Adsorption of hydrogen on graphite, or in zeolites, has been investigated at a molecular level for primarily fundamental reasons long before this became an urgent practical issue for the ultimate realization of a hydrogen economy. The use of neutron scattering techniques in both of these examples illustrates two basic requirements that can make neutrons a surface-sensitive probe, the first is that the adsorbate needs to have a significantly greater neutron scattering cross-section (e. g. H) than the host material (e. g. C or silica), and the second is that adsorbent must have a very large surface-area so that the total number of surface-bound molecules is sufficiently large. This is the case for most carbon materials, such as exfoliated graphite, which was used [94] to determine the two-dimensional structural phase diagram of monolayers of deuterium. The highest coverage was found for an incommensurate structure of 0.078 molecules A-2 which corresponds to about

1.3 x 10-5 mol. m-2. This number indicates a limit of the storage capacity for simple physisorption on carbons, namely that a material with a maximum surface area of 1,500 m2g-1 will bind close to 0.02 mol of H2 (0.04 g, 4 wt.%), which still falls short of what is required. Surface areas for the interior pores in zeolites are typically considerably less than that because of the high density of the silicate framework.

It has therefore long been recognized that surface areas have to be considerably increased, and this has been accomplished with the synthesis of a large number of metal-organic framework (MOF) materials with as much as 5,000 m2g 1 or more. Capacities around 8 wt.% have been achieved in these materials albeit under unfavourable conditions (77 K and pressures of approximately 60 bar), but the amount of hydrogen stored in MOFs at ambient temperature is far below those required for practical goals. The reason for this is, of course, that the binding energies for hydrogen provided by physisorption are too low (of the order of 5 kJmol 1) and hence vapour pressures are too high for the operation of such a storage system at ambient conditions.

Materials therefore need to be designed to not only have very high surface-areas but also binding energies for hydrogen at least three times as high as that for normal physisorption (e. g. on carbons) in order for such a system to operate at room temperature and modest pressures. Heats of adsorption must also be approximately constant over the entire range of hydrogen loading, i. e. of similar magnitude for all accessible binding sites.

An enormous effort to design and synthesize new and better porous materials guided by chemical or structural principles, including the use of combinatorial methods, has resulted in a huge variety of metal-organic materials not only for applications as hydrogen storage materials, but also for the capture of CO2, gas separation, and catalysis (see Chaps. 2 and 3). Binding energies of hydrogen molecules have more than doubled from that of simple physisorption in some of these materials, but are still quite far away from what is needed for a practical storage medium operating at ambient conditions.

The use of neutrons makes it possible to employ a systematic, molecular-level approach to the development of improved hydrogen storage materials, particularly when combined with advanced computational analysis. Such experiments can provide the location of the adsorbed hydrogen molecules at different sites (neutron diffraction) and give some measure of the strength of the interaction with the host material at each site (inelastic neutron scattering), whereas conventional thermo­dynamic measurements (adsorption isotherms) only give average properties for all sites that are occupied at a given loading. Results of such experiments should then serve as the basis for detailed computational analysis, which can rationalize both the locations of the hydrogen molecules and their binding strength in terms of the interatomic guest-interactions. This type of information should then in principle be useful in the design and synthesis of improved storage materials.

A detailed knowledge of the binding sites of hydrogen molecules in crystalline porous materials can be obtained by neutron diffraction methods. Adsorbed hydrogen molecules cannot be “seen” in single crystal X-ray diffraction experi­ments even at very low temperature, but this method has been used to obtain a very detailed picture of the adsorption and binding sites of Ar and N2, i. e. molecules with many electrons, in the prototypical metal-organic material Zn4O(bdc)3 (where bdc = 1,4-benzenedicarboxylate), also known as MOF-5. The primary binding site at 30 K for both types adsorbates was found to be in the pocket of the ZnrO cluster in the corner of the framework (a-site) [95]. A number of other binding sites were clearly identified, including ones that are triply or doubly bridging the carboxylate oxygens (P and у sites, respectively), which connect the cluster the organic linker. Interaction with the benzene ring leads to two binding sites, one near the two aromatic hydrogens in a side-on configuration, and the other on top of the ring.

Ideally a similar approach with single-crystal neutron diffraction should provide similar detail for hydrogen binding sites. This has, however, been accomplished just once, namely for hydrogen in MOF-5 [96], since the sizes of MOF single crystals are usually too small for the relatively-low intensities in neutron diffraction experiments. This experiment was carried out on a crystal that barely was of adequate size. Nonetheless, the most favourable binding site for H2 in MOF-5 at 5 K was clearly identified to be in the pocket of the Zn4O cluster (a-site) with one end of the molecule on the three-fold axis pointing at the central oxygen atom. The H-H bond of the molecule is at an angle relative to the three-fold axis of the crystal so that the other end is disordered over three equivalent orientations (Fig. 8.7) in a similar manner as N2. This site is essentially fully occupied at 5 K. Hydrogen was also located on the P-site, but could not be refined as two separate atoms because of rather large atomic displacement parameters.

Because of the aforementioned difficulties in obtaining sufficiently-large crystals of MOFs, a number of studies to characterize hydrogen binding sites have been carried out by neutron powder diffraction, including also on MOF-5 [97]. For neutron powder diffraction it is highly advantageous, although not always neces­sary, to have all Hs in the system replaced by deuterium (D) in the synthesis of the material, which can, of course, be rather difficult and expensive, and to use D2 as the adsorbate. The strong background of the incoherent scattering by H can thereby

Fig. 8.7 H2 binding in MOF-5 as determined using single-crystal neutron diffraction. From Ref. [96]

be reduced. Diffraction patterns can be collected at different loadings of D2 and interpreted under the assumption that sorption sites become occupied roughly in order of their respective binding energies. The molecule itself cannot, however, be refined as two separate atoms, so that only its centre-of-mass is found. In addition to the locations of the D2 molecules the degree that these sites are occupied is refined in the analysis, along with isotropic atomic displacement parameters all within the symmetry of the parent structure.

Cu3(btc)2 (where btc = 1,3,5-benzenetricarboxylate) and also known as HKUST-1, was one of the earliest MOFs reported and has been extensively characterized since then, including several neutron diffraction and inelastic neutron-scattering studies, as well as computational studies. Perhaps the most interesting structural feature of this material is the well-known dicopper acetate “paddlewheel” unit, which has subse­quently been used in many MOFs. The apical ligand on the Cu usually is a weakly bound water molecule, which can easily be removed, and thereby opens up a metal site for presumably stronger adsorption of hydrogen. The most favourable binding site in HKUST-1 was indeed found to be, using neutron powder diffraction, adjacent to the open Cu site (D2(1) in Fig. 8.8) [98, 99]. This would be expected on the basis of the relatively-strong interaction of the hydrogen molecule with the metal, so that these sites are largely responsible for the improved heats of adsorption found in this material. The hydrogen molecule was located [98] at a distance of 2.39(1) A from the Cu site, which is, however, far too great to allow for a direct electronic interaction with the metal. Additional binding sites at various points in the host structure were iden­tified in this study, some of which are shown in Fig. 8.8. Site 2, for example, is adjacent to the benzene rings of the btc unit, while D2 in site 3 is close to six btc oxygen atoms in the small window leading to the 5 A pores.

The enhanced interaction of H2 with such open-metal sites has been demonstrated in a number of other MOFs, most notably MOF-74 [100, 101], also known as CPO — 27 [102], for which a number of isostructural variants with different metal centres

Fig. 8.8 Binding of D2 in HKUST-1, with the sites labelled sequentially in order of binding strength. H atoms are omitted from the framework for clarity. From Ref. (V. K. Peterson, Y. Liu, C. M. Brown, C. J. Kepert, J. Am. Chem. Soc. 128, 15578 (2006)) [98]. Copyright (2006) American Chemical Society

(Zn, Mg, Mn, Co, Ni) can be synthesized [102—104]. The materials consist of a linear chain of inorganic metal clusters connected with organic linkers to form hexagonal channels, M2(dhtp), (dhtp = dihydroxyterephthalate). H2 binding energies indeed depend strongly on the type of metal centre, and do reach about 13 kJ/mol for Ni, but only until all these sites are filled. A powder neutron diffraction study [137] on the Zn analog finds that the D2 molecule is close enough (d(Zn-D2) * 2.6 A) to be affected by polarization from the metal, but again too far for a direct electronic interaction. A somewhat closer metal-H2 distance of 2.27 A was found [105] by neutron powder diffraction in the Mn-based MOF formed from Mn4Cl clusters and 1,3,5-benzenetristetrazolate bridging ligands. Desolvation of the parent material opens up one of the coordination sites on Mn2+, which in turn interacts rather strongly with H2 (10.1 kJ/mol near zero-loading), but again, not by direct electronic interaction, which requires H2-mteal distances in the order of 1.8 A.

One of the more important aspects of experimental efforts to locate the adsorbed hydrogen molecules in these systems is that this sort of information is critical for benchmarking the extensive theoretical studies of these systems that have been undertaken. A neutron powder diffraction study on hydrogen adsorbed in the zeolitic imidazolate framework ZIF-8 [106] was the first to rationalize the experi­mental findings with parallel DFT calculations of binding energies at various sites. The preferred adsorption site in this material in fact is not near the metal cluster but on top of the metal imidazolate linker. Most computational characterization of MOFs for hydrogen adsorption is accomplished by grand-canonical Monte Carlo simulations, or by molecular dynamics, both of which readily yield pair-distribution functions (PDFs) for the adsorbate molecules. These can be directly compared with diffraction experiments without any model dependent structure refinement, if the diffraction data are converted to direct space, i. e. to PDF form, as has been reported for D2 in IRMOF-1 [107]. This approach would be especially important for those MOFs, where the adsorption sites are poorly defined, MOFs which have defect structures, or are in fact amorphous.

A much greater number of neutron scattering studies on sorbent hydrogen storage materials has been carried out by inelastic scattering from the bound hydrogen molecules, perhaps because of the relative ease with which these can be carried out, and because of the sensitivity of the method. A detailed interpretation of such data is, however, very challenging, and ultimately requires extensive com­putational studies. The rotational dynamics of the adsorbed hydrogen molecule are a highly sensitive measure of its interactions with the host material and are readily accessible by the inelastic scattering of neutrons. The reason for this is that the rotational energy levels of the quantum rotor H2, E, are strongly perturbed from those of the free quantum rotor (E = BJ(J + 1), where B is the rotational constant of the molecule and J is the quantum number for total rotational angular momentum) by the barrier to rotation resulting from the influence of the adsorbent.

Transitions between these hindered-rotor levels can be directly observed by incoherent INS and give an indication of the height and shape of the barrier to rotation. Such a quantitative analysis of these measurements must however employ a phenomenological model for this barrier, whose parameters may be determined from the pattern of observed transitions.

The simplest such model is one of a double-minimum potential with the two angular degrees of freedom for the H2 molecule,

V(в, Ф) = 1/2V2(1 — cos20)

The energy levels for this potential can then be obtained by solving the Schrodinger equation:

for various values of V2, with the result that the observed transition(s) yield a value for the height of the barrier. It is important to note that the energy-level diagram (Fig. 8.9) shows that the spacing between the two lowest levels decreases with increasing barrier height, which is indicative of rotational tunnelling, where the tunnelling probability has an approximately-exponential inverse dependence on the barrier height. Therein lies the great sensitivity of this type of measurement.

A number of more elaborate phenomenological models for the H2 potential — energy surface (PES) have also been used to interpret the INS data, and these include separate barriers to in plane and out of plane reorientation (or some ratio between the two) plus a separable centre-of-mass vibration perpendicular to the surface [109]. Such models, however, require the observation of several transitions in order to determine the parameters in the form of the potential. While the use of such a phenomenological potential is essential for the assignment of the observed rotational transitions to hydrogen molecules absorbed at different sites, it does not, however provide direct insight into the atomic level origins of the rotational bar­riers. What we would like to learn from these experiments is the origin and nature of the interatomic interactions at each binding site that give rise to the PES, which

in turn gives rise to the observed rotational energy levels. This can, of course, be done by ab initio computational studies as long as the structural details of the binding sites are known, preferably from a neutron diffraction experiment. The dynamics of the hydrogen molecule within such a theoretical PES then has to be treated exactly, i. e. with quantum dynamics for both the rotational and (centre of mass) translational excitations. This has been accomplished for H2 in clathrates, various kinds of fullerenes, and just one MOF, namely MOF-5, which will be described below.

The seminal work of this type [110] on H2 adsorbed in MOF-5 helped to rationalize the large storage capacity of this material in terms of the availability of several different types of binding site. The INS spectra are collected at 10 K (or less) as a function of loading (carried out at 77 K) in a direct relationship to the numbers of equivalent sites of particular types available to facilitate a tentative identification of specific spectral features with binding sites (Fig. 8.10) by making use of the phenomenological PES described above. Three additional sites could be inferred from the INS spectra, which may now be matched with information available from the diffraction experiments as well as computational studies, i. e. the в, у, and 5 sites.

Similar experiments have been carried out on hydrogen on various forms of carbon, some of which were at one time considered to be promising hydrogen storage materials. The INS spectra, however, generally show just one broad peak centred near the free rotor value of 14.7 meV, which is representative of a broad

Fig. 8.11 INS spectra of H2 adsorbed on (top to bottom) two forms of activated carbon, nanotubes, and nanofibers. Reprinted from (H. G. Schimmel, G.

J. Kearley, M. G. Nijkamp, C. T. Visser, K. P. Dejong, F. K. Mulder, Chem. Eur. J. 9, 4764 (2003)) [111] with permission distribution of binding sites with very low barriers to rotation, or weak interaction with the host. These spectra clearly show that there is very little difference in the interaction of H2 and various forms of carbons, be it activated carbon, or nanotubes [111, 112] (Fig. 8.11).

These results may be contrasted with those for MOF-5, where the INS spectrum exhibits a number of sharp transitions indicative of different, well-defined binding sites, some of which have higher barriers to rotation of H2 than do carbons.

A considerable number of metal-organic materials (MOMs) have been investi­gated using rotational-tunnelling INS spectroscopy of the adsorbed H2 in an effort to determine which structural features may improve the observed heats of adsorption. Here we utilize the fact that a smaller rotational tunnel splitting implies a higher barrier to rotation, which in turn should, in general, signal a stronger interaction between the adsorbed H2 and the host and a particular binding site. This type of information can only be extracted if particular spectral features can be associated with H2 a known sites, which can be quite difficult in the presence of simultaneous multiple-site occupancy, as will be described below.

Previous studies of H2 adsorption in zeolites [113] had shown the strong effect of a charged framework along with the presence of charge-balancing extra framework cations on the binding energies of hydrogen molecules. The vast majority of MOMs, however, possess neutral frameworks, but some MOFs with anionic or cationic frameworks have indeed been characterized. Zeolitic MOFs, ZMOFs, are such a class of anionic MOFs built from metal centres with imidazole dicarboxylate ligands to form zeolitic structures (e. g. rho-ZMOF, Fig. 8.12) with much larger pores than their SiO2 analogs. This structural similarity arises from the fact that the Si-O-Si bond angle in the zeolite is nearly the same as the angle between the two N atoms in the imidazole dicarboxylate. The negatively-charged framework is compensated by exchangeable extra framework cations, as in zeolites. In the case of ZMOFs, how­ever, these cations are hydrated and bound to the framework by H-bonds from the water ligands to carboxylate oxygens. INS studies on rho-ZMOFs with different

Fig. 8.12 Left Zeolite rho. Right A rho-ZMOF. Taken from [114]

Fig. 8.13 INS spectra of H2 in rho-ZMOF with Li (left), as synthesized with DMA (right) [108]

cations revealed rather broad spectra with few well-defined peaks, which do not appear to depend very much on the nature of the cation. Much of the intensity in these spectra at lower loadings may be found well below 12 meV, the main peak in the spectrum for H2 in MOF-5 and therefore indicate stronger interaction (higher barriers to rotation) with the charged framework (Fig. 8.13). A wide range of binding sites seem to be occupied by hydrogen at all loadings, which appears to be characteristic of this environment of large, hydrated cations in an anionic framework. The isosteric heats of adsorption are indeed some 50 % greater than in the neutral framework MOF-5 at low loadings, but virtually independent of the nature of the cation.

Many of the original MOFs were noted for their very large pore sizes and hence extremely-low densities. This property is not, however, optimal for storing large amounts of hydrogen under ambient conditions because much of the interior of the pores may only bind H2 to other H2 molecules. If, however, the pore size of a MOM were to be reduced to values not much bigger than that the size of a hydrogen molecule, it could interact with many framework atoms all around rather than just those on a surface. We can therefore readily conclude that a MOM with uniformly small pores would be highly beneficial to hydrogen binding. This was convincingly demonstrated in recent work by Pham et al. [115] on a MOM, which occurs in two forms, one with interpenetrating identical frameworks, the other in a non-inter­penetrating form. SiF6 units and bipyridyl acetylene (bpa) linkers connect the Cu centres in this structure (Fig. 8.14). We note that the channel dimensions of the interpenetrating structure of 3.5 x 3.5 A are only slighty larger than the size of the hydrogen molecule of 2.9 A, which would suggest that H2 will have appreciable interaction with many of the surrounding framework atoms.

The INS spectrum in fact is rather unique for H2 in pores in that it only exhibits one strong peak at all loadings with a shoulder on the low energy side (Fig. 8.15), both of which simply increase intensity until the pores are filled. Moreover, this peak occurs at a lower energy (6.3 meV) than that for hydrogen near open-metal sites and hence indicates an appreciably-stronger overall interaction with the framework. The two similar binding sites found in a computational study [115] are

adjacent to each other in the narrow channels, and hence differ little in their interactions with framework atoms.

The analogous binding sites in the non-interpenetrating form are open towards the large pore, so that binding energies and barriers to rotation are much lower, and hence the rotational-tunnel transitions in the usual range of MOFs without open — metal sites, i. e. above 10 meV. The isosteric heat of adsorption [115] in the interpenetrating structure (8.3 kJmol-1) is nearly twice that of the non-interpene­trating structure and constant over most of the range of gas loading. The latter is an
important requirement for an adsorption-based hydrogen storage system, and one that few such materials to date can satisfy.

A direct electronic interaction of H2 with metal centres is, of course, known from inorganic and organometallic chemistry to be very strong and does in fact lead to dissociative binding of H2 in most cases. The ability to create coordinatively — unsaturated metal sites in a MOM by removal of a weakly-bound ligand might therefore have been expected to significantly enhance hydrogen binding-energies. INS studies of H2 adsorbed at such sites in a variety of MOFs have indeed revealed stronger interaction by the observation of larger barriers to rotation with smaller rotational-tunnelling splittings. The effect is not, however, nearly as large as anticipated, since the H2 molecule does not approach the metal site very closely. This was shown directly by powder neutron diffraction experiments on HKUST-1 and on MOF-74 [98, 99, 104, 137].

INS spectra of H2 in HKUST-1 [111] do show evidence for the stronger inter­action with this open-metal site, with a peak at about 9 meV (Fig. 8.16), which can then be associated with sorption near the open Cu site. This value is appreciably lower (and hence the barrier to rotation higher) than in MOF-5, which in turn reflects the respective isosteric heats of adsorption of 6.8 and 4.8 kJmol-1, respectively, at the lowest loadings.

Additional features in the INS spectrum have been attributed to secondary binding-sites in accordance with the structural information, although this process is not unambiguous. For example, the peak at 12.5 meV may be attributed to hydrogen at the so-called small pore site, while at higher loadings a peak near the free-rotor transition may be indicative of several binding sites farther inside the pores. The precise attribution of these spectral features has, however, been subject to different interpretations, which can be difficult to resolve.

Fig. 8.16 INS spectra for two loadings of H2 in HKUST-1. Reprinted with permission from (C. M. Brown, Y. Liu, T. Yildirim, V. K. Peterson, C. J. Kepert, Hydrogen adsorption inHKUST-1: a combined inelastic neutron scattering and first-principles study, Nanotechnology 20, 204025 (2009)) [118]

Fig. 8.17 5-tetrazolylisophthalic acid ligands, copper paddlewheel and Cu3O trimer units in rht-MOF-1 [120]

The Cu paddlewheel structural building unit (Fig. 8.8, left) can be found in many MOFs, some of which have been investigated by INS on adsorbed H2. The rota­tional transitions for H2 at this open Cu site vary somewhat from that in HKUST-1 depending on structural details of the environment, as, for example, the close proximity of another paddlewheel unit in Cu6(mdip)3 (mdip = 5,5’-methylene diisophthalate), also known as PCN-12 [119], where these are observed 7.7 and

8.6 meV.

A remarkable new class of ‘designer MOFs’ have structures with so-called rht net, which are synthesized from metal ions coordinated to a C3 symmetric ligand with three coplanar isophthalate moieties. Each isophthalate group is linked to metal ions to form a square paddlewheel cluster. In rht-MOF-1 a trigonal Cu3O trimer is liked to three 5-tetrazolylisophthalic acid ligands (Fig. 8.17). This com­pound therefore contains at least two different open Cu sites, which were identified from the rotational-tunnelling spectra and computational studies [120]. In this case the transition associated with the Cu paddlewheel was found to be at 9.7 mev, compared with about 7 meV for binding sites near the Cu3O trimer.

The increased interaction strength of hydrogen with these open-metal sites arises primarily from polarization, but falls far short of what one might expect if the molecule were to coordinate to the metal as in organometallic complexes. To date this has definitively been achieved only in the post-synthesis modified zeolite NanAlnSi96-nOi92 where 0 < n < 27, also known as ZSM-5. Highly undercoordi­nated Cu sites can be created by deposition of Cu in some form from the vapour or

Fig. 8.18 H2 in Cu-ZSM-5 a coordinated the Cu(I) and b physisorbed on the walls of chabazite. The rotational-tunnelling transitions in the two cases differ by a factor of nearly 100. Figure taken from [123]. Reprinted with permission from (A. J. Ramirez-Cuesta, P. C.H. Mitchell, Cat. Today 120, 368 (2007)) [124]. Copyright (2007) Elsevier

Fig. 8.19 Ab initio computational model for H2 in the zeolite chabazite. For details see [122]

solution in the pores of the material, whereby Cu-ZSM-5 is created. Cu (II) ions evidently coordinate to framework O atoms and can be reduced to Cu(I) by thermal activation. H2 can coordinate to these open Cu ions and form a dihydrogen complex similar to some organometallic compounds [121], which tend to show very small rotational tunnelling splittings (Fig. 8.18a) because of the chemical bond formed with the metal centre. This also requires that the H2 be able to approach the metal to within at least 1.7—1.8 A as is indeed shown in the computational model developed for the open Cu site using the zeolite chabazite (Fig. 8.19) [122].

The coordinated H2 is located at 1.653 A from the Cu centre, and the H-H bond has been activated as shown by an elongated H-H distance of 0.804 A. Heats of adsorption for binding to Cu(I) in the zeolite have been observed and calculated [122, 123], to be as high as 70 kJmol 1, more than ten times the typical H2 physisorption energies. The rotational-tunnelling splitting of about * 1 cm 1 for coordinated H2 is about 1/100 of that of hydrogen physisorbed on the walls (*11.5 meV) in the same zeolite [124]. However, only about 40 % of the Cu sites in the zeolite appear to be able to molecularly chemisorb H2, so that the overall isosteric heat of adsorption decreases rapidly when the H2 loading increases much beyond that. Nonetheless, this result may be viewed as a proof of principle in that it demonstrates that this type of molecular chemisorption offers the greatest hope to boost binding energies in sorption-based systems to the point where they can store adequate amounts of hydrogen for ambient temperature operation.

For MOMs to enable this type of binding the metal sites must be more open than what is provided by just having one vacant binding site. This may be more easily achieved in the presence of large numbers of defects, or in an amorphous material, and efforts to this end are well underway [125].

A hybrid type of hydrogen storage medium, which combines sorption of H2 and binding of H in the solid utilizes the so-called hydrogen spillover effect [126]. While neutron scattering methods would again seem to offer the best approach for the study of this phenomenon, experiments carried out to date have encountered difficulties because of the complexity of such media, e. g. metal particles on carbon supports, and the spillover process itself. The approach taken has been to monitor changes in the rotational transition band for H2 on the carbon support, and attempt to detect the dissociated H by INS vibrational spectroscopy [127].

One of the great strengths of neutron scattering methods is that the experimental results can be connected in a very direct manner to computational studies on account of the simple form of the interaction of neutrons with the atomic nuclei. This aspect does not play a role when it comes to determining to location of the adsorbate molecules, as the experimental probe does not enter the picture, but it does so in spectroscopy. In this case one may compare actual intensities for the various transitions or entire spectral profiles with experiment, in addition to their frequencies.

A number of approaches have been taken in the present case of adsorbed H2, the simplest of which may be that of rotational barriers. Energies are calculated for stepwise rotation of the hydrogen molecule with fixed centre-of-mass to obtain the rotational barrier height. This value is compared with a barrier height given by a assuming a phenomenological potential which is adjusted to account for the experimentally observed transitions. For H2 in Cu-ZSM-5 the barriers are 15 and 9 kJmol-1, respectively.

It is, of course, preferable to avoid having to choose a phenomenological potential for interpreting the experimental transition-frequencies by calculating those directly from a theoretical potential and making the comparison on the basis of frequencies, instead of derived barrier-heights. This requires not only an extremely accurate PES for the adsorbed H2, but also a full quantum dynamical treatment of all of its five degrees of freedom (two rotational and three translational). A simplification of this rather challenging computation can be achieved if one makes the assumption that the rotational degrees of freedom are separable from the centre-of-mass translational motions. This approach has been taken a number of times, including for HKUST-1 and MOF-74. Translational motion is often treated separately using a harmonic oscillator model as an approximation. For H2 in MOF-74, for example, the PES was derived from a periodic DFT calculation using a van der Waals density functional method [128], from which the calculated rotational transitions for the main binding sites near the open metal, as well as two other sites, were obtained in reasonable agreement with experiment. Translational excitations were calculated separately in a harmonic model with anharmonic corrections.

The computational methodology for the full five-dimensional, coupled transla­tional-rotational quantum dynamics for a bound H2 was developed by Bacic and collaborators [129] and originally applied to cases where the H2 was essentially trapped in a cage, and where the interaction potentials are well known, such as H2 in C60 [130], and in ice clathrates [131]. In the more general case for molecules with six degrees of freedom, such as methane, the Schrodinger equation with the fol­lowing Hamiltonian is solved numerically:

h=-Ь (I?+b+@?)+8,2+V (xyzфv) (8-8)

where the potential V has to be determined either from empirical forces, or high — level ab initio calculations, or some combination which includes all relevant interactions between the adsorbed molecule and the host. To do this for all five or six degrees of freedom is, however, a lengthy undertaking for many reasons, not the least of which is the fact the H2 interacts very weakly with its surroundings.

The first application of this methodology to hydrogen adsorbed in a MOF used PESs for MOF-5 derived in two different ways [132]. While both potentials reproduced the structural aspects (binding geometries) fairly well, they were found that a low barrier could facilitate translational motion from the a to the neigh­bouring у sites. This is, however, in disagreement both with diffraction experiment, and INS, which show single site occupancy at low temperature and low loadings. The full five-dimensional PES was therefore calculated with the у sites blocked by other H2 molecules, and translational-rotational transitions calculated within the PES surface shown (Fig. 8.20). The transition energies obtained agree well with the experimental. spectrum, and do confirm the conclusion reached from infrared experiments [133] that the lowest energy peak at 10 meV in the INS spectrum is in fact a translational excitation. Translational and rotational excitations were found to be coupled only for transitions to higher levels, as has generally been assumed in the analysis of the INS spectra described above at low frequencies.

While the comparison based on observed and calculated transition frequencies described above is indeed rather powerful, the ideal connection between experiment and theory is one where the computation is extended to produce what is actually observed in an experiment, namely in this case the INS spectrum itself. This is quite commonly done for INS vibrational spectroscopy, but has only recently been accomplished [135] for the full five-dimensional translational-rotational quantum dynamics of H2, in this case when trapped in the small cage of clathrate hydrate (Fig. 8.21).

This is a relatively more favourable case for application of this methodology that for H2 on interior surface sites in MOF-5, as the molecule is trapped in a well — defined site, and its interaction with the host, i. e. ice, is rather better known than with those of MOFs. Nonetheless, this approach holds significant promise for extracting the maximum amount of information on the interaction of H2 with the host material from the INS spectra. The significant computational effort involved in obtaining accurate, multi-dimensional PESs for each binding site, along with solving the five-dimensional quantum dynamics problem for the energy levels and displacements does, however, limit its application to select cases at the present time.

While quasielastic neutron-scattering (QENS) studies have been crucial in the extensive efforts to reach an understanding of the function of complex hydrides such experiments have rarely been carried out on sorption based systems. Translational diffusion of hydrogen in MOFs can readily be observed by QENS [136], but this may not be of sufficient interest for these system, as it is not in any way a rate — limiting factor in the operation of a typical sorption-based storage medium. Inves­tigations of materials with very small pores, or those including hydrogen spillover, may well require detailed QENS studies in the future for their understanding.