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1 Introduction

Reliable storage of hydrogen with an easy release on demand is a bottleneck problem of hydrogen-based energy solutions. Solid metal hydrides (e.g. MgH2) offer high capacity but are so far problematic due to strong metal-H bonds needing high temperature (>300 °C for MgH2) for releasing H2 [1].

Corresponding clusters face similar problem, while being smaller and less rigid, both factors reducing the hydrogen-desorption temperature—see, e.g., recent advances for MgH2-related species [2, 3]. Another class of such systems is represented by mixed/doped metal clusters M n A k (M=Be, B, Al; A=Li, Na, Mg, B, P, etc.), both physi- and chemisorbing hydrogen [4–7]. Here the hydrogen binding energies can be close to estimated ideal ∼0.5 eV [1], larger or smaller up to negative values (corresponding to metastable systems, e.g. for CAl12 or SiAl12 substrates [6, 7]), usually with ad-/desorption barrier of ∼1 eV.

In particular, H2 encapsulated in small Mg n cluster cages can form weakly bound or even metastable species, although with a low storage capacity (∼1 weight-%) [8]. In order to try to improve the situation, the present work has two aims: (1) investigate similar systems of a lighter metal, Be; (2) evaluate feasibility of merging them into assemblies as a step to material.

Previous relevant work includes modeling adsorption of atomic H on solid Be (see [9] and references therein). The desorption temperature for molecular hydrogen has been predicted as ∼450 °C, with the desorption energy of ∼1 eV, which is even higher than for bulk MgH2. Another family of systems studied have been BeH2 aggregates and “polymers” [10], for which species, however, the hydrogen desorption temperature or energy has not been specified. The present work employs the earlier results indicating cage isomers of small Be n clusters as most stable [11].

2 Computational Methods and Tools

Calculations have been carried out at the MP2/aug-cc-pvtz level, followed by the standard counterpoise BSSE correction [12]. This level of theory is employed as implemented in the NWChem ab initio package [13], and is preferred due to capability to deal reliably with anticipated non-covalent interactions and strong charge-transfer in the systems studied.

The system geometries have been fully optimized for all atomic coordinates, with no constraints (for the C1 symmetry). Vibrational frequencies have been calculated to verify local minima of energy. The Be n cages have been preoptimized, then hydrogen molecules have been put inside with different orientations, and the system then reoptimized. The potential energy barriers (e.g. for hydrogen exit from the cages) were estimated by pulling H in proper direction (e.g. through a gap between Be atoms): an appropriate Be–H distance was fixed at a series of values and all the other coordinates were reoptimized for each displacement.

Higher-spin states have been checked to confirm the ground state multiplicity. Natural charges on atoms have been calculated using the natural bonding orbital formalism [14].

3 Results and Discussion

3.1 H2@Be n

It is found that H2 molecule can be trapped inside small Be n clusters starting from n=8 [15]. In particular, this and n=10 cages generally preserve their shapes enveloping the dihydrogen oriented along their symmetry axis. This is different from the analogous case of Mg8 cage which changes its shape [8]. Inside the Be9 cage, however, the inserted molecule is trapped perpendicular to the original symmetry axis of the cage which therefore transforms slightly, adopting a matching shape (capped square antiprism) now axially symmetric around the molecule. In all these cases, the H2 molecule dissociates into H atoms suspended electrostatically (noncovalently) in the cage (Fig. 12.1).

Fig. 12.1
figure 1

Optimized geometries of H2@Be8, H2@Be9, H2@Be10

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For n=9 one H atom protrudes from the cage, while adding another axial Be atom pushes it inside for n=10. A higher-energy (by 1.7 eV relative to the isomer with the protruding H atom), “sunken” isomer of H2@Be9 has both H atoms inside the cage (Fig. 12.2), with a very low (∼0.04 eV) potential energy barrier separating this isomer from the more stable one. Another, “radial” isomer of H2@Be10 with the H2 molecule perpendicular to the cage axis is 0.7 eV higher in energy (relative to the isomer with the axial orientation of H2), with the potential energy barrier for turning the dihydrogen into axial orientation (in the lower-energy isomer) of only ∼0.06 eV.

Fig. 12.2
figure 2

Optimized geometries of higher-energy isomers of H2@Be9 and H2@Be10

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The apparent reason for the H2 molecule to dissociate is a considerable charge-transfer from the beryllium cage. According to calculations, each H atom is charged by −1.5e and −1.3e for n=8 and 10, respectively. The high negative charge is apparently due to the number of Be atom neighbours donating the electron density. This charge is much larger than −0.15e for analogous H2@Mg10 [8], contrary to the opposite relation in the BeH and MgH diatoms as well as to the higher ionization energy of Be, but consistent with considerably shorter Be–H distances in H2@Be n . Accordingly, each Be atom is charged by +0.2e to +0.4e (Table 12.1).

Table 12.1 Equilibrium parameters (in eV and Å) and atomic charges (in e) of H2@Be n
Full size table

For n=9, the charge drops to −0.7e on the protruding H atom while remaining at −1.2e on the other one, the system asymmetry producing an axial dipole moment of 0.76 D. The two charges become about equal for the “sunken” isomer which, however, exhibits a fourfold dipole value of 3.1 D. The “radial” isomer of H2@Be10 is also slightly polar (0.35 D) due to the introduced slight asymmetry.

The Coulomb explosion of H2 upon the shell-to-core charge-transfer is confined by the Be n cage, which is the reason of the metastability of H2@Be n for n=8 and 10. These systems are, respectively, 1.5 and 1.7 eV higher in energy relative to the isolated relaxed molecule and cage. The n=9 system, however, allows one H atom to stick outside, and, as a result, exhibits stability of 1.4 eV to such a dissociation. The “sunken” isomer of H2@Be9 is thus nearly iso-energetic with the dissociation products, such a stabilization relative to n=8 and 10 being consistent with the “magic” number (20) of valence electrons in H2@Be9.

Calculations predict a low barrier of ∼0.06 eV for H atom to escape from the Be8 cage which then opens up and lets both hydrogen atoms to surface. The resulting system is only marginally lower in energy (by 0.2 eV) than the original one, hence still metastable. For n=9, the further axial withdrawal of the protruding H atom shows a similar barrier (∼0.08 eV), with the other H atom escaping to the surface as well. The Be9 cage recovers its shape and the system further stabilizes by 1.3 eV relative to the original one. For n=10, however, the barrier experienced by H atom on its way to the cage surface reaches 0.6 eV, the resulting system having almost the same (marginally higher) energy as original H2@Be10. However, the cage distorts and, as a result, the other hydrogen atom can leave the cage with almost no barrier. This lowers the system energy by 2.2 eV, thereby making it stable by about 1 eV to dissociation into Be10 and H2.

The relative stabilities of H2@Be n to dissociation into H2+Be n are reflected in their total dissociation energies \(\mathrm{D}_{e}^{\mathrm{total}}\) (into H2+n Be) as compared to those for respective Be n . Metastable H2@Be8 and H2@Be10 have \(\mathrm{D}_{e}^{\mathrm{total}}\) decreased relative to the original cages by about 0.2 eV per Be atom, while H2@Be9 is stabilized by a similar amount relative to relaxed Be9. The overall trend of \(\mathrm{D}_{e}^{\mathrm{total}}\) increasing with n (in the range of 1.4–1.9 eV per Be atom) is, however, preserved (see Table 12.1).

3.2 (H2)2@Be n

Both H2@Be8 and H2@Be10 are employed as units in cluster assemblies [16]. Merging two n=8 systems axially via two shared atoms produces (H2)2@Be14 with slightly zig-zagging chain of four H atoms inside the Be14 cage (Fig. 12.3). The system is triply more metastable, being 4.3 eV above the relaxed cage plus two hydrogen molecules, as compared to H2@Be8, while the barrier for escape of H remains about same.

Fig. 12.3
figure 3

Optimized geometries of (H2)2@Be14 and (H2)2@Be18

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A similar situation is found when axially merging two n=10 units while removing two Be atoms. In fact, the resulting system resembles two H2@Be9 units joined by their open ends, with protruding H atoms being pushed back inside (Fig. 12.3). The system is twice as high in energy (by 3.4 eV) relative to relaxed Be18+2H2 as H2@Be10 relative to Be10+H2, the difference from the (H2)2@Be14 versus H2@Be8 case being due to longer distances between the hydrogen anions (Table 12.2).

Table 12.2 Equilibrium parameters (in eV and Å) and atomic charges (in e) of (H2)2@Be n
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The higher-energy isomers of H2@Be10 can also be merged in a similar way, in which case, however, the resulting (H2)2@Be18 isomer is only 2.4 eV above Be18+2H2, i.e. significantly lower in energy than the previous isomer. The apparent reason of this is a larger distance between the hydrogen diatoms oriented perpendicularly, even though each “radial” H2@Be10 unit is higher in energy.

Finally, two H2@Be10 units have been merged via a shared triatomic Be3 face at their ends, with the units twisting relative to one another to adopt a staggered arrangement of atoms at the other ends. The resulting (H2)2@Be17 system is composed of a C-shaped chain of four H atoms inside a bent Be17 cage (Fig. 12.4). An interesting feature of this species is its almost identical metastability (about 1.7 eV higher in energy than relaxed Be17+2H2) as compared to that of H2@Be10.

Fig. 12.4
figure 4

Optimized geometries of “radial” (H2)2@Be18 and (H2)2@Be17

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In all above systems, the charges on the H atoms only slightly reduce (by 0.2e for (H2)2@Be14 and 0.1e for the H2@Be10-based species) relative to those in the respective units (Table 12.2). This, together with shorter distances between the hydrogen anions in (H2)2@Be17 as compared to “radial” (H2)2@Be18, could not explain such a stabilization of the former. One possible interpretation could be based on the specific structure with Be atoms located between H atoms and resulting in a charge distribution “negative-positive-negative” creating a significant quadrupole moment of such a layered centre polarizing the outer beryllium atoms.

In addition, the (H2)2@Be17 system exhibits a significant stability to the hydrogen escape from the cage, with the potential energy barrier of 0.3 eV. This is a half of the barrier for the H2@Be10 unit, the reduction likely being due to increased repulsion between the larger number of closely-spaced hydrogen anions.

Further 1D structural extension of the above (H2)2@Be14 and (H2)2@Be18 systems into a beryllium “nanotube” with a hydrogen “wire” inside by adding more units would lead to the hydrogen storage capacity up to 3.6 and 2.7 weight-% (one H per three and four Be), respectively. This value could be increased via merging the units by their sides as well, i.e. for 2D and 3D extensions. In particular, when the (H2)2@Be10 units are merged by a shared Be3 face at their ends, as in the above (H2)2@Be17 system, the storage capacity is evaluated to have an upper limit of about 8 weight-%.

3.3 H k @Be n

Since H2 dissociates in Be n , encapsulation of separate H atoms in such cages has been considered as well [17]. The Be6 cluster is found to be the smallest one able to accommodate an H atom inside. The centrally positioned hydrogen atomic core transforms the beryllium shell from a bipyramid into a perfect octahedron (Fig. 12.5). Unlike the metastable H2@Be10 counterpart, the system is stable to dissociation into relaxed Be6+H by appreciable 1.7 eV. The isomers with H attached to Be6 outside are more stable (bound by 3 eV), both Be2-edge and Be3-face sites being nearly degenerate (within 0.1 eV). The barrier for the hydrogen exit from the cage is found to be 0.5 eV.

Fig. 12.5
figure 5

Optimized geometries of endohedral H@Be6 and two HBe6 isomers

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In endohedral H@Be6, the H atom is charged by −0.95e, which is only slightly more negative than −0.8e for the other, HBe6 isomers. The charge on H is smaller than in H2@Be10, even though in the latter case each hydrogen atom has less neighbouring electron-density donors.

The stability of the H@Be6 species suggests a possibility of building larger assemblies from such blocks. Merging two H@Be6 units via a shared Be3 face produces a (2H)@Be9 species with the units distorted but generally preserved (Fig. 12.6). The repulsion of two hydrogen anions makes the system metastable, 1.2 eV above relaxed Be9+H2, hence a higher-energy isomer relative to the above H2@Be9. When empty, the relaxed beryllium cage generally preserves its shape of two face-merged Be6 units, as a higher-energy isomer of Be9. Due to strain in the system, the barrier for H exit from the cage reduces to 0.2 eV. The two hydrogen centres close to one another in (2H)@Be9 carry significantly increased negative charges as compared to H@Be6, −1.4e on each, in spite of the smaller number of Be atoms per H. The system has a small dipole moment of 0.2 D, close to that for the (H2)@Be17 counterpart.

Fig. 12.6
figure 6

Optimized geometries of (2H)@Be9, (2H)@Be10, and (2H)@Be11

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When two H@Be6 units are merged via a shared Be2 edge, a (2H)@Be10 system is produced (Fig. 12.6), with the shared edge stretched. The H anions are slightly further apart as compared to (2H)@Be9 (see Table 12.3), which stabilizes the system which is now 0.8 eV above relaxed Be10+H2. This value is a half that for (H2)@Be10, but is relative to the relaxed empty beryllium cage preserving its shape of two Be6 edge-sharing units, which is a higher-energy isomer of Be10. As a result, (2H)@Be10 remains 1.9 eV higher in energy as compared to (H2)@Be10. The potential energy barrier between the two isomers is 0.15 eV, and this isomerization occurs along the pathway followed by H atom exiting the cage.

Table 12.3 Equilibrium parameters (in eV and Å) and atomic charges (in e) of (kH)@Be n
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An attempt to separate the H anions even further by merging two H@Be6 units via a shared atom results in a collapse of the units into a Be11 cage encapsulating both H atoms (Fig. 12.6). The (2H)@Be11 system is metastable, being higher in energy than relaxed Be11+H2 by 1.5 eV which is the largest value among those for (2H)@Be n , n=9–11. The Be–Be interactions holding the cage together thus overcome the repulsion of the hydrogen anions. The charges on the H atoms slightly decrease with increasing n (Table 12.3), to −1.2e for n=11, opposite to the number of the donating Be atoms. The asymmetry of the cage leads to a small dipole moment of 0.34 D.

As a next step, a third H@Be6 unit is merged to (2H)@Be9 via two shared Be3 faces, leading to a D3h -symmetric (3H)@Be11 assembly with three H atoms forming an equilateral triangle (Fig. 12.7). The beryllium frame keeps its shape when empty, the (3H)@Be11 system being weakly stable by 0.4 eV to dissociation into relaxed Be11+H2+H. Such a stabilization of (3H)@Be11 can be viewed as being due to combination of metastable (2H)@Be9 and stable H@Be6. An alternative channel is represented by dissociation into (2H)@Be11+H, in which case the above n=11 species with two H atoms is recovered. These products correspond to the dissociation energy of 0.6 eV. The barrier for H atom exit from the cage is back to 0.5 eV (as for H@Be6), apparently due to a lower strain in the beryllium cage. The charge on each of the H atoms is −1.4e, same as for (2H)@Be9.

Fig. 12.7
figure 7

Optimized geometry of (3H)@Be11

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Similar to (H2)@Be n , the relative stabilities of the above (kH)@Be n systems to separation into the hydrogen and beryllium components translate into their total dissociation energies \(\mathrm{D}_{e}^{\mathrm{total}}\) (corresponding to Be n dissociated into atoms). Inserting H into Be6 and H+H2 into Be11(D3h ) stabilizes these clusters, significantly (by 0.3 eV per Be atom) for the former and slightly for the latter. While insertion of H2 into Be9, Be10 and Be11 destabilizes them by about 0.1 eV per Be atom. Overall, the \(\mathrm{D}_{e}^{\mathrm{total}}\) values increase with the system size, as for (H2)@Be n , from 1.6 to 2 eV per Be atom (Table 12.3).

4 Conclusions

Beryllium cluster cages form metastable (by a few eV) core-shell systems when endohedrally doped by molecular hydrogen dissociating due to a strong electron donation from, and confined in, the cage. When small H2@Be n units are merged together, they can generally preserve shapes and integrity in the larger assemblies. In some cases, this can result in a further stabilization of the system, as for (H2)2@Be17. Another feature is a possible higher stability of the assemblies composed of higher-energy isomers of the units, as for (H2)2@Be18. Energy barriers to extraction of H2 can be low (∼0.1 eV for n=8, 9, 14) to appreciable (∼0.6 eV for n=10), suggesting low-temperature conditions for stabilization. Extraction of both H atoms at once is more likely.

The metastability may offer two benefits: (1) easier release of hydrogen, (2) direct storage of extra energy (∼0.5–2 eV per H2 molecule here). Hydrogen storage capacity can be increased in cluster assemblies/materials, e.g. nanofoams. This is confirmed for face-sharing H2@Be10 units, with extrapolated upper bound of ∼8 weight-%. Feasibility of such materials is supported experimentally by a recent progress reported for Mg [18].

Such an endohedral doping may also offer options for modification of mechanical and electronic characteristics (shape, dipole moment) of clusters. This suggests potential applications in nanomaterials and molecular electronics.

Be6 is able to accommodate H atom inside and is significantly stable (by ∼2 eV) to its release. This stability can be reduced by design via merging such units into assemblies which are metastable to release of molecular H2, with a desorption barrier of ∼0.5 eV (matching the suggested ideal binding energy [1]). Hydrogen storage capacity of such systems extrapolated to a nanofoam material, composed of face-sharing H@Be6 units filling space, can reach ∼10 weight-%. This may exceed the value for the counterparts with encapsulated H2 (e.g. inside Be10 units) due to higher symmetry and better packing.

Such cluster-assembled materials could be more straightforward to develop than macro-assemblies of clusters preserving multiple surface sites for external binding of H2 molecules. Firm conclusions, however, would benefit from relevant experiments as well as modelling of molecular dynamics related to hydrogen entering and exiting the cages, in particular for refilling the nanofoam. It is hoped that the present work will stimulate such studies in the near future.