Toward Efficient Hydrogen Storage: A Quantum Chemical Study of Beryllium Clusters

Abstract

The development of lightweight materials capable of reversibly storing hydrogen under practical conditions remains a central challenge for hydrogen-based energy technologies. In this work, a systematic density functional theory (DFT) investigation is carried out to evaluate the structural stability and hydrogen adsorption properties of beryllium clusters (Be _n , n = 3–25). Benchmark calculations against DLPNO-CCSD­(T) confirm that the ωB97X-3c composite functional provides an accurate and efficient description of Be–Be bonding and H₂ adsorption energetics across the full-size range. Single-molecule adsorption reveals predominantly molecular physisorption for most clusters, with adsorption energies lying in the optimal window for reversible hydrogen storage and negligible H–H bond activation. Detailed multi-H₂ adsorption studies on representative clusters (Be₃, Be_{1 4}, and Be_{2 4}) highlight a pronounced size dependence. Among them, the medium-sized Be_{1 4} cluster exhibits exceptional performance, accommodating up to 20H₂ molecules with smoothly decreasing average adsorption energies from −0.35 to −0.10 eV per H₂. This behavior enables a high theoretical gravimetric hydrogen density of 24.2 wt % and a reversible working capacity of 17.86 wt %, significantly exceeding the DOE 2025 target. Thermodynamic and kinetic analyses predict favorable desorption temperatures (127–259 K) and ultrafast desorption kinetics. Electronic structure analyses, including energy decomposition, density of states, and IGMH methods, reveal that hydrogen uptake is governed by cooperative, noncovalent interactions dominated by electrostatic, polarization, and dispersion contributions. These results identify Be_{1 4} as a promising nanoscale motif for reversible hydrogen storage and provide fundamental insights into size-dependent hydrogen adsorption in light-element clusters.

1. Introduction

Hydrogen is increasingly recognized as a key element in the global energy transition, playing a vital role in mitigating climate change. Green hydrogen is produced by electrolyzing water using renewable energy. Steam methane reforming (SMR) of natural gas produces blue hydrogen with carbon sequestration and gray hydrogen without. Green hydrogen is essential for energy conversion and storage, to decarbonize long-distance trucks, rail, air transportation, and heavy industries.

For effective onboard applications, hydrogen must be stored under optimal pressures and temperatures, ensuring easy reversibility and rapid kinetics. The use of hydrogen as a fuel is limited by the lack of cost-effective storage solutions. Recent years have seen growing interest in the solid-state storage of hydrogen via the adsorption/desorption mechanism. , Solid-state hydrogen storage is safer and more compact than compressed gas and liquid forms, which need high pressure and cryogenic temperatures. Atomic clusters serve as simplified models to study the formation of active sites on metal and alloy surfaces and their interactions with H₂ in energy storage.

Molecular hydrogen is weakly physisorbed onto the substrate surface, resulting in limited adsorption strength and low storage capacity, with hydrogen readily desorbing. By contrast, atomic hydrogen chemisorbs strongly onto the substrate, enabling higher storage capacity but making H₂ desorption thermodynamically more difficult. ,

The systematic design of a suitable hydrogen storage medium necessitates rigorous evaluation of both gravimetric storage capacity and practical manufacturability, in accordance with established scientific protocols. An effective hydrogen storage system requires (i) suitable adsorption kinetics, with average adsorption energies of −0.10 to −0.60 eV/H₂ to balance reversibility and binding strength, and (ii) a storage capacity of at least 5.5 wt % under operating conditions. These parameters have been established by the United States Department of Energy (DOE) as benchmark targets for onboard hydrogen storage systems by 2025. ,

Computational chemistry serves as a powerful approach for elucidating the fundamental nature and energetics of molecule surface interactions. In particular, density functional theory (DFT) can offer detailed insights into the adsorption mechanisms governing hydrogen interactions with nanomaterial surfaces. Light element materials are natural candidates for hydrogen adsorbents because their low atomic weight provides high surface area to mass ratios. Extensive research has been devoted to the development of carbon-based, boron-based, magnesium-based and beryllium-based materials, all of which demonstrate significant potential for high-capacity hydrogen storage. For instance, Yuan et al. performed a density functional theory (DFT) study on doped porous boron nitride, revealing its promise for reversible hydrogen storage applications. The authors found that 2D BNC can store 10.7 wt % of hydrogen by adsorbing up to 12 H₂ molecules. Koukaras et al. used DFT calculations to examine how hydrogen adsorption and desorption energetics vary with magnesium nanocluster size, offering a theoretical explanation for these trends. Similarly, Shinde et al. employed DFT calculations to evaluate the hydrogen storage capabilities of beryllium oxide clusters, Be _n O _n (n = 2–8). Their results revealed that the (BeO) _n cluster can adsorb between 8 and 12 H₂ molecules, with adsorption energy values ranging from −0.10 and −0.12 eV that fall within the optimal range for reversible hydrogen storage. Notably, the (BeO)₃ cluster achieves a gravimetric hydrogen density of 24.4 wt %, surpassing the U.S. DOE’s ultimate target.

Robust computational methods are essential for quantitatively evaluating hydrogen storage in emerging nanomaterials. In this work, we develop a DFT-based protocol to elucidate the electronic structure and adsorption behavior of beryllium clusters (Be _n ) across a broad size range. Although bulk beryllium is known to be toxic and experimentally challenging to manipulate, small Be_n clusters have been detected in the gas phase, providing a solid experimental foundation for theoretical investigations. Mass spectrometry has confirmed the formation and stability of Be _n clusters, while photoelectron spectroscopy has characterized the electronic structures of Be _n ^– species for n = 2–5. These observations validate the existence of finite beryllium clusters and support their use as fundamental model systems for exploring hydrogen adsorption at the nanoscale.

Herein, we establish a systematic correlation between cluster size and hydrogen adsorption energetics, enabling predictive insight into gravimetric storage capacity as a function of structural evolution. The investigated size range (Be₃–Be_{2 5}) corresponds to subnanometer finite clusters that remain far from the bulk regime. In this domain, discrete electronic structures and size-dependent geometrical effects dominate, in contrast to bulk beryllium, where periodic metallic bonding governs the physical behavior. The results obtained not only advance the fundamental understanding of hydrogen storage in small clusters but also provide transferable design principles for developing high-performance, experimentally viable light-element materials suited to clean and sustainable energy technologies.

2. Computational Methodology

The starting geometries of the beryllium clusters were obtained from previously reported literature sources. To evaluate the suitability and select the most appropriate exchange-correlation functionals for the present study, the geometries of several smaller Be _n clusters (n = 3–6) were further optimized, compared, and reassessed using ab initio methods, including second-order Mo̷ller–Plesset perturbation theory (MP2) and higher-level correlated approaches such as DLPNO-CCSD­(T). For the DFT calculations, a diverse selection of widely used and recently developed functionals was employed, encompassing generalized gradient approximation (GGA), hybrid GGA, and hybrid meta-GGA functionals, including CAM-B3LYP, M06-2X, PBE, ωB97X-V, and ωB97X-3c. All computations utilized the correlation-consistent cc-pVTZ basis set of triple-ζ quality, except for those employing the ωB97X-3c functional, which relies on a polarized valence double-ζ (vDZP) Gaussian basis set specifically optimized for molecular DFT applications. The vDZP basis set yields negligible basis set superposition error (BSSE), achieving accuracy comparable to that of triple-ζ basis sets, with residual errors effectively corrected by the D4 scheme. All calculations were carried out using the ORCA program, employing the DefGrid3 integration grid and TightOpt settings.

The detailed characteristics of charge transfer were analyzed using the charge decomposition analysis (CDA) and its extended version (ECDA) were conducted on all H₂/Be _n complexes. The calculations were carried out using the Multiwfn program.

The fully ab initio results for representative small clusters, such as Be _n (n = 3–6), were systematically compared with corresponding geometry optimizations and single-point energy calculations performed using DFT. The primary properties of interest in the present study are the binding energy per atom E _b and the average bond length D_Be–Be, which act as a criterion for assessing the accuracy of both energetic and geometric predictions. The binding energy per atom is defined by the following relation:

$$E_{\mathrm{b}}=\frac{E_{\mathrm{cluster}}-n\times E_{\mathrm{Be}}}{n}$$ 1

where n is the number of beryllium atoms in the cluster, E _cluster is the total energy of the cluster, and E _Be is the total energy of a single isolated beryllium atom.

To ensure that the optimized structures correspond to true minima on the potential energy surface, vibrational frequency calculations were performed for all functionals and MP2, confirming the absence of imaginary frequencies.

To quantify the interaction strength between single H₂ and beryllium clusters, the adsorption energies E _ads (in eV) of H₂ on Be _n were evaluated using the following formula:

$$E_{\mathrm{ads}}=E_{{\mathrm{H}}_2/\mathrm{cluster}}-[E_{\mathrm{cluster}}+E_{{\mathrm{H}}_2}]$$ 2

where E _ad represents the hydrogen adsorption energy, E _(H2/cluster) is the total electronic energy of the Be _n cluster with adsorbed hydrogen molecules, E _(cluster) is the total energy of the isolated Be _n cluster, E _(H2) is the total energy of an isolated H₂ molecule, and m is the number of adsorbed H₂ molecules.

3. Results and Discussion

3.1. Structural Benchmarking and Stability Analysis of Be _n Clusters

A rigorous assessment of hydrogen storage properties requires a reliable description of the underlying structural and electronic characteristics of the storage medium. For beryllium clusters, achieving this accuracy is particularly challenging due to strong electron correlation, delocalized bonding, and slight multireference character inherent to small Be systems, as further supported here by the calculated T1 diagnostic values for Be₃–Be₆. These features make the optimized geometries highly sensitive to the level of theory, and consequently, the predictive value of adsorption energetics relies directly on the robustness of the computational method selected. Because experimentally resolved geometries for Be _n clusters remain unavailable, although the existence of small gas-phase Be clusters has been confirmed by mass spectrometry and photoelectron spectroscopy studies, theoretical benchmarking is essential to ensure the validity of computed structures. To establish a reliable methodological foundation, we evaluated the performance of several widely used density functional theory (DFT) approaches (CAM-B3LYP, M06-2X, PBE, ωB97X-V, and ωB97X-3c) alongside MP2, and compared them to high-accuracy DLPNO-CCSD­(T)/cc-pVTZ results. DLPNO-CCSD­(T) is well recognized as a near-benchmark method for small metal clusters but becomes prohibitively expensive for larger systems. Therefore, the benchmarking was performed on Be₃–Be₆, which represent the essential local bonding motifs encountered across larger clusters while remaining computationally tractable at the coupled-cluster level. Representative results for the average Be–Be bond lengths are summarized in Table .

1. Average Be–Be Bond Lengths (D_Be–Be, Å) for Be₃–Be₆ Clusters Calculated Using Selected DFT Functionals, MP2, and DLPNO-CCSD­(T).

cluster	property	MP2	CAM-B3LYP	M06-2X	PBE	ωB97X-V	ωB97X-3c	DLPNO-CCSD(T)
Be3	DBe–Be	2.178	2.165	2.202	2.190	2.187	2.181	2.175
Be4	DBe–Be	2.050	2.029	2.055	2.059	2.049	2.045	2.044
Be5	DBe–Be	2.030	2.009	2.034	2.041	2.033	2.041	2.031
Be6	DBe–Be	1.972	1.965	1.982	2.006	1.985	1.976	1.976

To further assess the reliability of the coupled-cluster reference calculations, the T1 diagnostic was examined for the benchmark clusters Be₃–Be₆ at the DLPNO-CCSD­(T)/cc-pVTZ level. The obtained values are 0.0219 for Be₃, 0.0210 for Be₄, 0.0201 for Be₅, and 0.0223 for Be₆, indicating a slight to moderate multireference character for these very small beryllium clusters. Despite this, the close agreement observed between DLPNO-CCSD­(T) and ωB97X-3c bond lengths confirms that the selected DFT protocol remains reliable for describing the structural and adsorption trends investigated in this work.

Across all systems, ωB97X-3c exhibits the closest agreement with DLPNO-CCSD­(T), maintaining deviations within 0.000–0.010 Å despite the use of a compact double-ζ basis (vDZP). Remarkably, its accuracy surpasses that of functionals using the larger def2-TZVP basis set. For example, ωB97X-V shows deviations up to +0.012 Å, M06-2X yields positive deviations up to +0.027 Å, CAM-B3LYP systematically underestimates Be–Be distances (−0.022 to −0.010 Å), and PBE displays the largest spread (+0.010 to +0.030 Å), consistent with its tendency to overdelocalize electron density in metallic systems. MP2 performs moderately well but remains less consistent overall. The combination of accuracy, size-independence, and computational efficiency clearly identifies ωB97X-3c/vDZP as the optimal choice for describing the structures of Be _n clusters. Consequently, this functional is adopted for all subsequent geometry optimizations and hydrogen adsorption calculations in this study.

Beryllium clusters (Be _n ) exhibit a rich isomeric structure, with the number of possible isomers increasing rapidly with cluster size. These systems have attracted significant attention due to their intriguing bonding characteristics and the challenges they pose to theoretical modeling. Extensive computational studies have focused on small Be _n clusters, facilitated by the relatively low atomic number of berylliums, which allows the application of high-level ab initio methods. Despite its apparent simplicity, beryllium presents substantial computational challenges due to strong electron correlation and configuration interaction effects. Consequently, Be _n clusters have become valuable benchmark systems for evaluating the performance of various quantum chemical methods.

Abyaz et al. used the USPEX technique (Universal Structure Predictor: Evolutionary Xtallography) and DFT simulations to find the global minimum structures of beryllium clusters (Be _n , n = 3–25). Based on the preliminary structures identified in this study, DFT calculations were performed using the ωB97X-3c functional to determine the ground-state geometries of Be _n clusters. Geometry optimizations were carried out for each cluster size, and the resulting lowest-energy configurations are presented in Figure .

1.

Optimized ground-state geometries of Be _n clusters (n = 3–25) with their exact symmetries and states (S: singlet and T: triplet states).

The structures of Be _n clusters exhibit various symmetries and configurations. The ground-state geometry of Be₃ adopts an equilateral triangular configuration with D_2h symmetry. The computed Be–Be bond length is 2.18 Å, in excellent agreement with the previously reported value of 2.17 Å by Abyaz et al. Be₄ adopts a compact tetrahedral geometry, representing a transition from planar to three-dimensional structures. Beyond Be₄, the clusters exhibit three-dimensional architectures. For instance, Be₅ features a slightly distorted trigonal bipyramidal structure, while Be₆ adopts a C_2v-symmetric double pyramidal configuration.

Be₇ adopts a pentagonal bipyramidal ground state, and Be₈ can be viewed as Be₇ with one additional surface atom. Clusters Be₉–Be_{1 5} share a triangular-prism core motif. Beyond this size, Be₁₇ and Be_{1 8} evolve from Be_{1 6} through successive atom additions and structural relaxation. The ground-state structures of Be₂0 and Be_{2 1} adopt double-icosahedral frameworks, each featuring a capping atom and exhibiting C_2v symmetry. Notably, for cluster sizes in the range of n = 15–22, one beryllium atom is fully encapsulated within the cluster framework. As the cluster size increases to n = 23, a second interior atom appears, a feature that persists in Be_{2 4} and Be_{2 5} as well. Thus, the structures of Be_{2 3}, Be_{2 4}, and Be_{2 5} contain two internal atoms within double-icosahedral architectures. Among them, the most stable configuration of Be_{2 4} corresponds to a bicapped truncated octahedron with S₄ symmetry, while Be_{2 5} adopts a distorted icosahedral geometry. Overall, the ground-state evolution of Be _n clusters (3 ≤ n ≤ 25) evolves from planar structures (e.g., Be₃) to three-dimensional layered motifs with internal atoms for n > 14, predominantly based on icosahedral building blocks.

We now assess the thermodynamic stability of Be _n clusters by examining their intrinsic energetic properties. The stability of the ground-state structures is evaluated using the binding energy per atom (E _b), the second-order energy difference (Δ² E) and the fragmentation energy (E _frg). The calculated values of E _b are provided in Table S1 (supplementary data) and plotted as a function of cluster size n in Figure a. As observed, the average binding energy per atom increases with cluster size, suggesting that larger clusters are more thermodynamically stable and may form more readily than smaller ones. The binding energies for the studied beryllium clusters range from −0.44 to −2.29 eV, which aligns well with the values reported by Abyaz et al. Moreover, the predicted binding energies of Be _n structures are significantly higher than those reported for other metal clusters such as, Mg _n (−0.45 to −0.82 eV) and Na _n (−0.56 to −0.63 eV), highlighting the remarkable stability of beryllium clusters.

2.

Calculated binding energies per atom (a), second difference in the total energy (b) and fragmentation energies (c) of neutral Be _n clusters as a function of cluster size n.

The fragmentation energy of studied beryllium clusters is computed using the following equation:

$$E_{\mathrm{frg}}=E_{{\mathrm{Be}}_{n-1}}+E_{\mathrm{Be}}-E_{{\mathrm{Be}}_n}$$ 3

A negative fragmentation energy indicates that the reaction is exothermic and feasible under experimental conditions. Elevated fragmentation energy (E _frg) values indicate that greater energy is necessary to dissociate a Be atom from the host cluster, signifying enhanced Be–Be interactions and increased stability. The computed fragmentation energy of the most stable structures as a function of cluster size n are presented in Figure c and Table S1, where local maxima occur at n = 4, 10, 14, 17, 21, and 24. The same trend of variation was found by Abyaz et al., thereby supporting the reliability and consistency of our computational methodology.

The second-order energy difference (Δ² E) serves as a key indicator of cluster stability. The size dependence of Δ² E for Be _n clusters is illustrated in Figure c, and the values are listed in Table S1. Δ² E is defined as

$${\mathrm{\Delta }}^{2}E(n)=E({\mathrm{Be}}_{n+1})+E({\mathrm{Be}}_{n-1})-2E({\mathrm{Be}}_n)$$ 4

As shown in Figure c, Δ² E exhibits an oscillatory behavior with increasing cluster size. The peaks at n = 4, 10, 12, 14, 17, 21 and 22, indicate enhanced relative stability for Be₄, Be₁0, Be_{1 2}, Be_{1 4}, Be₁₇, Be_{2 1}, and Be_{2 2} compared to their neighboring clusters. This trend in the Δ² E profile is consistent with the variations observed in the binding energy E _b, and fragmentation energy E _frg plots.

The benchmarking analysis and stability evaluation establish a robust methodological foundation for the subsequent investigation of hydrogen adsorption on Be _n clusters. With accurate geometries and well-characterized stability trends, the following sections explore how cluster size modulates H₂ binding energies, reversible adsorption behavior, and practical hydrogen storage performance.

3.2. Single Hydrogen Adsorption on Be _n Clusters (n = 3–25)

The interaction of a single hydrogen molecule with beryllium clusters provides the fundamental basis for evaluating the hydrogen storage potential of these systems. Because the energetics and structural response associated with the first adsorbed H₂ molecule strongly influence subsequent uptake, a thorough understanding of single-H₂ adsorption across the entire Be _n series is essential. All adsorption geometries were optimized at the ωB97X-3c level, whose accuracy was validated through DLPNO-CCSD­(T) single-point calculations (Table S3). The resulting correlation plot (Figure S1) exhibits an excellent linear correlation (R ² = 0.98) with a very low RMSD of 0.032 eV, demonstrating that ωB97X-3c reproduces high-level coupled-cluster energetics at a negligible fraction of the computational cost. This confirmation establishes ωB97X-3c as a reliable and efficient method for modeling Be–H₂ interactions.

The Be₃ cluster offers a representative starting point for understanding the adsorption behavior. As illustrated in Figure , 13 initial adsorption configurations were generated by sampling top (T), bridge (B), and hollow (H) surface sites with both parallel (∥) and perpendicular (⊥) orientations of the H₂ molecule. After relaxation, all configurations converged to weakly bound physisorption complexes, with adsorption energies spanning −0.01 to −0.13 eV (Table S2). Throughout the Be₃ adsorption landscape, the H–H bond remains nearly unperturbed (0.74–0.77 Å) relative to the experimental value of 0.74 Å as well as previously reported computational results, , and the Be–Be framework shows negligible distortion. The most stable configuration occurs when H₂ adopts a parallel orientation above the triangular hollow site (H _∥‑2), with a Be···H distance of 2.40 Å (Figure ), illustrating that even small clusters provide electronically favorable system for nondissociative physisorption.

3.

Possible adsorption sites of H₂ on the Be₃ cluster, namely, T (top of the Beryllium atom), B (bridge of the Be–Be bond), and H (hollow is the triangle center).

4.

Optimized configurations of molecular hydrogen interacting with Be _n clusters, showing adsorption sites and interaction patterns.

Extending the analysis to larger clusters (Be₄–Be_{2 5}) reveals a rich and size-dependent evolution of adsorption behavior, as shown in Figure . Despite the structural diversity of the Be _n series from planar motifs (Be₃–Be₄) to bicapped icosahedral frameworks (Be_{2 4}–Be_{2 5}), the majority of clusters retain clear signatures of physisorption. In these systems, adsorption energies fall within the range −0.10 to −0.52 eV, H–H distances remain in the interval 0.76–0.82 Å, and Be–H separations lie between 1.52 and 1.80 Å, indicating modest polarization of the H₂ molecule without bond activation. Importantly, cluster geometries remain largely unaltered upon adsorption, demonstrating that the interaction is surface-localized and does not compromise the structural integrity of the Be skeleton. A strong correlation emerges between moderate adsorption energies, slight H–H elongation, and enhanced electron donation from the cluster (Table ), especially for Be₆, Be₇, Be₈, Be_{1 2}, Be_{1 4}, Be_{1 5}, Be_{1 8}, and Be_{2 4}. These clusters exhibit the largest charge donation values (0.16–0.27 e), consistent with their stronger yet still reversible adsorption energies and short Be–H distances, reflecting enhanced polarization without triggering dissociation.

2. Thermodynamic Parameters (eV) and Charge Transfer analysis (CDA/ECDA) for H₂ Adsorption on Be _n Clusters .

	thermodynamic parameters			CDA				ECDA
system	E ads	ΔH°	ΔG°	d	b	d – b	r	net electrons obtained by Be n
H2/Be3	–0.13	–0.09	–0.19	0.065	0.026	0.039	–0.197	0.035
H2/Be4	–0.02	0.00	–0.05	–0.001	0.009	–0.010	–0.015	–0.012
H2/Be5	–0.01	–0.02	–0.02	–0.002	0.000	–0.002	–0.009	–0.002
H2/Be6	–0.32	–0.24	–0.46	0.245	0.041	0.204	–0.184	0.193
H2/Be7	–0.67	–0.56	–0.85
H2/Be8	–0.43	–0.34	–0.58	0.269	0.077	0.191	–0.227	0.194
H2/Be9	–0.33	–0.21	–0.50
H2/Be10	–0.14	–0.10	–0.23	0.157	–0.002	0.159	–0.159	0.178
H2/Be11	–0.31	–0.34	–0.34
H2/Be12	–0.38	–0.28	–0.52	0.256	0.069	0.187	–0.136	0.199
H2/Be13	–0.34	–0.25	–0.48	0.221	0.048	0.173	–0.147	0.191
H2/Be14	–0.35	–0.23	–0.51	0.223	0.058	0.165	–0.139	0.180
H2/Be15	–0.34	–0.27	–0.49	0.250	0.081	0.169	–0.150	0.197
H2/Be16	–0.19	–0.12	–0.31	0.182	0.013	0.170	–0.157	0.190
H2/Be17	–0.02	–0.01	–0.05	0.139	0.002	0.137	–0.186	0.148
H2/Be18	–0.42	–0.45	–0.41	0.265	0.097	0.168	–0.152	0.199
H2/Be19	–1.97	–1.66	–2.38
H2/Be20	–0.16	–0.05	–0.31	0.167	–0.004	0.171	–0.175	0.191
H2/Be21	–0.19	–0.07	–0.27	0.159	0.023	0.136	–0.165	0.148
H2/Be22	–0.22	–0.06	–0.50	0.153	0.027	0.126	–0.169	0.135
H2/Be23	–3.03	–2.83	–3.33
H2/Be24	–0.52	–0.32	–0.81	0.251	0.062	0.189	–0.144	0.231
H2/Be25	–0.16	–0.08	–0.31	0.203	0.030	0.173	–0.185	0.197

^a d: the number of electrons donated from H₂ to Be _n nanoclusters; b: the number of electrons back-donated from the Be _n nanocluster to H₂; r: the number of electrons involved in repulsive polarization.

A small subset of clusters, namely Be₄, Be₅, and Be₁₇, exhibit extremely weak interactions (−0.02 to −0.05 eV), accompanied by long Be···H distances (>3.3 Å) and invariant H–H bond lengths. These weakly bound systems are unable to stabilize H₂ at practical storage conditions and therefore do not contribute effectively to reversible uptake. At the opposite extreme, Be_{1 9} and Be_{2 3} depart sharply from the physisorption trend and undergo dissociative chemisorption. Their optimized geometries display complete H–H bond cleavage with large separations (2.72–2.89 Å) and formation of robust Be–H bonds (1.45–1.47 Å), which are close to the experimental Be–H bond length in beryllium dihydride (1.33 Å). The corresponding adsorption energies (−2.0 to −3.0 eV) fall well outside the reversible storage window, indicating strong covalent interactions reminiscent of Kubas-type bonding. Such chemisorption, while thermodynamically favorable, is associated with high desorption barriers and slow kinetics and is therefore not suitable for practical hydrogen release. A similar observation was reported by Kevin et al. for the adsorption of H₂ on a cobalt cluster (Co₆), where the hydrogen molecules dissociate into individual H atoms, resulting in a binding energy of −1.51 eV and an elongated H–H distance of 0.80 Å.

In the absence of experimental or benchmark data for validation, close agreement between interaction energy and binding enthalpy is typically used to lend credibility to theoretical calculations. Furthermore, the standard enthalpy change (ΔH°) and Gibbs free energy change (ΔG°) are essential thermodynamic descriptors for evaluating the driving forces behind complex formation. Accordingly, thermodynamic parameters, including ΔH° and ΔG°, were calculated at a temperature of 298.15 K and a pressure of 1 atm, with zero-point energy (ZPE) corrections applied. The thermochemical descriptors presented in Table reinforce these structural and energetic trends. For most Be _n clusters, both ΔH° and ΔG° values are negative, indicating that adsorption is exothermic and spontaneous under ambient conditions. Physisorptive systems show ΔH° values ranging from −0.01 to −0.52 eV and ΔG° values between −0.02 and −0.80 eV, consistent with stable yet reversible complex formation. In contrast, the chemisorptive Be_{1 9} and Be_{2 3} clusters yield exceptionally large negative ΔH° (up to −2.82 eV) and ΔG° (up to −3.33 eV), further confirming H–H bond rupture and strong Be–H covalent bonding.

Charge-transfer analysis (Table ) provides deeper insight into the electronic origin of the observed adsorption trends and reveals a clear and systematic correlation between interaction strength, structural perturbation, and electron redistribution. In all physisorptive complexes, the amount of charge donated from the σ­(H₂) orbital toward the empty orbitals of the Be _n cluster (d) is consistently larger than the corresponding back-donation from Be _n into σ*­(H₂) (b). This imbalance (d > b) is characteristic of polarization-dominated physisorption, where the H₂ molecule remains electronically intact and only experiences mild distortion. The negative values of the repulsive polarization term (r) further confirm that electrostatic induction, rather than covalent interaction, governs the stabilization of these weakly interacting complexes.

Clusters exhibiting the highest physisorption energies, specifically Be₆, Be₈, Be_{1 2}, Be_{1 4}, Be_{1 5}, Be_{1 8}, and Be_{2 4} also display the largest charge-donation values, typically exceeding 0.20 e. This enhanced donation correlates strongly with (i) the shortest Be···H distances (1.52–1.57 Å) and (ii) moderate H–H elongations (0.79–0.82 Å). Such elongation, while modest, indicates that the electron density accumulated on the H₂ antibonding region remains insufficient to break the H–H bond, but contributes to a slight weakening consistent with strong physisorption at the edge of the reversible adsorption window. These clusters represent the optimal balance between interaction strength and molecular integrity and are therefore the most promising motifs for reversible hydrogen storage. In contrast, weakly interacting clusters such as Be₄, Be₅, and Be₁₇ show negligible total charge transfer, with d – b values approaching zero. Correspondingly, their adsorption energies are extremely small (−0.01 to −0.05 eV), and both H–H and Be–Be bond lengths remain essentially unchanged. These characteristics confirm the absence of meaningful electronic interaction and highlight that their surfaces do not provide the electrostatic environment necessary for stabilizing H₂ adsorption.

Overall, the combined analysis of optimized geometries, adsorption energies, thermodynamic quantities, and charge-transfer metrics across Figures , , Tables , S2 and S3 reveals a clear structure–property relationship governing H₂ interaction with Be _n clusters. Planar or open three-coordinate motifs (Be₃–Be₅) support only weak adsorption, while the majority of intermediate and larger clusters (Be₆–Be_{1 6}, Be_{1 8}, Be₂₀–Be_{2 2}, Be_{2 4}–Be_{2 5}) fall within the ideal reversible physisorption range (−0.10 to −0.60 eV). Very large clusters containing interior atoms can either preserve physisorption or transition into chemisorption depending on local electron density distribution, as illustrated by Be_{1 9} and Be_{2 3}. Among the entire series, Be_{1 4}, Be_{1 8}, and Be_{2 4} are identified as the most promising molecular-scale motifs, combining moderate binding energies, small structural perturbations, favorable thermodynamic parameters, and substantial yet nondissociative charge redistribution. These findings provide a robust foundation for understanding the subsequent multihydrogen adsorption behavior and for identifying cluster sizes with optimal characteristics for reversible hydrogen storage.

3.3. Multiple H₂ Adsorption and Practical Reversible Storage Performance

To evaluate the practical hydrogen storage capability of beryllium clusters, it is essential to determine how these systems behave under incremental hydrogen loading and whether they preserve reversible adsorption–desorption characteristics under operational conditions. Based on the stability trends and the electronic adsorption profiles described in previous sections, three representative clusters were selected: Be₃, Be_{1 4} and Be_{2 4}. This choice provides a systematic sampling of the structural evolution of Be _n systems from small planar units to medium-sized 3D motifs and finally to large, highly stable cage-like architectures. Be₃ represents the smallest stable geometry and constitutes an important reference case for minimalistic hydrogen binding. Be_{1 4} belongs to the set of “magic-number” clusters with enhanced thermodynamic stability (Δ₂ E and E _frg maxima), while Be_{2 4} exhibits one of the most stable 3D frameworks in the series, with internal atoms and a multifaceted surface. This size diversity allows directly probing how geometric complexity, surface curvature, and electronic structure govern H₂ uptake and reversibility. These systems also exhibit low, medium, and high hydrogen adsorption energies.

Sequential adsorption was carried out by adding H₂ molecules one at a time to the most favorable sites identified with single H₂, followed by full structural optimization. For each loading level, the average adsorption energy ${\stackrel{\circledR }{E}}_{\mathrm{ads}}$ was computed using eq , providing a quantitative descriptor of the thermodynamic evolution of binding strength as the cluster approaches saturation. Adsorption was continued until the threshold ${\stackrel{\circledR }{E}}_{\mathrm{ads}}$ = −0.10 eV/H₂ was reached, defining the lower limit of the reversible physisorption interval.

$${\stackrel{\circledR }{E}}_{\mathrm{ads}}(N)=\frac{E(N.{\mathrm{H}}_2/{\mathrm{Be}}_n)-E({\mathrm{Be}}_n)-NE({\mathrm{H}}_2)}{N}$$ 5

where E(N. H₂/Be _n ) is the total energy of the hydrogen-saturated complex and N is the number of adsorbed hydrogen molecules.

To determine the hydrogen uptake capacity, the gravimetric hydrogen density (wt %) was calculated using the following equation:

$$\mathrm{wt}\%=\frac{N{M}_{{\mathrm{H}}_2}}{N{M}_{{\mathrm{H}}_2}+M_{\mathrm{cluster}}}\times 100$$ 6

where, M _H2, and M _cluster are the masses of H₂ and Be _n cluster, respectively. This expression directly enables the determination of both the total gravimetric capacity (C _T), corresponding to the maximum number of adsorbed H₂ molecules at saturation, and the reversible working capacity (C _E), defined as the fraction of hydrogen that can be stored and released under practical operating conditions. Table summarizes the average adsorption energies ${\stackrel{\circledR }{E}}_{\mathrm{ads}}$ ­(in eV/H₂) and the corresponding H–H bond lengths obtained for multi-H₂ adsorption on the selected Be _n clusters using the ωB97X-3c/vDZP level of theory.

3. Calculated Adsorption Energy ${\stackrel{\circledR }{E}}_{\mathrm{ads}}$ in eV/H₂, the Corresponding H–H Bond Lengths in Å, and the Desorption Energy E _des in eV, Using the wB97x-3c/vDZP Level of Theory.

	Be 14			Be 24
N (H 2 )	${\stackrel{\circledR }{E}}_{\mathrm{ads}}$	E des	H–H	${\stackrel{\circledR }{E}}_{\mathrm{ads}}$	E des	H–H
1	–0.35		0.79	–0.52		0.80
2	–0.33	–0.31	0.79	–0.36	–0.26	0.79
3	–0.23	–0.21	0.77	–0.25	–0.22	0.77
4	–0.22	–0.20	0.77	–0.20	–0.16	0.76
5	–0.22	–0.21	0.77	–0.16	–0.13	0.76
6	–0.20	–0.16	0.77	–0.14	–0.11	0.76
7	–0.18	–0.11	0.77	–0.13	–0.12	0.76
8	–0.16	–0.11	0.77	–0.12	–0.09	0.75
9	–0.15	–0.10	0.77	–0.11	–0.08	0.75
10	–0.14	–0.10	0.76	–0.11	–0.06	0.75
11	–0.13	–0.10	0.76	–0.10	–0.07	0.75
12	–0.12	–0.10	0.76	–0.10	–0.03	0.75
13	–0.12	–0.08	0.76
14	–0.11	–0.07	0.76
15	–0.11	–0.06	0.76
16	–0.10	–0.05	0.75
17	–0.10	–0.04	0.75
18	–0.10	–0.04	0.75
19	–0.10	–0.04	0.75
20	–0.10	–0.02	0.75

For the Be₃ cluster, the average adsorption energies exhibit a gradual decrease in magnitude as additional H₂ molecules are adsorbed. The first few adsorption events fall well within the reversible physisorption regime, with ${\stackrel{\circledR }{E}}_{\mathrm{ads}}$ values ranging from −0.12 to −0.10 eV/H₂. Beyond this point, the stabilizing interaction weakens sharply, and the mean adsorption energy rises above the practical threshold (−0.10 eV/H₂), indicating the onset of steric and electronic saturation. The maximum number of H₂ molecules that can be stored while remaining within the reversible physisorption window is therefore limited to N = 2 H₂ molecules (Figure S2), yielding a total gravimetric storage capacity of C _T = 13 wt % for Be₃ (Table ). The H–H bond lengths for all adsorbed molecules remain close to the free-molecule value (0.74 Å), with only slight elongation (0.75 and 0.76 Å), confirming that adsorption proceeds strictly through nondissociative physisorption. No significant distortion of the Be₃ geometry is observed throughout loading, indicating that this small planar cluster possesses a sufficiently open and electronically isotropic surface to accommodate several H₂ molecules through weak dispersion-dominated interactions.

4. Calculated Thermodynamic and Kinetic Desorption Parameters (T _D in K, τ in ns) and Reversible Storage Metrics (N _A, N _D, N _P, C _E) for Be₃, Be₁₄, and Be₂₄ .

system	Be 3	Be 14	Be 24
T D (1 bar)	155.60	127.59	130.74
T D (3 bar)	177.01	145.17	148.75
T D (30 bar)	248.86	204.09	209.13
T D (100 bar)	315.91	259.08	265.48
τ (233 K)	0.43	0.14	0.16
τ (298 K)	0.11	0.05	0.05
τ (358 K)	0.05	0.03	0.03
τ (400 K)	0.03	0.02	0.02
N T	2	20	12
C T (wt %)	13.0	24.2	10.1
N A	0.06	19.85	11.99
N D	0.01	6.24	11.78
N P	0.05	13.61	0.21
C E (wt %)	0.43	17.86	0.20

The adsorption behavior changes markedly for the medium-sized Be_{1 4} cluster, which consistently emerges as the most efficient hydrogen host in this study. Unlike the small Be₃ cluster, which provides only a limited number of adsorption sites, or the large Be_{2 4} cluster, whose gravimetric efficiency decreases due to its higher mass, Be_{1 4} offers an optimal balance of structural openness, accessible surface area, and electronic stabilization. As successive H₂ molecules are added, the average adsorption energy decreases progressively and smoothly from −0.35 eV (N = 1) to −0.10 eV (N = 20), reflecting a gradual filling of equivalent or near-equivalent sites rather than a rapid destabilization of the adsorption process (Table ). This monotonic and moderate decline is characteristic of systems in which the adsorption sites possess similar local electronic environments, suggesting that hydrogen molecules interact predominantly through uniform physisorption governed by dispersion and polarization forces.

Importantly, all adsorption energies fall within the reversible physisorption window (−0.10 to −0.60 eV) up to N = 20, without crossing into excessively weak or too strongly bound regimes. This indicates that the Be_{1 4} cluster features a large number of energetically equivalent adsorption pockets capable of stabilizing H₂ molecules without inducing dissociation. Consistent with this interpretation, the H–H bond lengths remain in the molecular range (0.75–0.79 Å) for all adsorption events, showing minimal elongation and confirming that no bond activation or weakening occurs even at high loadings (Figure.S3). The fact that the structural integrity of both the H₂ molecules and the Be_{1 4} framework is preserved throughout the adsorption sequence reflects the robustness of the cluster and the dominance of nondissociative interactions.

Because Be_{1 4} sustains strong enough but fully reversible interactions up to 20 adsorbed H₂ molecules, it achieves an exceptionally high theoretical gravimetric capacity C _T of 24.2 wt % (Table ), the highest among the clusters examined.

The Be_{2 4} cluster, with its double-icosahedral geometry, offers several adsorption sites but displays a faster decline in interaction strength. The first two H₂ molecules adsorb strongly (−0.52 and −0.36 eV), after which the average interaction energy progressively weakens, reaching −0.11 eV at N = 11 and −0.10 eV at N = 12. At higher coverages, the interaction falls below the physisorption threshold, indicating that reversible adsorption is limited to the first 12 molecules. The H–H distances remain in the molecular regime (0.75–0.80 Å) for all adsorbed species. Owing to its larger intrinsic mass, Be_{2 4} reaches a theoretical gravimetric capacity of 10.1 wt %, lower than Be_{1 4} despite having more adsorption sites.

Overall, the results reveal a clear size-dependent trend. Small clusters such as Be₃ are limited by the number of available adsorption sites, while very large clusters such as Be_{2 4} suffer from reduced gravimetric performance. Intermediate-sized clusters exemplified by Be_{1 4} offer the best balance between adsorption strength, structural stability, and mass efficiency. Be_{1 4} uniquely maintains average adsorption energies in the optimal physisorption window up to high hydrogen loadings and delivers the highest theoretical storage capacity in the series.

While adsorption energies provide an initial indication of whether hydrogen binds within the desirable reversible range, they are not sufficient on their own to assess the operational viability of a hydrogen storage material. For practical onboard applications, a suitable storage medium must not only adsorb H₂ with moderate interaction energies but must also release it under realistic temperature–pressure conditions and within acceptable kinetic time scales. Accordingly, a comprehensive evaluation of desorption behavior is essential to determine whether the stored hydrogen can be delivered efficiently when needed. To this end, three complementary descriptors were examined: the desorption energy (noted E _des), the desorption temperature (T _D) and the desorption time (τ).

The desorption energy, defined as the energy required to remove a single H₂ molecule from the beryllium cluster surface, is calculated using the following equation:

$$\begin{array}{cccc}& E_{\mathrm{des}}=E_{N{\mathrm{H}}_2/\mathrm{cluster}}-(E_{(N-1){\mathrm{H}}_2/\mathrm{cluster}}+E_{{\mathrm{H}}_2})& & \end{array}$$ 7

E _{(N–1)H2/cluster} represents the total energy of the cluster with (N – 1) adsorbed hydrogen molecules.

The desorption energies (E _des) reported in Table reveal distinct release behaviors for the three selected clusters. For Be₃, the only accessible physisorption pockets yield uniformly weak desorption energies (≈ −0.11 eV), indicating that H₂ is easily released but that the cluster cannot retain more than two molecules, resulting in negligible storage capacity. In contrast, Be_{1 4} exhibits a smooth and well-behaved desorption-energy profile, decreasing from moderate values at low loading (−0.31 to −0.16 eV for N = 2–6) to weak physisorption (−0.10 to −0.05 eV) as the cluster approaches saturation (N = 10–20). This continuous but reversible weakening of Edes confirms that Be_{1 4} offers a large set of uniformly accessible adsorption sites while maintaining fully physisorptive, nonactivated H₂ binding across all 20 molecules. For the larger Be_{2 4} cluster, moderately strong desorption energies are found only for the first few adsorbed molecules (−0.26 to −0.16 eV), after which Edes rapidly decreases to very weak values (−0.08 to −0.03 eV), reflecting steric crowding and limited availability of effective adsorption pockets. Overall, Be_{1 4} displays the most favorable desorption-energy landscape, combining sufficient binding strength for retention with fully reversible release conditions necessary for practical hydrogen-storage operation.

The desorption temperature of H₂ is a critical parameter in evaluating the practical performance of hydrogen storage materials. If the desorption temperature is either too high or too low, it can negatively impact the effectiveness of these materials. Therefore, an optimal desorption temperature is essential for balancing storage stability and release efficiency. Typically, the desorption temperature, T _D, can be estimated using the Van’t Hoff equation:

$$\begin{array}{cccc}& T_D=-\frac{E_{\mathrm{ads}}}{k_{\mathrm{B}}(\frac{\Delta S}{R}-\ln P)}& & \end{array}$$ 8

T _D incorporates variables such as adsorption energy (E _ads), the equilibrium pressure (P), and the entropy change (ΔS = 75.44 J mol^–1 K^–1) of hydrogen molecules transitioning from gaseous phase to liquid phase. Additionally, k _B and R represent the Boltzmann constant and the ideal gas constant, respectively.

Desorption time (τ) is another critical parameter in evaluating the performance of hydrogen storage materials, as it reflects how rapidly hydrogen molecules can be released from the adsorbent surface. The desorption time can be estimated using the following equation:

$$\begin{array}{cccc}& \tau =\frac{\exp (-\frac{E_{\mathrm{ads}}}{k_{\mathrm{B}}T})}{{\nu }_0}& & \end{array}$$ 9

where v ₀ is the attempt frequency (10¹² s^–1).

The desorption temperatures (T _D) reported in Table provide key insight into the operational conditions required for hydrogen release from the three clusters. For Be₃, T _D at 1 bar (155.6 K) indicates that hydrogen is only weakly retained and would desorb readily even at low cryogenic temperatures, consistent with its very small Edes values. Increasing pressure to 100 bar raises T _D to 315.9 K, but the cluster still remains too weakly binding to ensure significant retention under practical storage conditions. In contrast, the Be_{1 4} cluster exhibits highly favorable desorption characteristics, with T _D values of 127.6 K (1 bar) and 259.1 K (100 bar). These temperatures fall squarely within the operational window of cryo-compressed and near-ambient hydrogen-storage conditions, confirming that the moderate adsorption energies observed for Be_{1 4} translate directly into efficient and controllable release. The Be_{2 4} cluster follows a similar trend, with T _D values of 130.7–265.5 K over the 1–100 bar range, but its lower uptake capacity limits its practical relevance compared to Be_{1 4}.

Kinetic desorption times (τ) complement the thermodynamic picture by quantifying the rate at which hydrogen can escape from the clusters at a given temperature. All three systems exhibit extremely rapid desorption kinetics, with τ values in the subnanosecond regime across the temperature range 233–400 K. For Be₃, τ decreases from 0.43 ns at 233 K to 0.03 ns at 400 K, reflecting the weak H₂ binding. Be_{1 4} and Be_{2 4} display even faster desorption kinetics, with τ values of 0.14–0.02 and 0.16–0.02 ns, respectively, over the same temperature range. These ultrafast kinetics demonstrate the absence of activation barriers and confirm that all adsorbed hydrogen molecules remain in a purely physisorbed state. Among the three systems, Be_{1 4} again exhibits the optimal balance between sufficiently strong retention (moderate T _D values) and instantaneous desorption kinetics, reinforcing its status as the most promising candidate for practical reversible hydrogen storage.

While adsorption energies, desorption temperatures, and kinetic release times characterize the intrinsic thermodynamic behavior of the H₂/cluster interaction, they do not directly quantify how much hydrogen can be reversibly cycled under realistic operating conditions. To evaluate the operational storage performance, we calculate the grand canonical partition function z, the hydrogen chemical potential μ­(P, T), and the resulting average number of adsorbed molecules N _avg (P, T), which together determine the reversible uptake N _A, the desorbed fraction N _D, and ultimately the usable storage N _P = N _A – N _D and C _E (wt %). In this framework, the adsorption probability of each configuration enters through the grand canonical partition function (z). The equation to calculate (z) is defined as follows:

$$z=1+\sum _{i=1}^n\exp \left(\frac{E_{i,\mathrm{ads}}-\mu (P,T)}{k_{\mathrm{B}}T}\right)$$ 10

where n is the maximum number of adsorbed H₂ molecules, E _i,ads is the adsorption energy of the ith H₂ molecule, k _B is the Boltzmann constant (1.38 × 10^–23 J K^–1), and μ­(P, T) is the chemical potential of molecular hydrogen in the gas phase at a given pressure (P) and temperature (T).

The chemical potential μ­(P, T) can be determined from thermodynamic relations or from experimental data, and it is given by

$${\mu }_{{\mathrm{H}}_2}(P,T)=\mathrm{\Delta }H+T\mathrm{\Delta }S+k_{\mathrm{B}}\mathrm{Tln}\frac{P}{P_0}$$ 11

where ΔH, ΔS, and P ₀ represent the enthalpy change, entropy change, and standard atmospheric pressure (1.01 × 10⁵ Pa), respectively. In this expression, the term ΔH + TΔS describes the free energy contribution of the hydrogen gas phase, while the logarithmic term accounts for the dependence on external pressure. The values of ΔH and ΔS were taken from the experimental database, which provides reliable thermodynamic parameters for H₂ adsorption and desorption on metal-decorated systems. The average number of adsorbed hydrogen molecules at a given pressure and temperature, N _avg(P, T), is derived from the partition function and is expressed as

$$N_{\mathrm{avg}}(P,T)=N_{\mathrm{T}}\left[\frac{z-1}{z}\right]$$ 12

As shown in Table , the three representative clusters exhibit sharply contrasting behaviors despite displaying similar desorption-temperature trends and ultrafast release kinetics in Figure . The small Be₃ cluster stores only trace amounts of hydrogen reversibly (N _P = 0.05), corresponding to a negligible reversible capacity of 0.43 wt %, which is consistent with its limited surface area and the very weak adsorption energies observed in Table . For the larger Be_{2 4} cluster, although the total uptake is moderate, the number of molecules released under practical operating conditions remains extremely small (N _P = 0.21), yielding a reversible capacity of just 0.20 wt %. This is attributed to the presence of deeper adsorption sites and local trapping effects, which hinder efficient desorption despite favorable kinetics.

5.

Average number of H₂ molecules (N _ave) adsorbed on (a) Be₃, (b) Be₁₄, and (c) Be₂₄ as a function of practical temperatures and pressures.

In striking contrast, the medium-sized Be_{1 4} cluster displays a uniquely advantageous balance between adsorption capacity, desorption thermodynamics, and kinetic accessibility. It releases 13.61 H₂ molecules under operating conditions, the highest among the studied systems, which translates into an exceptional reversible capacity C _E of 17.86 wt % (Table ). This superior performance is also reflected in the nearly ideal desorption-response contours in Figure , where Be_{1 4} exhibits smooth desorption transitions across the full P, T domain without entering strongly bound or kinetically hindered regions. The combination of uniformly distributed adsorption sites, moderate adsorption energies (−0.10 to −0.35 eV), and structural stability under multi-H₂ loading makes Be_{1 4} the only cluster in this series capable of achieving practical, high-density, and fully reversible hydrogen cycling.

To further assess the practical reversible storage performance of Be_{1 4} under near-ambient conditions, additional gravimetric capacities were evaluated at 298 and 303 K under 1, 3, and 30 bar, and the corresponding values are summarized in Table S4 (Supporting Information). Using the same grand-canonical statistical framework and working-capacity definition applied throughout this study, the calculated C _E values at 298 K are 14.00, 16.55, and 17.87 wt % at 1, 3, and 30 bar, respectively. At 303 K, the corresponding values are 12.94, 16.11, and 17.82 wt %. These results confirm that Be_{1 4} retains high reversible hydrogen-storage efficiency close to ambient conditions, while pressure remains the dominant factor governing usable storage capacity.

To contextualize the reversible hydrogen storage performance of Be_{1 4}, Table compares its calculated reversible capacity (C _E) with a range of state-of-the-art adsorbent materials evaluated using the same grand-canonical statistical thermodynamic framework. This benchmarking provides a consistent basis for assessing the relative performance of Be_{1 4} within the broader landscape of lightweight hydrogen storage media.

5. Comparison of Reversible Hydrogen Storage Capacities (C _E, wt %) of Be_{1 4} with Representative Adsorbent Materials.

adsorbent	C E (wt %)	ref
Be14	17.86	present work
Li–C2N	10.00
Sc-graphene	5.00
GR-MOF	11.95
Ti-PCP222	7.37
Sc-IG	21.61
Li-BP	2.92
Li-BPL	6.76
Ca-2dpa	6.91
Mg–C8	21.72
Ti–C3B2	20.10
DOE target	5.50

The comparison clearly shows that Be_{1 4} belongs to the highest-performing lightweight hydrogen-storage systems reported to date. Its reversible capacity of 17.86 wt % significantly exceeds the DOE 2025 target (5.5 wt %) and remains markedly higher than many established adsorbents, including Li–C₂N (10.00 wt %), GR-MOF (11.95 wt %), Ti-PCP222 (7.37 wt %), Li-BPL (6.76 wt %), Sc-graphene (5.00 wt %), and Ca-2dpa (6.91 wt %). Although several recently reported systems, such as Sc-IG (21.61 wt %), Mg–C8 (21.72 wt %), and Ti–C₃B₂ (20.10 wt %), exhibit somewhat higher theoretical capacities, Be_{1 4} distinguishes itself through a particularly favorable balance between storage density and adsorption reversibility. Unlike heavier transition-metal-decorated systems, the low atomic mass of beryllium preserves gravimetric efficiency while maintaining adsorption energies within the optimal reversible window (−0.10 to −0.35 eV per H₂). In addition, hydrogen remains fully molecular throughout the adsorption sequence, and desorption occurs within a narrow thermodynamic window without kinetic trapping, which is essential for practical cycling performance. These combined characteristics indicate that Be_{1 4} offers not only high storage capacity but also a particularly favorable thermodynamic profile among lightweight cluster-based hydrogen-storage candidates.

3.4. Electronic and Interaction Properties of Be₁₄ upon Single and Multiple H₂ Adsorption

To rationalize why the Be_{1 4} cluster supports both stable single-molecule physisorption and an exceptionally high reversible multi-H₂ uptake, a comprehensive electronic-structure analysis was performed combining Energy Decomposition Analysis (EDA), Density of States (DOS), and Independent Gradient Model based on Hirshfeld partitioning (IGMH). These complementary approaches provide a unified understanding of the physical interactions governing hydrogen binding at low and high coverage.

To explore the factors governing the fundamental forces driving the interaction between a single H₂ molecule and the Be₁₄ cluster, as well as the interactions within an ensemble of 20H₂ molecules with Be₁₄, a comprehensive EDA was performed using the GAMESS-US software. The calculations were performed using the M06-2X functional in conjunction with the def2-TZVP basis set for hydrogen atoms and the def2-SVP basis set for beryllium atoms, employing the localized molecular orbital energy decomposition analysis (LMOEDA) method developed by Su and Li. This approach allowed the interaction to be separated into its principal energetic components, namely electrostatic, exchange, polarization, repulsion, and dispersion contributions, with the values summarized in Table .

6. Energy Decomposition Analysis (EDA) of the H₂/Be₁₄ and 20H₂/Be₁₄ Systems at the DFT/M06-2X Level of Theory.

energy component (eV)	H 2 /Be 14	attractive energy (%)	20H 2 /Be 14	attractive energy (%)
electrostatic	–0.70	20.6	–5.45	20.8
exchange	–1.20	35.3	–9.55	36.5
polarization	–1.10	32.3	–7.14	27.3
dispersion	–0.40	11.8	–4.05	15.4
total attractive energy	–3.40		–26.20
repulsion	+2.98		+23.95
total interaction energy	–0.42		–2.25

The EDA results reveal that the interaction between Be_{1 4} and a single H₂ molecule is dominated by noncovalent attractive forces, with exchange (35.3%), polarization (32.3%), electrostatic (20.6%), and dispersion (11.8%) contributions summing to −3.40 eV. These attractive terms are almost perfectly counterbalanced by Pauli repulsion (+2.98 eV), yielding a small net interaction energy of −0.42 eV, consistent with weak, fully reversible physisorption. When 20 hydrogen molecules are adsorbed, the total attractive interaction becomes significantly larger (−26.20 eV), driven by the accumulation of many independent weak interactions. The distribution of attractive contributions remains similar (exchange ≈ 36.5%, polarization ≈ 27.3%, electrostatic ≈ 20.8%, dispersion ≈ 15.4%), while repulsion increases proportionally (+23.95 eV). Despite this, the resulting interaction remains moderately stabilizing (−2.25 eV), confirming that even at saturation, Be_{1 4} maintains a noncovalent binding regime with no transition toward chemisorption. Thus, the EDA reveals that the Be_{1 4} cluster can accommodate multiple weak adsorption sites, facilitating noncovalent interactions with H₂ molecules without leading to H₂ activation.

Electronic-structure analysis based on the total and projected density of states (TDOS/PDOS) further supports the physisorptive nature of hydrogen binding on Be_{1 4}. Figure a,b, which display the PDOS of the pristine Be₁₄ cluster and the H₂/Be₁₄ system, show that the electronic structure of the cluster remains essentially unchanged upon H₂ adsorption. The pristine cluster Be₁₄ exhibits HOMO and LUMO energies of −6.94 and −1.30 eV, respectively, resulting in a HOMO–LUMO gap of 5.64 eV. Similarly, the H₂/Be₁₄ system shows HOMO and LUMO levels of −6.77 and −1.10 eV, yielding a similar gap of 5.67 eV. The PDOS plot further reveals that the orbital contributions associated with the adsorbed H₂ molecule appear as localized states that lie well away from the frontier molecular-orbital region of the Be₁₄ cluster, indicating negligible orbital interaction between H₂ and Be₁₄. The close agreement between the HOMO–LUMO gaps and frontier orbital energies of the two systems indicates that H₂ adsorption induces only a negligible perturbation to the electronic structure of Be₁₄, supporting the conclusion that the interaction is characteristic of a physisorption mechanism.

6.

TDOS/PDOS analysis for (a) the pristine Be₁₄ cluster, (b) Be₁₄ with a single adsorbed H₂ molecule, and (c) Be₁₄ with 20 adsorbed H₂ molecules.

When the number of adsorbed H₂ molecules increases to 20, their overall PDOS intensity becomes more pronounced (Figure c); however, the contributions from their σ bonding and σ antibonding states remain far below and well above the frontier orbitals of 20H₂/Be₁₄, respectively. These observations demonstrate that, as in the case of single H₂ adsorption, the adsorption of 20H₂ molecules results in minor changes to the electronic structure of Be₁₄, confirming that the interaction remains fundamentally noncovalent and consistent with a physisorption mechanism. Combined with EDA, which shows that exchange and polarization remain the dominant attractive contributions from N = 1 to N = 20, the DOS results confirm that even at saturation the Be_{1 4} cluster maintains the same electronic nature and adsorption mechanism. These PDOS trends align closely with previously reported PDOS-based interpretations of physisorptive hydrogen storage mechanisms in light-element nanostructures and metal-decorated systems, confirming that adsorption proceeds without significant orbital hybridization.

Finally, to assess the noncovalent intermolecular interactions between the Be₁₄ cluster and a single and multiple H₂ molecule, the independent gradient model based on Hirshfeld partitioning (IGMH) was applied. The corresponding isosurfaces were generated using the Multiwfn program and subsequently visualized with the VMD package. For the IGMH isosurfaces, the conventional coloring scheme was adopted, wherein a deeper blue coloration indicates stronger attractive interactions, whereas green regions correspond to weak noncovalent interactions such as dispersion-dominated van der Waals forces. As illustrated in Figure , the IGMH isosurface shows that the interaction region between the H₂ molecule and the Be₁₄ cluster exhibits a biconcave, disc-like morphology with green–blue coloring between H₂ and the nearest Be atom, reflecting the presence of moderately strong noncovalent interactions.

7.

IGMH isosurface plots (isovalue = 0.004 au) for the H₂/Be₁₄ interaction and the 20H₂/Be₁₄ systems. Atom color code: teal green for beryllium and white for hydrogen.

For the Be₁₄ cluster interacting with 20H₂ molecules, the IGMH isosurface reveals a heterogeneous interaction profile. Eight H₂ molecules exhibit moderately strong noncovalent interactions with the Be₁₄ cluster, analogous to the interaction observed with a single H₂ molecule. These H₂ molecules are adsorbed near the top apical Be atom and the three Be atoms of the first upper layer, corresponding in total to eight Be···H contacts ranging from 1.62 to 1.81 Å. These distances lie above typical Be–H covalent bond lengths, indicating that the interactions are noncovalent in nature, though stronger than typical van der Waals contacts. In contrast, the remaining 12H₂ molecules interact primarily through weak to very weak van der Waals forces, as indicated by the predominantly green regions of the IGMH isosurface. Collectively, these interactions yield an overall moderate interaction between the Be₁₄ cluster and the ensemble of 20H₂ molecules.

Overall, the IGMH analysis corroborates the EDA and PDOS findings, demonstrating that hydrogen storage in Be_{1 4} is driven by collective, spatially distributed physisorption, and that the adsorption mechanism remains fully reversible even at maximum loading. This explains the exceptional reversible capacity (C _E = 17.86 wt %) achieved by Be_{1 4}, positioning it among the most efficient molecular hydrogen adsorbents reported to date.

Conclusions

A systematic DFT study was conducted to evaluate the hydrogen storage properties of beryllium clusters (Be _n , n = 3–25) and to clarify the role of cluster size on stability and hydrogen adsorption behavior. Benchmark calculations against DLPNO-CCSD­(T) confirm the reliability of the ωB97X-3c composite functional for describing Be–Be interactions and H₂ adsorption energetics. Most Be _n clusters bind hydrogen molecularly through physisorption, with adsorption energies lying in the optimal range for reversible storage and minimal H–H bond activation. Pronounced size effects are observed in multi-H₂ adsorption. Small clusters are limited by site availability, while larger clusters suffer from reduced gravimetric efficiency. In contrast, the medium-sized Be_{1 4} cluster provides an optimal balance, accommodating up to 20H₂ molecules with smoothly decreasing adsorption energies (−0.35 to −0.10 eV per H₂) and preserving full reversibility. This results in a high theoretical gravimetric hydrogen capacity of 24.2 wt % and a reversible working capacity of 17.86 wt %, exceeding the DOE 2025 target. Thermodynamic and kinetic analyses indicate favorable desorption temperatures (127–259 K) and rapid hydrogen release. Overall, this work identifies Be_{1 4} as a promising nanoscale hydrogen-storage motif and provides general insights into size-dependent hydrogen adsorption in light-element clusters.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.6c01490.

• Computational details; optimized geometries; adsorption energies; thermodynamic data; grand-canonical analysis; and additional tables and figures (PDF)

S.R.: Writing original draft, visualization, supervision, methodology, investigation, formal analysis, data curation, and conceptualization. Y.B.: Writing original draft, conceptualization, visualization, supervision. methodology, writing review and editing, and investigation. R.B.S.: Software, validation, investigation, writing review and editing, and visualization.

The authors declare no competing financial interest.