5D calculations of the TR energy levels and wave functions of H2 confined in the individual cages and the clathrate hydrate domains11,12,20 is outlined in the Supporting Information. The 5D intermolecular PES for the interaction between the entrapped H2 and the water molecules within the clathrate domain considered, denoted as VH2 −domain , is assumed to be pairwise additive.11–13,19,21–24 For N H2 O molecules within the domain (with N ranging from 20 to 1872), VH2 −domain can be written as VH2 −domain (qh ) =

N X

VH2 −H2 O (qh , Ξw ),

(1)

w=1

where qh are the coordinates (x, y, z, θ, φ) of H2 , VH2 −H2 O is the pair interaction between H2 and a framework H2 O molecule, and the index w runs over the water molecules of the domain, whose coordinates Ξw are fixed. VH2 −domain is based on the interaction potential by Alavi et al..35 The pair interaction VH2 −H2 O in eq 1 combines the Coulomb interactions between the three point charges on the H2 O molecule, taken from the SPC/E effective pair potential model for water,34 and the three point charges on the H2 molecule, chosen to reproduce its gas-phase quadrupole moment, with a Lennard-Jones (LJ) interaction between the O atom of H2 O and the c.m. of H2 .12,35 More details are given in the Supporting Information. Figure 2 shows the correlation plots and the distributions of the j = 1 splittings calculated for 432 small cages when they are (i) isolated, (ii) in their respective condensed-phase environments (domain radius r = 25.65 ˚ A, 1872 water molecules), and (iii) in their respective intermediate-size clathrate domains (r = 9.0 ˚ A, 72 water molecules). The splittings ∆E are calculated as the difference between the highest- and lowest-energy components of the j = 1 triplet, labeled using the quantum numbers (j, |m|)11 as (1, 1)u and (1, 1)l , respectively. The top panel of Figure 2 provides several important pieces of information. First, the distribution of j = 1 splittings is broad, already spread over ∼ 30 cm

−1


for the isolated small cages,

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implying that the j = 1 splitings depend strongly on the configuration of the cage water protons. The explanation for this sensitivity is straightforward: the angular anisotropy of the H2 -cage interaction arises primarily from the Coulomb interactions between the point charges placed on both H2 and H2 O. Consequently, different water proton distributions generate a broad range of angular anisotropies, which result in j = 1 splittings of widely different magnitudes. Next, for the small cages embeded in the condensed-phase environment (1872 water molecules), the width of the j = 1 splitting distribution is larger, ∼40 cm−1 . Moreover, this distribution peaks around 30 cm−1 , in contrast to that for the isolated small cages, whose maximum is at ∼20 cm−1 . Taken together, these observations reveal that the long-range Coulomb interactions between H2 and water molecules beyond the confining small cage contribute significantly to the angular anisotropy, and on the average increase it markedly. Furthermore, the degree of correlation beteween the j = 1 splittings computed for H2 in each of 432 isolated small cages and in the cages surrounded by the condensed-phase environment, is low. Finally, the j = 1 splitting observed in the INS spectra14,15,19,22 of the binary sII clathrate hydrate is the average value of the broad distribution shown in Figure 2. The situation is analogous for the j = 2 splitting in the Raman spectra.16–18 Very interesting insight is provided by the bottom panel of Figure 2. It shows that the j = 1 splittings computed taking into account the 72 water molecules inside the intermediate-size domain with the 9.0 ˚ A radius are highly correlated with those calculated for the condensedphase environment of 1872 water molecules. The two j = 1 splitting distributions are also very similar in both their widths and the positions of their respective maxima. This means that the dominant contributions to the j = 1 splittings come from the 72 water molecules effectively forming three complete hydration shells around the H2 molecule [Figure 1c)], which thus capture the main effects of the condensed phase on this important observable. To gain a better understanding of this finding, in Figure S1 of the Supporting Information, the H2 -host electrostatic interaction potential is decomposed into contributions from each of the three hydration shells and the condensed-phase environment beyond them. The effects of the proton disorder on the j = 1 rotational level of the entrapped H2 are evident in Figure S2 of the Supporting Information, where the distribution of its three components calculated for the 432 small cages in their respective condensed-phase environments of 1872 H2 O molecules shows a distinctive triplet structure. Figure 3 displays the correlation between the splittings of the H2 translational fundamenACS Paragon 7 Plus Environment


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tal calculated for each of the 432 isolated small cages and when these cages are embedded in their condensed-phase environments. The splittings ∆E are obtained as the difference between the highest- and lowest-energy sublevels of the translational fundamental, labeled using the quantum numbers (v, |l|, vz )12 as (0, 0, 1) and (1, 1, 0), respectively.11 What Figure 3 reveals differs qualitatively from our findings regarding the j = 1 splittings, which emerged from Figure 2. The translational splitting distributions obtained for the isolated cages and the condensed phase have very similar widths and maxima. Moreover, their widths, 2-3 cm−1 , are an order of magnitude smaller than the widths of the j = 1 splitting distributions. From this, we conclude that (a) the water molecules beyond the small cages contribute very little to the radial anisotropy responsible for the splitting of the translational fundamental, in contrast to the angular anisotropy causing the j = 1 splittings, and (b) the translational fundamental splitting is only weakly sensitive to the proton configurations, unlike the j = 1 rotational splitting. The explanation for both observations lies in the fact that the translational rattling dynamics of the guest H2 is governed mainly by the LJ interactions between its c.m. and the O atoms of the clathrate water molecules. Proton disorder therefore plays only a minor role, hence a narrow distribution of the translational fundamental splittings. In addition, the LJ interactions are short-range, and as a result condensed-phase water molecules have a negligible influence on the translational splittings. Having discussed the effects of the condensed-phase environment on the TR dynamics, we now explore their manifestations in the INS spectra of an H2 molecule inside the small cage of the binary sII clathrate hydrate. Figure 4 shows the low-energy portions of the INS spectra of the caged p-H2 and o-H2 , respectively, recorded at 20 K on the TOSCAII spectrometer, as described elsewhere.19,22 At this low temperature, only the ground TR states of p- and o-H2 are populated, and all INS transitions originate from them. Also shown are the corresponding simulated INS spectra, computed both for H2 in one particular isolated small cage (chosen so that the j = 1 splitting calculated for it lies at the center of the 2D distribution in Figure 2) and with that cage in the condensed-phase environment of 1872 water molecules. The INS spectra were calculated utilizing our quantum methodology,19,20 and convolved with the experimental resolution function of the TOSCA-II spectrometer.19,22 The experimental spectrum of p-H2 in Figure 4 shows the band associated with the rotational j = 0 → 1 transition. It is split into a triplet by the angular anisotropy of the guest-host interaction, as discussed earlier. The same rotational band simulated for p-H2 in ACS Paragon 8 Plus Environment

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the isolated small cage also exhibits the triplet structure, but its splitting is somewhat larger than the measured value, indicating that the 5D H2 -cage PES employed in the calculations overestimates the angular anisotropy of the H2 -cage interaction. However, this comparison is not definitive, since the INS spectrum is measured for H2 in the bulk sII clathrate hydrate, not in an isolated cage. Therefore, a much more relevant comparison with experiment is provided by the INS spectrum computed for p-H2 in the condensed-phase environment, also shown in Figure 4. This spectrum displays a resolved triplet structure of the rotational band as well, but the splitting of its three components significantly exceeds that calculated for the isolated cage. This is in line with the observations made earlier in connection with Figure 2, that the j = 1 splitting distribution calculated for the condensed-phase environment is broader and peaks at a higher value than that for the isolated small cage. The transition present in the experimental spectrum of o-H2 in Figure 4 is the rotationally elastic (j = 1 → 1) translational fundamental excitation. In principle, the radial anisotropy of the cage should split this transition into a triplet. However, this band actually consists of 27 components,22 and due to the limited instrumental resolution, the triplet is only partially resolved into two rather broad components. As Figure 4 shows, the INS spectra calculated for o-H2 in the isolated small cage and the condensed-phase environment reproduce very well the two peaks of the translational fundamental. Moreover, the translational bands in the two simulated spectra have very similar positions and shapes, implying that the condensed-phase environment beyond the small cage containing the H2 molecule contributes very little to them. This is consistent with our finding based on Figure 3, that the condensed phase has a minor role in the splitting of the translational fundamental. Some residual quantitative differences between experiment and theory are evident from Figure 4. One of them is the slightly larger experimental splitting of the translational fundamental relative to the calculated one. The other involves the lower-energy peak of the translational band, which hides two unresolved (1, 1, 0) components12 of the fundamental; its experimentally determined width is greater than that from theory. Both discrepancies suggest that the radial anisotropy of the H2 -clathrate PES employed is not sufficiently strong. However, a definitive conclusion on this point cannot be reached yet. Comparison to measured INS spectra requires taking proton disorder into account, which the present calculations of the INS spectrum, for H2 in a single small cage embedded in the condensed-phase environment, do not capture fully. Accounting for proton disorder can be accomplished by averaging ACS Paragon 9 Plus Environment


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over calculated INS spectra of H2 inside many small cages of the supercell, each with a distinct hydrogen-bond topology and surrounded by the 1872 water molecules mimicking the condensed-phase environment. Such simulations are planned for the near future. In summary, we have presented the first systematic investigation of the effects of the condensed-phase environment of the sII clathrate hydrate on the TR dynamics and INS spectra of an H2 molecule in the small dodecahedral cage. Our goal has been to elucidate the degree to which its properties are affected by the interactions with the clathrate water molecules beyond the confining small cage and the proton disorder of the water framework. Rigorous quantum calculations of the 5D TR eigenstates and INS spectra were performed for the H2 molecule confined inside several spherical clathrate domains with gradually increasing radius and a growing number of water molecules in them, from the smallest, encompassing only the single small cage with 20 water molecules, to the largest domain possible for the 3 × 3 × 3 supercell employed, with a total of 1872 water molecules in it. For each domain size, 432 hydrogen-bonding topologies were considered, aiming to capture the effects of the proton disorder. Our focus has been on the clathrate-induced splittings of the j = 1 rotational level and the translational fundamental of the guest H2 , which appear prominently in the experimental INS spectra. The quantum 5D TR eigenstate calculations and the simulated INS spectra of the nanoconfined p- and o-H2 reveal that these two types of splittings are affected very differently by the condensed-phase environment. The j = 1 rotational splitting is increased significantly by the interactions of H2 with the water molecules beyond the central small cage, and it depends strongly on the distribution of the water protons. In contrast, the splitting of the translational fundamental changes very little with the inclusion of the condensed-phase water and is only weakly sensitive to the proton disorder. These striking differences can be traced to the different types of interactions chiefly responsible for the splittings: long-range Coulomb interactions in the case of the j = 1 splittings, and the short-range Lennard-Jones interactions for the splitting of the translational fundamental. A clathrate domain consisting of 72 water molecules forming three complete solvation shells around an H2 molecule accounts for the main condensed-phase effects on the TR dynamics. The insights gained in the present study are applicable to molecular hydrogen in other clathrate hydrates structures, and are relevant for other small molecules, e.g., CH4 , in clathrate hydrates and, likely, in other inclusion compounds.

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ACKNOWLEDGMENTS Z.B. thanks the National Science Foundation for financial support of this research through the Grant CHE-1112292. SUPPORTING INFORMATION Computational methodology for the quantum 5D calculation of the TR eigenstates of H2 confined in the individual clathrate cages and domains, detailed description of the 5D intermolecular H2 -clathrate PES and its numerical evaluation. Decomposition of the electrostatic H2 -clathrate potential into contributions from distinct solvation shells, and the effect of the proton disorder on the three components of the j = 1 rotational level. This material is available free of charge via the Internet at http://pubs.acs.org .

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M. Xu, F. Sebastianelli, and Z. Baˇcić. 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. J. Chem. Phys., 128:244715, 2008.

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M. Xu, F. Sebastianelli, and Z. Baˇcić. 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. J. Phys. Chem. A, 113:7601, 2009.

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M. Xu, L. Ulivi, M. Celli, D. Colognesi, and Z. Baˇcić. Quantum calculation of inelastic neutronscattering spectra of a hydrogen molecule inside a nanoscale cavity based on rigorous treatment of the coupled translation-rotation dynamics. Phys. Rev. B, 83:241403(R), 2011.

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M. Xu and Z. Baˇcić. 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, 84:195445, 2011. ACS Paragon12 Plus Environment

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M. Xu, L. Ulivi, M. Celli, D. Colognesi, and Z. Baˇcić. 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., 563:1, 2013.

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D. Colognesi, M. Celli, L. Ulivi, M. Xu, and Z. Baˇcić. Neutron scattering measurements and computation of the quantum dynamics of hydrogen molecules trapped in the small and large cages of clathrate hydrates. J. Phys. Chem. A, 117:7314, 2013.

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D. Colognesi, A. Powers, M. Celli, M. Xu, Z. Baˇcić, and L. Ulivi. The HD molecule in small and medium cages of clathrate hydrates: Quantum dynamics studied by neutron scattering measurements and computation. J. Chem. Phys., 141:134501, 2014.

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M. Celli, A. Powers, D. Colognesi, M. Xu, Z. Baˇcić, and L. Ulivi. Experimental inelastic neutron scattering spectrum of hydrogen hexagonal clathrate-hydrate compared with rigorous quantum simulations. J. Chem. Phys., 139:164507, 2013.

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F. Sebastianelli, M. Xu, and Z. Baˇcić. Quantum dynamics of small H2 and D2 clusters in the large cage of structure II clathrate hydrate: Energetics, occupancy, and vibrationally averaged cluster structures. J. Chem. Phys., 129:244706, 2008.

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FIG. 1: Structures of the sII clathrate hydrate domains with incrasing radii (r) from the center of the small cage chosen to be at the center of the supercell. (a) r = 5.0 ˚ A, 20 H2 O molecules (blue) of the small dodecahedral cage. (b) r = 7.5 ˚ A, additional 20 H2 O molecules (red), total of 40 H2 O molecules. (c) r = 9.0 ˚ A, additional 32 H2 O molecules (green), total of 72 H2 O molecules. (d) r = 25.65 ˚ A, additional 1800 H2 O molecules, total of 1872 H2 O molecules. FIG. 2: Top: Correlation plot of the j = 1 rotational splittings (∆E) calculated for 432 small cages, isolated (domain radius r = 5.0 ˚ A, 20 water molecules) and in their respective condensed-phase environments (r = 25.65 ˚ A, 1872 water molecules). Bottom: Correlation plot of the j = 1 splittings (∆E) calculated for 432 small cages, in their respective intermediate-size clathrate domains (r = 9.0 ˚ A, 72 water molecules) and in their respective condensed-phase environments (r = 25.65 ˚ A, 1872 water molecules). Also shown in the two panels are the distributions of the 432 j = 1 splittings calculated for each of the three clathrate domain sizes. For additional explanation see the text. FIG. 3: Correlation plot of the splittings (∆E) of the translational fundamental calculated for 432 small cages, isolated (domain radius r = 5.0 ˚ A, 20 water molecules) and in their respective condensed-phase environments (r = 25.65 ˚ A, 1872 water molecules). Also shown are the distributions of the 432 translational splittings calculated for each of the isolated cages and for their condensed-phase environments. For additional explanation see the text. FIG. 4: Comparison of the experimental INS spectra of p-H2 (red full line) and o-H2 (blue full line) in the small cage of the sII clathrate hydrate with the corrresponding INS spectra calculated for p-H2 (red) and o-H2 (blue) in the individual small cage (single cage) and the condensed-phase environment (full), respectively. For additional explanation see the text.



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Quantum dynamics of hydrogen in the condensed phase Quantum dynamics of hydrogen in the single cage


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Quantum dynamics of hydrogen in the condensed phase Quantum dynamics of hydrogen in the single cage


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Impact of the Condensed-Phase Environment on ... - ACS Publications

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