Thermodynamic Stability of Hydrogen Clathrates

Thermodynamic stability of hydrogen clathrates

Serguei Patchkovskii and John S. Tse*

Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, ON, Canada K1A 0R6

Edited by Russell J. Hemley, Carnegie Institution of Washington, Washington, DC, and approved October 13, 2003 (received for review February 14, 2003) The stability of the recently characterized type II hydrogen clath- and raised the prospect of using clathrate hydrate as an efficient rate [Mao, W. L., Mao, H.-K., Goncharov, A. F., Struzhkin, V. V., Guo, H2 storage medium. Q., et al. (2002) Science 297, 2247–2249] with respect to hydrogen Under normal conditions, clathrate hydrates are known to occupancy is examined with a statistical mechanical model in have three distinct crystalline structures (10). Both structure I conjunction with first-principles quantum chemistry calculations. It and II clathrate hydrates have a cubic structure and a guest:host is found that the stability of the clathrate is mainly caused by water ratio of Ϸ1:6 (10). Usually, very small guest (rare gas) dispersive interactions between H2 molecules and the water form- favors the formation of the type II structure (10). For a very large ing the cage walls. Theoretical analysis shows that both individual guest molecule, the hexagonal structure H with larger cavities is hydrogen molecules and nH2 guest clusters undergo essentially often formed (11). It was a general perception that hydrogen free rotations inside the clathrate cages. Calculations at the ex- molecules are too small to fit into the hydrate cages, and no kPa) and 250 K stable clathrate hydrate structure can be formed (10). Not until 100 ؍ perimental conditions ؊ 2,000 bar (1 bar confirm multiple occupancy of the clathrate cages with average recently, under high-pressure conditions, hydrogen clathrate 12 occupations of 2.00 and 3.96 H2 molecules per D-5 (small) and with the type II structure was synthesized and characterized (9). 12 4 H-5 6 (large) cage, respectively. The H2–H2O interactions also are More surprisingly, the hydrogen clathrate was found to have an O ͞ responsible for the experimentally observed softening of the H H unusually high H2 H2O ratio (1:2) by high-pressure x-ray dif- stretching modes. The clathrate is found to be thermodynamically fraction, Raman, and infrared spectroscopy (9). The chemical stable at 25 bar and 150 K. composition of these clathrates can only be accounted for if multiple (up to quadruple) occupancies of the clathrate cages are lathrate hydrates are a class of inclusion compounds in which assumed. Moreover, the new clathrate remains apparently stable Cguests (noble gases or small organic molecules) occupy, fully even at the normal atmospheric pressure as long as the temper- or partially, cages in the host framework made up of H-bonded ature is below 150 K. The experimental observations clearly water molecules (1). Clathrate hydrate research is of fundamen- indicate that hydrogen incorporation in these structures is tal and practical importance and involves a broad variety of associated with a low free-energy process, much lower than scientific disciplines. The behavior of clathrate hydrates under would be expected for the mechanical encapsulation of the pressure can provide valuable information on water–water in- hydrogen gas. Additionally, the experimentally observed HOH teractions and interactions of water with a wide range of guests. stretching modes in the new clathrate are softened as compared Methane hydrate is the most abundant natural form of clathrate. to the free molecule (9). This observation is in contrast to the CHEMISTRY An estimate of the global reserve of natural gas in the hydrate behavior of hydrogen gas under similar pressures (Ͻ1 GPa) (12). form buried in the permafrost and sediments underneath the Multiple occupancy of the clathrate cavities is a rare phenom- continental shelf is significantly larger than that from traditional enon. The only existing example is nitrogen clathrate synthesized fossil fuels and will be a valuable future energy resource (2). On under pressurized nitrogen atmosphere (13). The purpose of this the negative side, the blockage of natural gas pipeline by solid article is to investigate the stability of hydrogen clathrate, in hydrocarbon hydrates is a potentially hazardous and expensive particular, the effects of occupancy in the empty cavities, by problem that has not been fully resolved (3). Methane hydrates using statistical mechanical theory with first-principles quantum also represent a potential source of climate instability. As chemistry calculations. warming proceeds downward toward the seafloor and reaches the limits of hydrate stability, the hydrate will decompose, and Theoretical Background some of the methane gas will escape to the atmosphere and Hydrogen clathrate forms a type II structure (14) with 136 water increase the greenhouse effect. There are recent reports sug- molecules and 24 cages per unit cell. Sixteen of the cages are gesting that a cause of ancient global warming and mass extinc- pentagonal dodecahedra (D-512). The remaining eight cages are tion of many forms of life 183 million years ago may be traced 16-hedra (H-51264; see refs. 1 and 10 for nomenclature and to sudden eruption of oceanic methane hydrate (4, 5). Clathrate further information). To avoid possible confusion with the hydrates also may be the most abundant form of volatile hydrogen guest, we will refer to the D-512 and H-51264 as S (for materials in the solar system. The possible existence of gas small) and L (for large) cages, respectively. The structures of the hydrate is crucial to the modeling of bodies in the solar system. cages are shown in Fig. 1. Experimentally, it was found that the The identification of several high-pressure forms of methane new clathrate contains 61 Ϯ 7 (9) hydrogen molecules, distrib- hydrates has helped to reconcile the origin of the presence of uted among the cages, per unit cell. Placing two guest molecules large amount of methane in the atmosphere of Saturn’s moon in each S cage and four in each L cage would give an approximate ͞ Titan (6). Solid clathrate hydrate also has been suggested to exist 1:2 H2 H2O clathrate composition. in comets in order to explain the large difference between the The structure and stability of several multiple-occupancy ice latent heats of vaporization of the various ices in the Whipple’s clathrates have been examined with molecular dynamics meth- model (7). Recently, the discovery of new species of centipede- ods (15, 16). Here, we adopt a statistical thermodynamics ͞ like worms living on and within mounds of methane hydrate on approach to investigate the thermodynamics of H2 H2O clath- the floor of Gulf of Mexico challenges the conventional view that rate formation. Encapsulation of successive hydrogen molecules deep sea bottom is a monotonous habitat and indicates that methane hydrate may play a role in marine ecosystem (8). More This paper was submitted directly (Track II) to the PNAS office. recently, the synthesis of H2 hydrate, a hydrate with the smallest guest with multiple occupancy, questioned the conventional Abbreviations: DFT, density functional theory; MP2, second-order Mo¨ller-Plesset. theory for the prediction of the stability of clathrate hydrate (9) *To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org͞cgi͞doi͞10.1073͞pnas.2430913100 PNAS ͉ December 9, 2003 ͉ vol. 100 ͉ no. 25 ͉ 14645–14650 Downloaded by guest on September 25, 2021 Table 1. DFT and MP2 reaction energies* (kcal͞mol) for hydrogen encapsulation reactions at optimized DFT geometries

† naked ‡ Reaction ⌬rE0 (DFT) ⌬rE0 (MP2͞͞DFT) ⌬rE0 (MP2͞͞DFT)

§ 1H2ϩS ϭ 1H2@S Ϫ0.49 Ϫ1.84 ϩ0.04 2H2ϩS ϭ 2H2@S ϩ3.56 Ϫ9.00 ϩ0.99 3H2ϩS ϭ 3H2@S ϩ15.44 Ϫ6.09 ϩ3.63 ¶ 1H2ϩL ϭ 1H2@L Ϫ0.87 Ϫ0.54 ϩ0.01 2H2ϩL ϭ 2H2@L ϩ0.73 Ϫ3.90 ϩ0.36 3H2ϩL ϭ 3H2@L ϩ3.41 Ϫ5.77 ϩ0.93 4H2ϩL ϭ 4H2@L ϩ5.42 Ϫ11.00 ϩ0.92 5H2ϩL ϭ 5H2@L ϩ14.37 Ϫ7.79 ϩ5.67

MP2 formation enthalpies for the naked hydrogen clusters with the water cage walls removed are given for comparison. *Not including zero-point vibration corrections. †S, small (D-512) cage; L, large (H-51264) cage. 12 12 4 ‡ Fig. 1. Structure of the S (D-5 )andL(H-5 6 ) cages of the type II ice Free cluster of nH2 molecules by using geometry optimized inside the cage. § clathrate. Positions of the hydrogen atoms are omitted for clarity. The coor- H2 molecule at cage center. For the true, off-center MP2 minimum, ⌬rE0 dinate axes correspond to the orientation of the model cages used in the (MP2) ϭϪ2.85 kcal͞mol (see text). ¶ calculations. H2 molecule at cage center. For the true, off-center MP2 minimum, ⌬rE0 (MP2) ϭϪ2.54 kcal͞mol (see text).

is treated as a series of chemical reactions between gaseous hydrogen with an (initially empty) ice cage C: The small cage was modeled with a 20-molecule water cluster (Fig. 1). In this structure, 8 oxygen atoms are arranged in a ϩ ϭ 1H2(g) C 1H2@C [1] perfect cube (Oh) 3.83 Å from the cage center. The remaining ϩ ϭ 12 oxygen atoms are placed 3.96 Å from the center of the cage 2H2(g) C 2H2@C [2] and transform according to the Th subgroup of Oh. For the large ϩ ϭ cage, a 28-molecule model cluster was constructed in a distorted 3H2(g) C 3H2@C [3] Td symmetry. Oxygen atoms in this cluster are located between ϩ ϭ 4H2(g) C 4H2@C [4] 4.32 and 4.84 Å from the cage center. For both model cages, hydrogen atoms, with a fixed OOH bond distance of 1.0 Å, were ϩ ϭ 5H2(g) C 5H2@C [5] placed so as to maximize the hydrogen-bonded network. These structural models correspond closely to the average x-ray struc- ...,etc. ture of type II clathrates (14). At moderate pressures, reactions 1–5 are characterized by the ͞ corresponding equilibrium constants: Structure and Stability of the Model H2 H2O Complexes For the small clathrate cage, we considered complexes encap- K ϭ x ͞(x Pn ), p,n n 0 H2 sulated with one, two, and three hydrogen molecules. For the large cage, up to five hydrogen molecules were placed inside the where xn is the fraction of the cages of a given type with nH2 ¥ ϭ cage. As expected for weakly bound clusters, the calculated molecules inside them ( iϭ0 xi 1), and pH is the hydrogen gas 2 energy minima for fully optimized H2 clusters are rather soft pressure. The equilibrium constants Kp,n can be estimated with with many near-zero harmonic vibrational frequencies. For standard statistical thermodynamics (17) by using the energetics ϭ several structures (nH2@L, n 1, 2, 3, and 5), we obtained one obtained from first-principles calculations of the potential en- or more imaginary vibrational modes corresponding to rotations ergy surfaces. of the hydrogen molecules. Attempts to remove the imaginary Computational Details frequencies by following the corresponding normal mode did not lead to any significant decrease in the total energy. In most cases, Density functional theory (DFT) (18–20) calculations were the total energy is insensitive to the orientation of the individual performed with the Amsterdam density functional (ADF) pro- hydrogen molecules. Furthermore, the intramolecular HOH gram package (refs. 21–24 and www.scm.com) by using Cartesian distances of the clusters are very similar to the free molecule with space numerical integration (25) and analytical gradients for the the exception of 2H @S, where the HOH bond length is slightly ␨ 2 geometry optimization (26). An uncontracted double- basis of elongated by 0.004 Å. The calculated reaction enthalpies at 0 K Slater-type orbitals was employed for the 2s,2p shells of oxygen are collected in Table 1. and 1s orbitals of hydrogen atoms. The oxygen 1s shell was For H2@S and H2@L complexes, density functional calcula- treated with the frozen-core approximation (22). Closed-shell tions predict a shallow minimum near the center of the respective spin-restricted calculations were performed employing the cages. This very soft minimum is likely an artifact of the Perdew–Burke–Ernzerhof (27) Generalized Gradient Approxi- approximate exchange-correlation functional, which is unable to mation (GGA) functional with revised exchange parameteriza- properly describe the dispersive interactions (see below). In both tion of Zhang and Yang (revPBE) (28), applied self-consistently complexes, the hydrogen stretching vibration is softened by 87 Ϫ1 Ϫ1 (29). Unless otherwise stated, guest H2 molecules were fully cm (H2@S) and 21 cm (H2@L) with respect to that of the optimized in a fixed model water cage. All closed-shell ab initio free molecule. The translational displacement of the hydrogen second-order Mo¨ller-Plesset (MP2) calculations were per- molecules (radial displacement) from the center of both cages is formed by using GAMESS program (30) at optimized DFT associated with very low harmonic vibrational frequencies (120 Ϫ1 Ϫ1 geometries. These calculations used split valence Gaussian cm in H2@S and 70 cm in H2@L). ϭ 3–21G (31) atom basis sets, augmented by a single set of 2p For 2H2@C (C S, L) clusters, the distance between center ␨ ϭ polarization functions ( 1.1) on hydrogen atoms. Core 1s of mass of the H2 molecules in these clusters is found to be 2.58 electrons on oxygen were not correlated. Å (S) and 2.80 Å (L). DFT calculations predict both complexes

14646 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.2430913100 Patchkovskii and Tse Downloaded by guest on September 25, 2021 to be less stable than the separate hydrogen molecules and the cage by 4.0 and 3.4 kcal͞mol for the small and large cages, respectively. The hydrogen stretching frequency is again softened by 190 to 15 cmϪ1 in the small cage and by Ϸ10 cmϪ1 in the large cage. The very large spread in calculated stretching frequencies in the 2H2@S cluster indicates the sensitivity of frequency shifts to small changes in the cluster geometry. Because we model each nH2@C cluster with a single representative structure, only qualitative conclusions for the frequency shifts can be drawn. A direct comparison with experimental IR and Raman spectra would require a considerably more extensive molecular dynamics simulation. When a three-molecule cluster is placed inside a cage, the positions of the hydrogen molecules relax to form an approxi- O mate equilateral triangle. In the S cage, the H2 H2 separations are between 2.47 and 2.50 Å. In the L cage, the center of mass O H2 H2 distances increase to 2.82–2.86 Å. Hydrogen stretch Ϫ 12 4 frequency shifts range from Ϫ26 to Ϫ4cm 1 in the S cage and Fig. 2. Translational potential energy curve for rigid H2 inside an L (H-5 6 ) Ϫ 12 from Ϫ19 to ϩ14 cm 1 in the L cage. Both 3H @S and 3H @L cage (a)andS(D-5 ) cage (b).The H2 molecules are displaced from the center 2 2 ϭ are predicted to have positive formation enthalpies of ϩ15 and of the cage (R 0) along the (1, 1, 1) direction (Fig. 1). Calculated average occupancies of the S (H-51264) and L (H-51264) cages at 250 K (c) and 150 K (d). ϩ3 kcal͞mol, respectively. Because of its smaller size, the S cage cannot accommodate more than three hydrogen molecules under realistic conditions. occupancy structures is expected to enhance the dispersive The calculated DFT formation enthalpy for a model 4H2@S ϩ ͞ interactions. In view of the very low calculated DFT harmonic structure is in excess of 30 kcal mol. In contrast, in the L cage frequencies (Ͻ130 cmϪ1) for the radial translation of a single H four hydrogen molecules arrange to form a slightly distorted 2 O molecule in both model cages, it is likely that the equilibrium tetrahedron with H2 H2 distances between 2.90 and 3.13Å. position of the hydrogen will move off-center once dispersive Once again, the hydrogen stretch modes are predicted to be interactions are taken into account. Unfortunately, ab initio MP2 softened by Ϫ26 to Ϫ3cmϪ1. The calculated DFT formation ϩ ͞ full geometry optimization of complexes of this size would have energy of 5 kcal mol is somewhat endothermic. been prohibitively expensive. Because the cage models are Finally, we considered a cluster of five hydrogen molecules approximately spherically symmetric, to a first approximation, inside the L cage. For this structure, geometry optimization the interaction potential may be examined through a series of converges to a distorted tetrahedron of H molecules (center of 2 single-point MP2 calculations with the H molecule displaced mass distances between 3.74 and 4.23 Å). The fifth hydrogen 2 from the center of the cage. The results of these calculations are

molecule is situated approximately at the cage center and is CHEMISTRY compared with the DFT interaction energies evaluated at the 2.42–2.53 Å from the remaining H molecules at the corners of 2 same geometries in Fig. 2. the tetrahedron. The hydrogen stretch mode in this structure is For the S cage, the MP2 energy minimum is located at an broadened compared to the free H2 molecule with shifts be- ϭ tween Ϫ33 and ϩ37 cmϪ1. The DFT formation energy of this off-center displacement of Rmin 1.1 Å. The corresponding H2 ϩ ͞ incorporation energy of Ϫ2.9 kcal͞mol is 1.1 kcal͞mol lower structure is 14 kcal mol. Ϫ ͞ Except for the low-occupancy complexes (H @S, H @L, and than the 1.8 kcal mol calculated at the cage center. For the L 2 2 cage, the MP2 energy minimum is 2.0 Å off-center with a binding 2H2@L) the calculated reaction enthalpies (Table 1) for hydro- Ϫ ͞ gen molecule(s) encapsulation are too endothermic to account energy of 2.5 kcal mol. From the MP2 potential energy curves, ϭ ϭ one can deduce the harmonic vibrational frequency for hydro- for the clathrate formation at temperature T 250K(RT 0.5 Ϫ1 Ϫ1 kcal͞mol, where R is the universal gas constant) and 2,000 bar gen radial translation of 250 cm in the small cage and 160 cm (1 bar ϭ 100 kPa). The inconsistency is not entirely unexpected in the large cage. Notably, the shape of the repulsive part of the MP2 potential because all the popular approximate density functionals are Ͼ incapable of describing dispersive interactions (32, 33). Such energy curves (R Rmin) closely resembles their DFT coun- interactions are expected to be the dominant attractive contri- terpart (Fig. 2). From geometrical considerations, the average bution for the neutral, nonpolar hydrogen molecule. To correct distances between H2 molecules (H2–H2) and between H2 and for this deficiency, we performed single-point ab initio MP2 water (H2–H2O) forming the cage walls are closer in multiply calculations of the hydrogen clusters at optimized DFT geom- occupied cages. Therefore, apart from a constant energy shift, etries. The inclusion of dispersive interactions increases stability DFT potential energy surfaces can be expected to provide of the clathrate structures, particularly at higher hydrogen a reasonable description of clusters at higher hydrogen molecule occupancies (Table 1). For multiply occupied S and L occupancies. cages, the difference between DFT and MP2 energies ap- It is important to examine the origin of the stabilization of the proaches 7 and 4 kcal͞mol per hydrogen molecule, respectively. hydrogen clusters inside the ice cages. The relative importance In comparison, the difference between DFT and MP2 reaction of the H2–H2 and H2–H2O interactions can be estimated from energies is much smaller for low-occupancy complexes (1H2@S, the MP2 formation energies of the ‘‘naked’’ nH2 cluster evalu- 1H2@L, and 2H2@L). ated at the optimized geometry inside the model cages (Table 1). This apparently counterintuitive behavior can be explained in It is found that a ‘‘magic’’ size exists (n ϭ 2 for the S cage and Ϫ6 ϭ terms of the r dependence of the dispersive interaction energy. n 4 for the L cage) up to which the repulsive H2–H2 By using the standard Lennard–Jones 6–12 parameters for van interactions seem to be relatively unimportant and contribute Ͻ ͞ der Waals interactions (see, e.g., ref. 34), the H2–H2O interac- 0.5 kcal mol per H2 molecule to the reaction energy. At higher Ͼ tion potential is expected to be attractive for R0 2.8 Å with an cage occupancies, H2–H2 repulsion increases sharply. In con- energy minimum at Ϸ3.2 Å. Because the water cages’ radii are trast, the stability gained from attractive H2–water cage wall Ϸ Ϸ significantly larger than R0 (S 3.8 Å and L 4.0 Å), interactions reaches saturation at the magic occupancy. This displacement of H2 molecules from the cage center in multiple- trend is the consequence of the total area of H2–wall contacts,

Patchkovskii and Tse PNAS ͉ December 9, 2003 ͉ vol. 100 ͉ no. 25 ͉ 14647 Downloaded by guest on September 25, 2021 which reaches a maximum at the magic cluster size and remains almost constant for larger clusters (see Table 1). Therefore, the net result is that the hydrogen encapsulation energy shows a pronounced minimum at the magic occupancies. If a similar analysis is applied to H2 harmonic stretch frequencies, it is clear that the repulsive intracluster interactions lead to an increase in the stretch frequencies, of 9 to 60 cmϪ1, depending on the geometry. The only exception is in the ͞ 2H2 2H2@S cluster, where one of the vibrational modes is Ϸ Ϫ1 Ϫ1 softened by 90 cm as compared with 190 cm in 2H2@S. In this case, the contribution to the frequency shift arises mainly from the HOH bond elongation. Both energy and harmonic frequency decomposition suggest that at the experimental conditions of 2,000 bar and 250 K, H2O–H2 interactions dominate the structure and properties of the clathrate, whereas the H2–H2 interactions are of secondary importance. Consequently, caution is advised in applying the results of high-pressure studies on ͞ hydrogen gas to the H2 H2O clathrate. Estimation of Thermodynamic Parameters To assess the stability of the clathrate at experimental conditions, finite temperature and zero-point motion effects must be taken into account. By using the MP2 reaction energies at 0 K (Table 1) and DFT harmonic vibrational frequencies, standard statistical thermodynamics techniques (17) can be used to estimate the correction of zero-point vibrational energy and thermal contributions to reaction enthalpies and reaction entropies. The results for the energetics (HT-E0, ST) at 150 K, are shown in Fig. 3. Similar behavior is found at 250 K, with thermal contributions to the entropy and enthalpy increasing by 3–5 cal͞mol⅐K and 0.5–1.0 kcal͞mol, respectively. The calculated compositions of the clathrate are collected in Table 2. At the experimental conditions where the clathrate were synthesized (T ϭ 250 K, P ϭ 2,000 bar), the harmonic approximation leads to average cage occupancies of 1.8 (S cage) and 1.6 Fig. 3. Calculated dynamical contributions to enthalpy (Upper) and en- thropy (Lower) per mole of clathrated H at 150 K. The extreme left point (L cage). These guest occupancies are insufficient to prevent the 2 ϭ ϭ corresponds to H2 in ideal gas. Open and ﬁlled symbols denote the results clathrate structure from collapsing (10). At T 150 K and P obtained in harmonic and ‘‘improved’’ approximations, respectively. Quali- 25 bar, the average occupancy of the L cage drops to almost zero tatively similar dependence (not shown) is found at 250 K. (Table 2). Clearly, the harmonic vibrational approximation does not lead to an adequate description of the thermodynamics of the ͞ H2 H2O clathrate cages. The reason for this failure becomes to overestimation of the equilibrium constants (S.P. and S. clear on examination of the individual contributions to the Yurchenko, unpublished data). Any attempt to predict the calculated free energies of the hydrogen incorporation reactions. thermodynamics of the model clathrate cages with classical ϩ ϭ For example, the calculated reaction energy for 4L (4H2 L molecular dynamics techniques will suffer from the same prob- ⌬ ϭϪ ͞ 4H2@L) is exothermic at 250 K ( rE250 4.9 kcal mol). This lem (37). ⌬ energy gain is associated with a large decrease in entropy (T S250 To attain a computationally affordable semiquantitative de- ϭ ͞ 20.7 kcal mol). The entropic contribution is, in turn, domi- scription of the thermal contributions to the free energy of the ͞ nated by the loss of translational entropy of free H2 (27.2 cal mol H2 incorporation reaction, we propose the following scheme. We per K) with no contributions of similar magnitude appearing for will refer to this procedure as the ‘‘improved’’ approximation. the nH2 cluster inside the cage. Therefore, it appears likely that For each nH2 cluster inside a rigid model cage the 6n degrees the harmonic vibrational approximation failed to describe high- of freedom are split into three categories. amplitude motions of the nH2 clusters inside the model cages and led to underestimation of the entropies. A more realistic treat- 1. Two degrees of freedom for each individual hydrogen mol- ment of the vibrational entropy clearly is required. ecule are treated as a free rotor. At 250 K, this term For a spherically symmetric rigid host cage and freely rotating contributes 0.5n kcal͞mol to ⌬H, and 2.7n cal͞mol⅐Kto⌬S. (or monoatomic) guest, the classical partition function can be This assumption is justified by the observation that spacing reduced to a one-dimensional integral of the interaction poten- between rotational energy levels in H2 (experimental value, ϭ Ϫ1 ϭ Ϫ1 Ϸ ͞ tial. This leads to a simple expression for the first equilibrium Be 60.9 cm ; DFT, Be 60.1 cm )is 0.2 kcal mol, constant (35, 36): which is comparable to the orientational dependence of the H2–cage interaction energy. Experimental Raman spectra 4␲ v(r) ϭ ͵ 2 ͩϪ ͪ also suggest essentially free rotation of the clathrated H2 Kp,l r exp dr, kT kT molecules (9). Within this approximation, the H2 rotational contributions to reaction free energies cancel identically. where v(r) is the guest-cage interaction energy at displacement 2. Three degrees of freedom per nH2 cluster are treated as a r from the center of the cage. In the temperature range where rigid body rotation with moments of inertia calculated at the ͞ Ͻ H2 H2O clathrate is stable (T 250K), the assumption of equilibrium structure of the cluster relative to the center of classical motion breaks down for the light H2 molecule leading the cage. For the 4H2@L cluster, this term amounts to 0.8

14648 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.2430913100 Patchkovskii and Tse Downloaded by guest on September 25, 2021 Table 2. Calculated composition of H2͞H2O clathrate by using the harmonic and improved approximations S cage (D-512) L cage (H-51264)

† ‡ ‡ † ‡ ‡ ‡ ‡ Method P, bar T, K Ave* S 1H2@S 2H2@S P, bar T, K Ave* L 1H2@L 2H2@L 3H2@L 4H2@L

Harmonic 2,000 250 1.81 8.5 ϫ 10Ϫ2 2.2 ϫ 10Ϫ2 8.9 ϫ 10Ϫ1 2,000 250 1.61 1.6 ϫ 10Ϫ1 1.6 ϫ 10Ϫ1 6.4 ϫ 10Ϫ1 1.1 ϫ 10Ϫ2 3.8 ϫ 10Ϫ2 100 250 0.06 9.6 ϫ 10Ϫ1 1.2 ϫ 10Ϫ2 2.5 ϫ 10Ϫ2 100 250 0.07 9.4 ϫ 10Ϫ1 4.6 ϫ 10Ϫ2 9.5 ϫ 10Ϫ3 8.4 ϫ 10Ϫ6 1.4 ϫ 10Ϫ6 500 150 2.00 3.3 ϫ 10Ϫ4 7.7 ϫ 10Ϫ5 1.0 500 150 3.25 4.5 ϫ 10Ϫ2 1.0 ϫ 10Ϫ2 2.7 ϫ 10Ϫ1 2.9 ϫ 10Ϫ3 6.7 ϫ 10Ϫ1 25 150 1.77 1.1 ϫ 10Ϫ1 1.4 ϫ 10Ϫ3 8.8 ϫ 10Ϫ1 25 150 0.04 9.7 ϫ 10Ϫ1 1.1 ϫ 10Ϫ2 1.4 ϫ 10Ϫ2 7.8 ϫ 10Ϫ6 9.1 ϫ 10Ϫ5 Improved 2,000 250 2.00 7.0 ϫ 10Ϫ6 2.7 ϫ 10Ϫ4 1.0 2,000 250 3.96 4.8 ϫ 10Ϫ4 3.7 ϫ 10Ϫ3 1.3 ϫ 10Ϫ2 4.1 ϫ 10Ϫ3 9.8 ϫ 10Ϫ1 100 250 1.99 2.8 ϫ 10Ϫ3 5.4 ϫ 10Ϫ3 9.9 ϫ 10Ϫ1 100 250 0.39 6.8 ϫ 10Ϫ1 2.6 ϫ 10Ϫ1 4.7 ϫ 10Ϫ2 7.3 ϫ 10Ϫ3 8.8 ϫ 10Ϫ3 500 150 2.00 4.6 ϫ 10Ϫ9 1.1 ϫ 10Ϫ6 1.0 500 150 4.00 2.2 ϫ 10Ϫ7 1.6 ϫ 10Ϫ6 6.4 ϫ 10Ϫ5 2.6 ϫ 10Ϫ5 1.0 25 150 2.00 1.8 ϫ 10Ϫ6 2.1 ϫ 10Ϫ5 1.0 25 150 3.78 3.3 ϫ 10Ϫ2 1.2 ϫ 10Ϫ2 2.4 ϫ 10Ϫ2 4.8 ϫ 10Ϫ4 9.3 ϫ 10Ϫ1

The fraction of cages with higher occupancies (3H2@S and 5H2@L) is always below 2 parts per million and is not shown. *Average number of H2 molecules inside a cage. †Fraction of empty cages. ‡ Fraction of cages with nH2 molecules.

͞ ⌬ ͞ ⅐ Ͼ kcal mol for H250 and 19.8 cal mol K for S250. This assump- bar, whereas the L cage is already fully occupied at P 25 bar. tion is justified by the approximately spherical symmetry Again, the calculated results agree well with experiment and ͞ of the cages and weakness of the individual H2–H2O suggest that type II H2 H2O clathrate is thermodynamically interactions. stable with respect to H2 loss under mild temperature and 3. The remaining 4n-3 degrees of freedom are treated as pressure conditions. In the experiment, by warming the clathrate harmonic oscillators with frequencies taken from DFT har- from 78 K to 115 K (9), it was observed that the intensity of the monic vibrational analysis. The n high-frequency modes infrared absorption band assigned to vibrations of hydrogen in correspond to the HOH stretching mode. The remaining the small cages decreases rapidly. This observation may infer that modes correspond to the breathing modes of the cluster. For the hydrogen content in the small cage decreases faster than in ͞ 4H2@L, this contribution adds 11.9 cal mol per K to S250 and the large cages with increasing temperature, which is in apparent ͞ ⌬ 30.4 kcal mol to H250. contradiction with the theoretical prediction. However, it should be recognized that the vibrational frequency of hydrogen mol- An additional justification is required for treating the rattling ecule inside the cavities is very sensitive to the local environment. motion of the hydrogen molecule in a singly occupied cage as It may not be appropriate to associate that higher frequency rigid body rotation around the cage center. For the temperature vibrations with the motions of hydrogen in the more spatially range of interest (150–250 K), the RT value amounts to 0.3–0.5 confined smaller S cavity. In fact, the converse is observed for ͞ CHEMISTRY kcal mol. This value should be compared with the barrier height the COH stretching modes of methane in methane clathrate at the cage center: Ϸ1.0 kcal͞mol for the small cages and 2.0 O ͞ (38). Furthermore, because the H H stretching frequency is kcal mol for the large cages (see Fig. 1). At these temperatures very sensitive to the location of the molecule inside the cavity, the hydrogen molecule may be expected to be localized within we may expect a very pronounced thermal broadening effect, Ϸ the thin, spherical shell with R Rmin. leading to an apparent intensity decrease. Thermal contributions to formation energy and entropy cal- The analysis described above suggested that the hydrogen culated following this procedure (approximations 1–3) are molecule clusters are rigid rotors in a weakly perturbed envi- shown on Fig. 3. For singly occupied cages, the improved values ronment. We can then estimate the quantum rotation (tunnel- compare favorably with the results from the direct solution of ing) splittings in the limit of a free rigid rotor from the respective Schroedinger equations for the nuclear motion of the guest (S.P. rotational constants. For hydrogen dimer in the S cage 12 Ϸ Ϫ1 and S. Yurchenko, unpublished data). In general, the approach (2H2@5 ), the maximum energy splitting is 2cm , and for 12 4 outlined above leads to significantly lower calculated HT-E0 the hydrogen tetramer in the large cage (4H2@5 6 ), the contributions and higher entropies than the harmonic approxi- splitting is Ϸ0.9 cmϪ1. These splittings may be observable in mation. The bulk of the change in the enthalpic contribution is microwave spectroscopy experiments. attributable to the absence of the zero-point energy from the free rotation contributions. The increase in the calculated Conclusions and Outlook ͞ entropy compared to the harmonic oscillator approximation We have examined the stability of the type II H2 H2O clathrate derives largely from the rotations of the cluster. Treating these with respect to the hydrogen occupancy by using DFT and ab degrees of freedom as free rotations is essential for achieving initio MP2 calculations on model water cages. The experimen- agreement between theory and experiment. tally observed composition and formation conditions can only be Clathrate compositions at several (p, T) values calculated with explained if two assumptions are made: (i) individual hydrogen the improved approximation are collected in Table 2. The guest molecules undergo free rotation and (ii) the collective average occupancies of the S and L cages at two representative motions of the cluster of hydrogen molecules inside the individ- temperatures are depicted in Fig. 2. At 250 K, the S cage is ual host cages must be treated as rigid body rotations. Under already fully occupied at relatively low pressures (Ϸ100 bar). these assumptions, the clathrate reaches the observed Ϸ1:2 ͞ ϭ Ͼ However, the bigger L cage is still empty at this pressure. At H2 H2O composition at T 250 K, P 1,000 bar. Ͼ pressures 1,000 bar, both cages become fully occupied. This The incorporation of H2 into the cages is calculated to be ⌬ ϭϪ ͞ result agrees very well with the experimental formation condi- exothermic ( rH250 1.8 kcal mol), and the stability of the tions (T ϭ 250 K, P ϭ 2,000 bar). In view of the relatively crude clathrate increases at lower temperatures. Below 150 K, the approximations made in the calculations, this agreement should clathrate should be thermodynamically stable at near-ambient be considered excellent. pressure, making it a promising and easy way to handle stored At 150 K, calculations suggest that the S cage should be able hydrogen. The stabilization of the hydrogen guest clusters arises to retain both hydrogen molecules at H2 pressures well below 1 mainly from attractive dispersive interactions with the water on

Patchkovskii and Tse PNAS ͉ December 9, 2003 ͉ vol. 100 ͉ no. 25 ͉ 14649 Downloaded by guest on September 25, 2021 the cage walls. As the surface contact is maximized for similar occupancy. Although the occupancy may well be the dominating sizes of the guest and the host cage, sites within the smaller D-512 stability factor, other thermodynamic factors such as the stability ⌬ ϭϪ ͞ cage are more stable ( rH250 2.6 kcal mol) than in the of the ice framework at different cage occupations and kinetic H-51264 cage (⌬ H ϭϪ1.2 kcal͞mol). Thus, double clathrates r 250 (e.g., H2 diffusion rates) factors need to be examined. Moreover, with a suitable choice of guest in the larger cage (e.g., tetrahy- we only considered a single representative structure for each drofuran) should form under even milder conditions. If hydro- occupancy of the host cage. The structures of the guest clusters gen loss from the D-512 cage remains the stability-determining factor, such clathrates should be stable at hydrogen pressures of appear to be highly fluxional, leading to broad complex features Ϸ20 bar (T ϭ 250 K). in the experimental IR and Raman spectra. Understanding these Interactions with the cage walls are responsible for the exper- features would require a detailed quantum molecular dynamics imentally observed softening of the HOH stretching modes in simulation of the guest clusters (37). In this study, quantum the clathrate. If these interactions are ignored, the stretching effects have been included in the consideration of the rotational frequencies actually increase, as expected for moderately com- motions only. The quantum contributions to the translational pressed hydrogen gas (9). motions were neglected. It is expected that the equilibrium It should be noted that a more definitive determination of the constants will be somewhat reduced if this effect was included. occupancy number can be obtained from structural refinement Finally, it is clear that the discovery of the type II hydrogen of experimental neutron diffraction patterns. Furthermore, mi- clathrate is opening new exciting horizons in the field of clathrate crowave spectroscopy will be invaluable to characterize the tunneling splittings if the hydrogen molecule clusters were research. indeed behave as rigid rotors. In summary, we have performed a theoretical investigation of We thank Wendy Mao, David Mao, and Russell Hemley for communi- the stability of hydrogen clathrate with respect to the guest cating their results prior to publication and for many helpful discussions.

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