Thermodynamic stability of hydrogen clathrates

Abstract

The stability of the recently characterized type II hydrogen clathrate [Mao, W. L., Mao, H.-K., Goncharov, A. F., Struzhkin, V. V., Guo, Q., et al. (2002) Science 297, 2247–2249] with respect to hydrogen occupancy is examined with a statistical mechanical model in conjunction with first-principles quantum chemistry calculations. It is found that the stability of the clathrate is mainly caused by dispersive interactions between H₂ molecules and the water forming the cage walls. Theoretical analysis shows that both individual hydrogen molecules and nH₂ guest clusters undergo essentially free rotations inside the clathrate cages. Calculations at the experimental conditions – 2,000 bar (1 bar = 100 kPa) and 250 K confirm multiple occupancy of the clathrate cages with average occupations of 2.00 and 3.96 H₂ molecules per D-5¹² (small) and H-5¹²6⁴ (large) cage, respectively. The H₂–H₂O interactions also are responsible for the experimentally observed softening of the H—H stretching modes. The clathrate is found to be thermodynamically stable at 25 bar and 150 K.

Clathrate hydrates are a class of inclusion compounds in which guests (noble gases or small organic molecules) occupy, fully or partially, cages in the host framework made up of H-bonded water molecules (1). Clathrate hydrate research is of fundamental and practical importance and involves a broad variety of scientific disciplines. The behavior of clathrate hydrates under pressure can provide valuable information on water–water interactions and interactions of water with a wide range of guests. Methane hydrate is the most abundant natural form of clathrate. An estimate of the global reserve of natural gas in the hydrate form buried in the permafrost and sediments underneath the continental shelf is significantly larger than that from traditional fossil fuels and will be a valuable future energy resource (2). On the negative side, the blockage of natural gas pipeline by solid hydrocarbon hydrates is a potentially hazardous and expensive problem that has not been fully resolved (3). Methane hydrates also represent a potential source of climate instability. As warming proceeds downward toward the seafloor and reaches the limits of hydrate stability, the hydrate will decompose, and some of the methane gas will escape to the atmosphere and increase the greenhouse effect. There are recent reports suggesting that a cause of ancient global warming and mass extinction of many forms of life 183 million years ago may be traced to sudden eruption of oceanic methane hydrate (4, 5). Clathrate hydrates also may be the most abundant form of volatile materials in the solar system. The possible existence of gas hydrate is crucial to the modeling of bodies in the solar system. The identification of several high-pressure forms of methane hydrates has helped to reconcile the origin of the presence of large amount of methane in the atmosphere of Saturn's moon Titan (6). Solid clathrate hydrate also has been suggested to exist in comets in order to explain the large difference between the latent heats of vaporization of the various ices in the Whipple's model (7). Recently, the discovery of new species of centipede-like worms living on and within mounds of methane hydrate on the floor of Gulf of Mexico challenges the conventional view that deep sea bottom is a monotonous habitat and indicates that methane hydrate may play a role in marine ecosystem (8). More recently, the synthesis of H₂ hydrate, a hydrate with the smallest guest with multiple occupancy, questioned the conventional theory for the prediction of the stability of clathrate hydrate (9) and raised the prospect of using clathrate hydrate as an efficient H₂ storage medium.

Under normal conditions, clathrate hydrates are known to have three distinct crystalline structures (10). Both structure I and II clathrate hydrates have a cubic structure and a guest:host water ratio of ≈1:6 (10). Usually, very small guest (rare gas) favors the formation of the type II structure (10). For a very large guest molecule, the hexagonal structure H with larger cavities is often formed (11). It was a general perception that hydrogen molecules are too small to fit into the hydrate cages, and no stable clathrate hydrate structure can be formed (10). Not until recently, under high-pressure conditions, hydrogen clathrate with the type II structure was synthesized and characterized (9). More surprisingly, the hydrogen clathrate was found to have an unusually high H₂/H₂O ratio (1:2) by high-pressure x-ray diffraction, Raman, and infrared spectroscopy (9). The chemical composition of these clathrates can only be accounted for if multiple (up to quadruple) occupancies of the clathrate cages are assumed. Moreover, the new clathrate remains apparently stable even at the normal atmospheric pressure as long as the temperature is below 150 K. The experimental observations clearly indicate that hydrogen incorporation in these structures is associated with a low free-energy process, much lower than would be expected for the mechanical encapsulation of the hydrogen gas. Additionally, the experimentally observed H—H stretching modes in the new clathrate are softened as compared to the free molecule (9). This observation is in contrast to the behavior of hydrogen gas under similar pressures (<1 GPa) (12). Multiple occupancy of the clathrate cavities is a rare phenomenon. The only existing example is nitrogen clathrate synthesized under pressurized nitrogen atmosphere (13). The purpose of this article is to investigate the stability of hydrogen clathrate, in particular, the effects of occupancy in the empty cavities, by using statistical mechanical theory with first-principles quantum chemistry calculations.

Theoretical Background

Hydrogen clathrate forms a type II structure (14) with 136 water molecules and 24 cages per unit cell. Sixteen of the cages are pentagonal dodecahedra (D-5¹²). The remaining eight cages are 16-hedra (H-5¹²6⁴; see refs. 1 and 10 for nomenclature and further information). To avoid possible confusion with the hydrogen guest, we will refer to the D-5¹² and H-5¹²6⁴ as S (for small) and L (for large) cages, respectively. The structures of the cages are shown in Fig. 1. Experimentally, it was found that the new clathrate contains 61 ± 7 (9) hydrogen molecules, distributed among the cages, per unit cell. Placing two guest molecules in each S cage and four in each L cage would give an approximate 1:2 H₂/H₂O clathrate composition.

Fig. 1.

Structure of the S (D-5¹²) and L (H-5¹²6⁴) cages of the type II ice clathrate. Positions of the hydrogen atoms are omitted for clarity. The coordinate axes correspond to the orientation of the model cages used in the calculations.

The structure and stability of several multiple-occupancy ice clathrates have been examined with molecular dynamics methods (15, 16). Here, we adopt a statistical thermodynamics approach to investigate the thermodynamics of H₂/H₂O clathrate formation. Encapsulation of successive hydrogen molecules is treated as a series of chemical reactions between gaseous hydrogen with an (initially empty) ice cage C:

[1]

[2]

[3]

[4]

[5]

..., etc.

At moderate pressures, reactions 1–5 are characterized by the corresponding equilibrium constants:

where x_n is the fraction of the cages of a given type with nH₂ molecules inside them (∑_i=0 x_i = 1), and p_H2 is the hydrogen gas pressure. The equilibrium constants K_p,n can be estimated with standard statistical thermodynamics (17) by using the energetics obtained from first-principles calculations of the potential energy surfaces.

Computational Details

Density functional theory (DFT) (18–20) calculations were performed with the Amsterdam density functional (adf) program package (refs. 21–24 and www.scm.com) by using Cartesian space numerical integration (25) and analytical gradients for the geometry optimization (26). An uncontracted double-ζ basis of Slater-type orbitals was employed for the 2s, 2p shells of oxygen and 1s orbitals of hydrogen atoms. The oxygen 1s shell was treated with the frozen-core approximation (22). Closed-shell spin-restricted calculations were performed employing the Perdew–Burke–Ernzerhof (27) Generalized Gradient Approximation (GGA) functional with revised exchange parameterization of Zhang and Yang (revPBE) (28), applied self-consistently (29). Unless otherwise stated, guest H₂ molecules were fully optimized in a fixed model water cage. All closed-shell ab initio second-order Möller-Plesset (MP2) calculations were performed by using gamess program (30) at optimized DFT geometries. These calculations used split valence Gaussian 3–21G (31) atom basis sets, augmented by a single set of 2p polarization functions (ζ = 1.1) on hydrogen atoms. Core 1s electrons on oxygen were not correlated.

The small cage was modeled with a 20-molecule water cluster (Fig. 1). In this structure, 8 oxygen atoms are arranged in a perfect cube (O_h) 3.83 Å from the cage center. The remaining 12 oxygen atoms are placed 3.96 Å from the center of the cage and transform according to the T_h subgroup of O_h. For the large cage, a 28-molecule model cluster was constructed in a distorted T_d symmetry. Oxygen atoms in this cluster are located between 4.32 and 4.84 Å from the cage center. For both model cages, hydrogen atoms, with a fixed O—H bond distance of 1.0 Å, were placed so as to maximize the hydrogen-bonded network. These structural models correspond closely to the average x-ray structure of type II clathrates (14).

Structure and Stability of the Model H₂/H₂O Complexes

For the small clathrate cage, we considered complexes encapsulated with one, two, and three hydrogen molecules. For the large cage, up to five hydrogen molecules were placed inside the cage. As expected for weakly bound clusters, the calculated energy minima for fully optimized H₂ clusters are rather soft with many near-zero harmonic vibrational frequencies. For several structures (nH₂@L, n = 1, 2, 3, and 5), we obtained one or more imaginary vibrational modes corresponding to rotations of the hydrogen molecules. Attempts to remove the imaginary frequencies by following the corresponding normal mode did not lead to any significant decrease in the total energy. In most cases, the total energy is insensitive to the orientation of the individual hydrogen molecules. Furthermore, the intramolecular H—H distances of the clusters are very similar to the free molecule with the exception of 2H₂@S, where the H—H bond length is slightly elongated by 0.004 Å. The calculated reaction enthalpies at 0 K are collected in Table 1.

Table 1.

DFT and MP2 reaction energies^* (kcal/mol) for hydrogen encapsulation reactions at optimized DFT geometries

Reaction†	ΔrE0 (DFT)	ΔrE0 (MP2//DFT)	ΔrE0naked (MP2//DFT)‡
1H2+S = 1H2@S	-0.49	-1.84§	+0.04
2H2+S = 2H2@S	+3.56	-9.00	+0.99
3H2+S = 3H2@S	+15.44	-6.09	+3.63
1H2+L = 1H2@L	-0.87	-0.54¶	+0.01
2H2+L = 2H2@L	+0.73	-3.90	+0.36
3H2+L = 3H2@L	+3.41	-5.77	+0.93
4H2+L = 4H2@L	+5.42	-11.00	+0.92
5H2+L = 5H2@L	+14.37	-7.79	+5.67

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.

^‡Free cluster of nH₂ molecules by using geometry optimized inside the cage.

^§H₂ molecule at cage center. For the true, off-center MP2 minimum, Δ_rE₀ (MP2) = -2.85 kcal/mol (see text).

^¶H₂ molecule at cage center. For the true, off-center MP2 minimum, Δ_rE₀ (MP2) = -2.54 kcal/mol (see text).

For H₂@S and H₂@L complexes, density functional calculations predict a shallow minimum near the center of the respective cages. This very soft minimum is likely an artifact of the approximate exchange-correlation functional, which is unable to properly describe the dispersive interactions (see below). In both complexes, the hydrogen stretching vibration is softened by 87 cm^–1 (H₂@S) and 21 cm^–1 (H₂@L) with respect to that of the free molecule. The translational displacement of the hydrogen molecules (radial displacement) from the center of both cages is associated with very low harmonic vibrational frequencies (120 cm^–1 in H₂@S and 70 cm^–1 in H₂@L).

For 2H₂@C (C = S, L) clusters, the distance between center of mass of the H₂ molecules in these clusters is found to be 2.58 Å (S) and 2.80 Å (L). DFT calculations predict both complexes to be less stable than the separate hydrogen molecules and the cage by 4.0 and 3.4 kcal/mol for the small and large cages, respectively. The hydrogen stretching frequency is again softened by 190 to 15 cm^–1 in the small cage and by ≈10 cm^–1 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 approximate 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 H₂—H₂ distances increase to 2.82–2.86 Å. Hydrogen stretch frequency shifts range from –26 to –4 cm^–1 in the S cage and from –19 to +14 cm^–1 in the L cage. Both 3H₂@S and 3H₂@L are predicted to have positive formation enthalpies of +15 and +3 kcal/mol, respectively.

Because of its smaller size, the S cage cannot accommodate more than three hydrogen molecules under realistic conditions. The calculated DFT formation enthalpy for a model 4H₂@S structure is in excess of +30 kcal/mol. In contrast, in the L cage four hydrogen molecules arrange to form a slightly distorted tetrahedron with H₂—H₂ distances between 2.90 and 3.13Å. Once again, the hydrogen stretch modes are predicted to be softened by –26 to –3 cm^–1. The calculated DFT formation energy of +5 kcal/mol is somewhat endothermic.

Finally, we considered a cluster of five hydrogen molecules inside the L cage. For this structure, geometry optimization converges to a distorted tetrahedron of H₂ molecules (center of mass distances between 3.74 and 4.23 Å). The fifth hydrogen molecule is situated approximately at the cage center and is 2.42–2.53 Å from the remaining H₂ molecules at the corners of the tetrahedron. The hydrogen stretch mode in this structure is broadened compared to the free H₂ molecule with shifts between –33 and +37 cm^–1. The DFT formation energy of this structure is +14 kcal/mol.

Except for the low-occupancy complexes (H₂@S, H₂@L, and 2H₂@L) the calculated reaction enthalpies (Table 1) for hydrogen molecule(s) encapsulation are too endothermic to account for the clathrate formation at temperature T = 250 K (RT = 0.5 kcal/mol, where R is the universal gas constant) and 2,000 bar (1 bar = 100 kPa). The inconsistency is not entirely unexpected because all the popular approximate density functionals are incapable of describing dispersive interactions (32, 33). Such interactions are expected to be the dominant attractive contribution for the neutral, nonpolar hydrogen molecule. To correct for this deficiency, we performed single-point ab initio MP2 calculations of the hydrogen clusters at optimized DFT geometries. The inclusion of dispersive interactions increases stability of the clathrate structures, particularly at higher hydrogen molecule occupancies (Table 1). For multiply occupied S and L cages, the difference between DFT and MP2 energies approaches 7 and 4 kcal/mol per hydrogen molecule, respectively. In comparison, the difference between DFT and MP2 reaction energies is much smaller for low-occupancy complexes (1H₂@S, 1H₂@L, and 2H₂@L).

This apparently counterintuitive behavior can be explained in terms of the r^–6 dependence of the dispersive interaction energy. By using the standard Lennard–Jones 6–12 parameters for van der Waals interactions (see, e.g., ref. 34), the H₂–H₂O interaction potential is expected to be attractive for R₀ >2.8 Å with an energy minimum at ≈3.2 Å. Because the water cages' radii are significantly larger than R₀ (S ≈ 3.8 Å and L ≈ 4.0 Å), displacement of H₂ molecules from the cage center in multiple-occupancy structures is expected to enhance the dispersive interactions. In view of the very low calculated DFT harmonic frequencies (<130 cm^–1) for the radial translation of a single H₂ molecule in both model cages, it is likely that the equilibrium position of the hydrogen will move off-center once dispersive interactions are taken into account. Unfortunately, ab initio MP2 full geometry optimization of complexes of this size would have been prohibitively expensive. Because the cage models are approximately spherically symmetric, to a first approximation, the interaction potential may be examined through a series of single-point MP2 calculations with the H₂ molecule displaced from the center of the cage. The results of these calculations are compared with the DFT interaction energies evaluated at the same geometries in Fig. 2.

Fig. 2.

Translational potential energy curve for rigid H₂ inside an L (H-5¹²6⁴) cage (a)andS(D-5¹²) cage (b).The H₂ molecules are displaced from the center 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).

For the S cage, the MP2 energy minimum is located at an off-center displacement of R_min = 1.1 Å. The corresponding H₂ incorporation energy of –2.9 kcal/mol is 1.1 kcal/mol lower than the –1.8 kcal/mol calculated at the cage center. For the L cage, the MP2 energy minimum is 2.0 Å off-center with a binding energy of –2.5 kcal/mol. From the MP2 potential energy curves, one can deduce the harmonic vibrational frequency for hydrogen radial translation of 250 cm^–1 in the small cage and 160 cm^–1 in the large cage.

Notably, the shape of the repulsive part of the MP2 potential energy curves (R > R_min) closely resembles their DFT counterpart (Fig. 2). From geometrical considerations, the average distances between H₂ molecules (H₂–H₂) and between H₂ and water (H₂–H₂O) forming the cage walls are closer in multiply occupied cages. Therefore, apart from a constant energy shift, DFT potential energy surfaces can be expected to provide a reasonable description of clusters at higher hydrogen occupancies.

It is important to examine the origin of the stabilization of the hydrogen clusters inside the ice cages. The relative importance of the H₂–H₂ and H₂–H₂O interactions can be estimated from the MP2 formation energies of the “naked” nH₂ cluster evaluated at the optimized geometry inside the model cages (Table 1). It is found that a “magic” size exists (n = 2 for the S cage and n = 4 for the L cage) up to which the repulsive H₂–H₂ interactions seem to be relatively unimportant and contribute <0.5 kcal/mol per H₂ molecule to the reaction energy. At higher cage occupancies, H₂–H₂ repulsion increases sharply. In contrast, the stability gained from attractive H₂–water cage wall interactions reaches saturation at the magic occupancy. This trend is the consequence of the total area of H₂–wall contacts, 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 cm^–1, depending on the geometry. The only exception is in the 2H₂/2H₂@S cluster, where one of the vibrational modes is softened by ≈90 cm^–1 as compared with 190 cm^–1 in 2H₂@S. In this case, the contribution to the frequency shift arises mainly from the H—H 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 (H_T-E₀, S_T) 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 cal/mol·K and 0.5–1.0 kcal/mol, respectively. The calculated compositions of the clathrate are collected in Table 2.

Fig. 3.

Calculated dynamical contributions to enthalpy (Upper) and enthropy (Lower) per mole of clathrated H₂ at 150 K. The extreme left point corresponds to H₂ in ideal gas. Open and filled symbols denote the results obtained in harmonic and “improved” approximations, respectively. Qualitatively similar dependence (not shown) is found at 250 K.

Table 2. Calculated composition of H₂/H₂O 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 × 10-2	2.2 × 10-2	8.9 × 10-1	2,000	250	1.61	1.6 × 10-1	1.6 × 10-1	6.4 × 10-1	1.1 × 10-2	3.8 × 10-2
	100	250	0.06	9.6 × 10-1	1.2 × 10-2	2.5 × 10-2	100	250	0.07	9.4 × 10-1	4.6 × 10-2	9.5 × 10-3	8.4 × 10-6	1.4 × 10-6
	500	150	2.00	3.3 × 10-4	7.7 × 10-5	1.0	500	150	3.25	4.5 × 10-2	1.0 × 10-2	2.7 × 10-1	2.9 × 10-3	6.7 × 10-1
	25	150	1.77	1.1 × 10-1	1.4 × 10-3	8.8 × 10-1	25	150	0.04	9.7 × 10-1	1.1 × 10-2	1.4 × 10-2	7.8 × 10-6	9.1 × 10-5
Improved	2,000	250	2.00	7.0 × 10-6	2.7 × 10-4	1.0	2,000	250	3.96	4.8 × 10-4	3.7 × 10-3	1.3 × 10-2	4.1 × 10-3	9.8 × 10-1
	100	250	1.99	2.8 × 10-3	5.4 × 10-3	9.9 × 10-1	100	250	0.39	6.8 × 10-1	2.6 × 10-1	4.7 × 10-2	7.3 × 10-3	8.8 × 10-3
	500	150	2.00	4.6 × 10-9	1.1 × 10-6	1.0	500	150	4.00	2.2 × 10-7	1.6 × 10-6	6.4 × 10-5	2.6 × 10-5	1.0
	25	150	2.00	1.8 × 10-6	2.1 × 10-5	1.0	25	150	3.78	3.3 × 10-2	1.2 × 10-2	2.4 × 10-2	4.8 × 10-4	9.3 × 10-1

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.

At the experimental conditions where the clathrate were synthesized (T = 250 K, P = 2,000 bar), the harmonic approximation leads to average cage occupancies of 1.8 (S cage) and 1.6 (L cage). These guest occupancies are insufficient to prevent the clathrate structure from collapsing (10). At T = 150 K and P = 25 bar, the average occupancy of the L cage drops to almost zero (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 clear on examination of the individual contributions to the calculated free energies of the hydrogen incorporation reactions. For example, the calculated reaction energy for 4L (4H₂+L = 4H₂@L) is exothermic at 250 K (Δ_rE₂₅₀ = –4.9 kcal/mol). This energy gain is associated with a large decrease in entropy (TΔS₂₅₀ = 20.7 kcal/mol). The entropic contribution is, in turn, dominated by the loss of translational entropy of free H₂ (27.2 cal/mol per K) with no contributions of similar magnitude appearing for the nH₂ cluster inside the cage. Therefore, it appears likely that the harmonic vibrational approximation failed to describe high-amplitude motions of the nH₂ clusters inside the model cages and led to underestimation of the entropies. A more realistic treatment of the vibrational entropy clearly is required.

For a spherically symmetric rigid host cage and freely rotating (or monoatomic) guest, the classical partition function can be reduced to a one-dimensional integral of the interaction potential. This leads to a simple expression for the first equilibrium constant (35, 36):

where ν(r) is the guest-cage interaction energy at displacement r from the center of the cage. In the temperature range where H₂/H₂O clathrate is stable (T < 250K), the assumption of classical motion breaks down for the light H₂ molecule leading to overestimation of the equilibrium constants (S.P. and S. Yurchenko, unpublished data). Any attempt to predict the thermodynamics of the model clathrate cages with classical molecular dynamics techniques will suffer from the same problem (37).

To attain a computationally affordable semiquantitative description of the thermal contributions to the free energy of the H₂ incorporation reaction, we propose the following scheme. We will refer to this procedure as the “improved” approximation.

For each nH₂ cluster inside a rigid model cage the 6n degrees of freedom are split into three categories.

1. Two degrees of freedom for each individual hydrogen molecule are treated as a free rotor. At 250 K, this term contributes 0.5n kcal/mol to ΔH, and 2.7n cal/mol·K to ΔS. This assumption is justified by the observation that spacing between rotational energy levels in H₂ (experimental value, B_e = 60.9 cm^–1; DFT, B_e = 60.1 cm^–1) is ≈0.2 kcal/mol, which is comparable to the orientational dependence of the H₂–cage interaction energy. Experimental Raman spectra also suggest essentially free rotation of the clathrated H₂ molecules (9). Within this approximation, the H₂ rotational contributions to reaction free energies cancel identically.

2. Three degrees of freedom per nH₂ cluster are treated as a rigid body rotation with moments of inertia calculated at the equilibrium structure of the cluster relative to the center of the cage. For the 4H₂@L cluster, this term amounts to 0.8 kcal/mol for ΔH₂₅₀ and 19.8 cal/mol·K for S₂₅₀. This assumption is justified by the approximately spherical symmetry of the cages and weakness of the individual H₂–H₂O interactions.

3. The remaining 4n-3 degrees of freedom are treated as harmonic oscillators with frequencies taken from DFT harmonic vibrational analysis. The n high-frequency modes correspond to the H—H stretching mode. The remaining modes correspond to the breathing modes of the cluster. For 4H₂@L, this contribution adds 11.9 cal/mol per K to S₂₅₀ and 30.4 kcal/mol to ΔH₂₅₀.

An additional justification is required for treating the rattling motion of the hydrogen molecule in a singly occupied cage as rigid body rotation around the cage center. For the temperature range of interest (150–250 K), the RT value amounts to 0.3–0.5 kcal/mol. This value should be compared with the barrier height at the cage center: ≈1.0 kcal/mol for the small cages and 2.0 kcal/mol for the large cages (see Fig. 1). At these temperatures the hydrogen molecule may be expected to be localized within the thin, spherical shell with R ≈ R_min.

Thermal contributions to formation energy and entropy calculated following this procedure (approximations 1–3) are shown on Fig. 3. For singly occupied cages, the improved values compare favorably with the results from the direct solution of Schroedinger equations for the nuclear motion of the guest (S.P. and S. Yurchenko, unpublished data). In general, the approach outlined above leads to significantly lower calculated H_T-E₀ contributions and higher entropies than the harmonic approximation. The bulk of the change in the enthalpic contribution is attributable to the absence of the zero-point energy from the free rotation contributions. The increase in the calculated entropy compared to the harmonic oscillator approximation derives largely from the rotations of the cluster. Treating these degrees of freedom as free rotations is essential for achieving agreement between theory and experiment.

Clathrate compositions at several (p, T) values calculated with the improved approximation are collected in Table 2. The average occupancies of the S and L cages at two representative temperatures are depicted in Fig. 2. At 250 K, the S cage is already fully occupied at relatively low pressures (≈100 bar). However, the bigger L cage is still empty at this pressure. At pressures >1,000 bar, both cages become fully occupied. This result agrees very well with the experimental formation conditions (T = 250 K, P = 2,000 bar). In view of the relatively crude approximations made in the calculations, this agreement should be considered excellent.

At 150 K, calculations suggest that the S cage should be able to retain both hydrogen molecules at H₂ pressures well below 1 bar, whereas the L cage is already fully occupied at P > 25 bar. Again, the calculated results agree well with experiment and suggest that type II H₂/H₂O clathrate is thermodynamically stable with respect to H₂ loss under mild temperature and pressure conditions. In the experiment, by warming the clathrate from 78 K to 115 K (9), it was observed that the intensity of the infrared absorption band assigned to vibrations of hydrogen in the small cages decreases rapidly. This observation may infer that the hydrogen content in the small cage decreases faster than in the large cages with increasing temperature, which is in apparent contradiction with the theoretical prediction. However, it should be recognized that the vibrational frequency of hydrogen molecule inside the cavities is very sensitive to the local environment. It may not be appropriate to associate that higher frequency vibrations with the motions of hydrogen in the more spatially confined smaller S cavity. In fact, the converse is observed for the C—H stretching modes of methane in methane clathrate (38). Furthermore, because the H—H stretching frequency is very sensitive to the location of the molecule inside the cavity, we may expect a very pronounced thermal broadening effect, leading to an apparent intensity decrease.

The analysis described above suggested that the hydrogen molecule clusters are rigid rotors in a weakly perturbed environment. We can then estimate the quantum rotation (tunneling) splittings in the limit of a free rigid rotor from the respective rotational constants. For hydrogen dimer in the S cage (2H₂@5¹²), the maximum energy splitting is ≈2 cm^–1, and for the hydrogen tetramer in the large cage (4H₂@5¹²6⁴), the splitting is ≈0.9 cm^–1. These splittings may be observable in microwave spectroscopy experiments.

Conclusions and Outlook

We have examined the stability of the type II H₂/H₂O clathrate with respect to the hydrogen occupancy by using DFT and ab initio MP2 calculations on model water cages. The experimentally observed composition and formation conditions can only be explained if two assumptions are made: (i) individual hydrogen guest molecules undergo free rotation and (ii) the collective motions of the cluster of hydrogen molecules inside the individual host cages must be treated as rigid body rotations. Under these assumptions, the clathrate reaches the observed ≈1:2 H₂/H₂O composition at T = 250 K, P > 1,000 bar.

The incorporation of H₂ into the cages is calculated to be exothermic (Δ_rH₂₅₀ = –1.8 kcal/mol), and the stability of the clathrate increases at lower temperatures. Below 150 K, the clathrate should be thermodynamically stable at near-ambient pressure, making it a promising and easy way to handle stored hydrogen. The stabilization of the hydrogen guest clusters arises mainly from attractive dispersive interactions with the water on the cage walls. As the surface contact is maximized for similar sizes of the guest and the host cage, sites within the smaller D-5¹² cage are more stable (Δ_rH₂₅₀ = –2.6 kcal/mol) than in the H-5¹²6⁴ cage (Δ_rH₂₅₀ = –1.2 kcal/mol). Thus, double clathrates with a suitable choice of guest in the larger cage (e.g., tetrahydrofuran) should form under even milder conditions. If hydrogen loss from the D-5¹² cage remains the stability-determining factor, such clathrates should be stable at hydrogen pressures of ≈20 bar (T = 250 K).

Interactions with the cage walls are responsible for the experimentally observed softening of the H—H stretching modes in the clathrate. If these interactions are ignored, the stretching frequencies actually increase, as expected for moderately compressed hydrogen gas (9).

It should be noted that a more definitive determination of the occupancy number can be obtained from structural refinement of experimental neutron diffraction patterns. Furthermore, microwave spectroscopy will be invaluable to characterize the tunneling splittings if the hydrogen molecule clusters were indeed behave as rigid rotors.

In summary, we have performed a theoretical investigation of the stability of hydrogen clathrate with respect to the guest occupancy. Although the occupancy may well be the dominating stability factor, other thermodynamic factors such as the stability of the ice framework at different cage occupations and kinetic (e.g., H₂ diffusion rates) factors need to be examined. Moreover, we only considered a single representative structure for each occupancy of the host cage. The structures of the guest clusters appear to be highly fluxional, leading to broad complex features in the experimental IR and Raman spectra. Understanding these features would require a detailed quantum molecular dynamics simulation of the guest clusters (37). In this study, quantum effects have been included in the consideration of the rotational motions only. The quantum contributions to the translational motions were neglected. It is expected that the equilibrium constants will be somewhat reduced if this effect was included. Finally, it is clear that the discovery of the type II hydrogen clathrate is opening new exciting horizons in the field of clathrate research.