Sequential H₂ Adsorption on the Aromatic Li₆ Superatom: Field-Activated Physisorption and Thermodynamic Limits

Abstract

Understanding the intrinsic Li–H₂ interaction, decoupled from substrate effects, is essential to rationalize the performance of lithium-decorated hydrogen storage materials. To address the current lack of a clean theoretical baseline, we characterized the sequential H₂ adsorption on the gas-phase Li₆ superatomic cluster using high-level density functional theory (DFT), complemented by Energy Decomposition Analysis (EDA), QTAIM, and NICS(0) calculations. Li₆ acts as a structurally rigid platform (RMSD < 0.032 Å) where ligand-induced polarization progressively strengthens its σ-aromaticity (NICS(0) from −2.917 to −13.98 ppm) and increases the HOMO–LUMO gap up to 5.05 eV. EDA identifies the binding as field-activated physisorption, electrostatically dominated (65–67%) and mechanistically distinct from Kubas coordination, as confirmed by QTAIM closed-shell interaction parameters. Negative cooperativity governs an effective loading capacity of n = 2 molecules under cryogenic conditions (T_eq = 143.76 and 114.64 K), while an entropic bottleneck renders higher loading non-spontaneous at all temperatures. These results establish Li₆(H₂)_n as a foundational gas-phase reference, providing a systematic, contamination-free descriptor set for the intrinsic Li–H₂ interaction. This framework is essential for isolating the electronic role of the lithium superatom and unambiguously identifying substrate-induced modulations in supported hydrogen storage materials.

1. Introduction

Hydrogen storage in lithium-decorated nanostructured materials is a promising strategy for the sustainable energy economy [1,2,3]. The Li–H₂ interaction governs the thermodynamic performance of these systems. Lithium has been widely used as a dopant in substrates such as graphene, borophene, B₂S monolayers, and corannulene [4,5,6,7]. However, existing computational studies analyze lithium clusters on complex surfaces or decorated molecular frameworks [8,9,10], which introduce substrate effects (electronic charge transfer to/from the support), morphological changes (symmetry breaking and coordination geometry changes), and confinement-related effects that prevent unambiguous attribution of the Li–H₂ interaction descriptors to the cluster itself. In supported systems, it is impossible to determine a priori whether the adsorption enthalpy or the charge distribution reflects intrinsic Li–H₂ chemistry or artifacts of the substrate environment. A clean gas-phase theoretical reference on an isolated, electronically well-characterized cluster is therefore not merely convenient, but methodologically necessary [4,5,11], to decouple these contributions and establish transferable descriptors. Any deviation observed in supported systems can then be unambiguously attributed to substrate-specific modulation rather than to the intrinsic superatom chemistry. The Li₆ cluster is the ideal candidate for this purpose. It possesses a closed electronic shell, octahedral symmetry, and aromatic superatom character. Furthermore, Li₆ units have been shown to retain stabilizing aromatic motifs in assembled hydrogen storage materials [9,11,12,13,14,15].

While hydrogen adsorption on alkali-decorated systems has been extensively reported, existing studies often conflate the intrinsic chemistry of the lithium cluster with artifacts arising from the support, such as substrate-to-cluster charge transfer or confinement-induced distortions [16]. A systematic, gas-phase study at a high level of theory is therefore a methodological necessity to establish the clean limits of Li–H₂ coordination. By decoupling these effects, this work provides the intrinsic descriptors required to rationalize the performance of complex decorated materials. This work addresses that gap using density functional theory (DFT) at the ωB97X-D4rev/def2-TZVPPD level with ωB97X-2 single-point refinement. We evaluated sequential adsorption in the gas-phase Li₆(H₂)_n system (n = 1–4) through a multivariate characterization aimed at answering four fundamental questions: structural stability, electronic identity, interaction mechanism, and thermodynamic limits. This approach separates intrinsic electronic effects from thermodynamic constraints and establishes a clean theoretical reference for isolating Li–H₂ chemistry from substrate effects in the design of lithium-decorated hydrogen storage materials.

2. Materials and Methods

2.1. Software

We performed all electronic structure calculations using the ORCA 6.1 package (Max Planck Institute for Chemical Energy Conversion, Mülheim an der Ruhr, Germany) [17,18,19,20,21] with tight convergence criteria (TightOpt and TightSCF) and a high-density integration grid (DefGrid3), including the NICS(0) indices and the Energy Decomposition Analysis (EDA) within the Ziegler–Rauk scheme. We carried out all wavefunction analyses with Multiwfn 3.8 (Beijing Tamaoqian Technology Co., Ltd., Beijing, China) [22], including the Molecular Electrostatic Potential (MESP) and the Quantum Theory of Atoms in Molecules (QTAIM) topological analysis with bond critical point (BCP) parameters. We developed in-house Python 3.12 (Python Software Foundation, Wilmington, DE, USA) [23] scripts for the systematic placement of H₂ molecules on the Li₆ cluster and for RMSD calculation using the Kabsch algorithm with atomic permutation [24]. We visualized molecular structures, isomers, MESP maps, and molecular orbitals with Jmol 14.32.83 (Jmol Development Team, Northfield, MN, USA) [25], and generated thermodynamic plots with Gnuplot 6.0 (Gnuplot Development Team, online resource) [26].

2.2. Level of Theory

We performed geometry optimizations and vibrational frequency analyses using the range-separated hybrid functional ωB97X-D4rev [27,28]. This functional belongs to the fourth rung of Jacob’s Ladder and incorporates long-range corrections and fourth-generation dispersion (D4), both critical for describing physisorption in lithium clusters. We combined it with the triple-zeta basis set def2–TZVPPD [29,30], whose diffuse functions (PD suffix) are required to capture the polarizability of H₂ molecules under the superatom electric field and to describe the interstitial electron density at Non-Nuclear Attractors (NNAs). This level of theory overcomes the known limitations of pure GGA functionals such as PBE, which inadequately describe van der Waals forces and suffer from self-interaction errors. The single-reference character of the systems was verified via CCSD(T) calculations, yielding T₁ diagnostic values below 0.011 for both the bare cluster and the representative Li₆(H₂)₁ complex.

To achieve chemical accuracy, we refined electronic energies via single-point calculations using the double-hybrid functional ωB97X-2-D3BJ [31,32,33], which belongs to the fifth rung of Jacob’s Ladder and incorporates an MP2 correlation contribution. Its performance on the GMTKN55 benchmark database [34] yields a mean absolute error of ~0.5 kcal/mol for non-covalent interactions, surpassing canonical MP2 and approaching CCSD(T) quality at a fraction of the computational cost. This level of theory has been specifically validated for weakly bound complexes involving light metals and H₂, where double-hybrid functionals with dispersion corrections consistently reproduce CCSD(T) interaction energies within ~0.3 kcal/mol [31,34]. This combination is therefore appropriate for the sequential adsorption energies in the −2 kcal/mol regime studied here, where basis set truncation errors are further controlled by the Counterpoise correction. We note that all reported vibrational frequencies are harmonic; the known overestimation of H–H stretching frequencies at this level (~6–7% relative to the experimental fundamental of 4161 cm⁻¹) does not affect the relative trends in redshift across the series, which constitute the physically relevant quantity for mechanistic assignment. Raw energies and correction components are provided in Tables S3–S5.

2.3. Potential Energy Surface (PES) Sampling

We based the initial geometry of the Li₆ cluster (neutral, singlet state) on literature-reported parameters [35]: D₄h symmetry (compressed octahedral geometry, with four equatorial and two axial lithium atoms), with interatomic distances r(Li–Li)eq ≈ 2.98 Å and r(Li–Li)ax–eq ≈ 3.04 Å. To locate the low-lying isomers of the Li₆(H₂)_n complexes (n = 1–4), we implemented an exhaustive search protocol using in-house Python scripts, generating over 200 initial structures through three complementary strategies: symmetry-adapted combinatorial sampling, stochastic sampling, and redundancy filtering.

Symmetry-adapted combinatorial sampling: We systematically placed H₂ molecules at high-symmetry sites of the compressed octahedral (D₄h) core: vertex (atop), edge-center (bridge), and face-center (hollow) positions. We generated all possible occupancy combinations for each coverage n.

Stochastic sampling: We randomly distributed H₂ molecules at varying orientations and distances (2.0 to 5.0 Å) around the cluster to locate non-intuitive local minima.

Redundancy filtering: We screened all optimized structures through an RMSD analysis based on the Kabsch algorithm to discard redundant isomers.

We performed vibrational frequency calculations within the harmonic approximation to confirm the nature of the local minima and to derive thermodynamic corrections. Although we acknowledge that the harmonic approximation systematically overestimates H–H stretching frequencies by approximately 6–7% relative to experimental values, we found that the resulting redshifts (Δν) are consistent across the series and serve as a reliable descriptor for assigning the physisorption mechanism.

2.4. Energy Analysis

We construct the total adsorption energy (ΔE_ads,Total) additively from four contributions, each designed to isolate a distinct physical phenomenon. First, we obtain the zero-point energy correction as the difference between the ZPE of the complex and those of the isolated fragments:

ΔZPE = ZPE_complex − ∑ZPE_isolated,

(1)

while we correct the basis set superposition error (BSSE) using the Counterpoise method:

δ_CP = ∑(E_frag,dist − E_frag,ghost).

(2)

The interaction energy (E_int) quantifies the pure electronic attraction between fragments at the distorted geometry they adopt within the complex, excluding the cost of such distortion:

E_int = E_complex − (E_Li6,dist + E_nH2,dist).

(3)

We capture this geometric cost through the deformation energy (E_def), which represents the energetic penalty required to bring each fragment from its isolated equilibrium geometry to the geometry it adopts in the complex:

E_def = (E_Li6,dist − E_Li6,isolated) + (E_nH2,dist − E_nH2,isolated).

(4)

Finally, we integrate all four contributions into the total adsorption energy:

ΔE_ads,Total = E_int + E_def + δ_CP + ΔZPE.

(5)

2.5. Thermodynamic Analysis

We computed standard thermochemical properties within the Rigid Rotor-Harmonic Oscillator (RRHO) approximation at 298.15 K and 1 atm. We construct the internal energy of adsorption (ΔU) from four contributions:

ΔU = E_int + E_def + δ_CP + ΔZPE,

(6)

where E_int is the interaction energy between the cluster and the ligands, E_def is the geometric deformation energy of the fragments upon complexation, δ_CP is the Counterpoise BSSE correction, and ΔZPE is the zero-point energy difference. The internal energy of adsorption (ΔU) is numerically equivalent to Δ_Eads,Total defined in Equation (5), but is recast here within the thermodynamic state function formalism as the foundation for deriving ΔH, TΔS, and ΔG:

ΔH = ΔU + ΔnRT,

(7)

TΔS = T(Scomplex − ∑SisolatedFragments),

(8)

ΔG = ΔH − TΔS,

(9)

where Δn is the change in the number of gas-phase molecules (negative for adsorption), R is the universal gas constant, and T = 298.15 K. The TΔS term quantifies the loss of translational and rotational degrees of freedom of the H₂ molecules upon binding, and ΔG determines the spontaneity of the process: positive values indicate non-spontaneous adsorption under the reference conditions.

To compare isomers and identify the global minimum (GM), we calculated the relative free energy and the corresponding Boltzmann population:

ΔΔG = ΔG_isomer − ΔG_GM,

(10)

P_i = (exp(−ΔG_i/(RT))/∑_j exp(−ΔG_j/(RT)))100.

(11)

To decouple the contribution of each individual ligand and determine the saturation limits, we extended the analysis to sequential thermodynamic parameters (ΔX_seq), where X represents any thermodynamic property. We calculated the sequential change upon adsorption of the n-th molecule as

ΔX_seq(n) = ΔX_cum(n) − ΔX_cum(n − 1),

(12)

where ΔX_cum denotes the cumulative value for the complex with n ligands. We modeled the temperature dependence of spontaneity using the Gibbs–Helmholtz equation, assuming ΔH and ΔS remain constant over the temperature range studied:

ΔG(T) = ΔH_seq − TΔS_seq.

(13)

From this relation, we defined the equilibrium (crossover) temperature T_eq as the threshold at which the process transitions from endergonic to exergonic (ΔG = 0):

T_eq = ΔH_seq/ΔS_seq.

(14)

This protocol allows us not only to identify the structural saturation ceiling of the system, but also to define the thermodynamic operational window (temperature and pressure) required for spontaneous loading of the superatom.

2.6. Topological, Magnetic, and Electronic Characterization

We quantified the geometric distortion of the Li₆ cluster upon complexation through the RMSD of interatomic distances, computed using the Kabsch algorithm. We evaluated magnetic aromaticity via NICS(0) indices calculated at the geometric center of the cluster. In lithium superatoms, NICS(0) probes the magnetic response of the collective interstitial electron density generated by four Non-Nuclear Attractors (NNAs) located at the centers of four of the eight lateral triangular faces of the compressed octahedral D₄h core (one NNA per unique symmetry-equivalent set of faces under D₄h, positioned at the four faces sharing the equatorial plane). The geometric center of the cluster lies at the centroid of these four NNAs and constitutes a direct descriptor of the superatom shell stability, in contrast to planar organic systems where its interpretation differs substantially.

We performed all wavefunction analyses with Multiwfn using the electron density obtained at the ωB97X-D4rev/def2-TZVPPD level. We carried out three analyses. First, we performed QTAIM topological analysis, from which we extracted the bond critical points (BCPs) of the Li⋯H interactions, together with the electron density ρ(r) and its Laplacian ∇²ρ(r) at each BCP. Second, we performed Bader charge partitioning to obtain atomic charges and volumes. Third, we derived global reactivity descriptors (conceptual DFT, c–DFT) from the HOMO and LUMO orbital energies. We also computed Pearson correlation coefficients using an in-house Python script to quantify the linear relationships between energy components and structural descriptors of the system.

3. Results

3.1. Structural Stability and Superatom Identity

We identified the global minima (GM) through an exhaustive potential energy surface (PES) sampling of over 200 structures. As shown in Table S6, the Boltzmann populations for these GMs exceed 71% in all cases, reaching nearly 100% for the most stable complexes. Detailed energetic comparisons and geometries for all 20+ competitive isomers are available in Tables S1–S6 and Figure S1 of the Supplementary Material.

Across all complexes, the Li₆ cluster (Figure 1) preserves D₄h symmetry with minimal variations in Li–Li interatomic distances, reflected in RMSD values below 0.032 Å (

Table 1. Electronic, structural, and magnetic descriptors of the Li₆(H₂)_n global minima.

System (GM)	1 NICS(0) (ppm)	2 Gap (eV)	3 RMSD
Li6	−2.92	4.60	0.0000
Li6(H2)1	−10.74	4.89	0.0315
Li6(H2)2	−13.98	5.04	0.0178
Li6(H2)3	−13.85	5.04	0.0179
Li6(H2)4	−13.70	5.05	0.0232

¹ Nucleus-Independent Chemical Shift. ² HOMO–LUMO Energy Gap. ³ Root-Mean-Square Deviation.

and Table S7). In contrast, higher-energy isomers exhibit more severe structural distortions (Table S7). We analyzed the redshift of the vibrational frequencies of the adsorbed H₂ molecules (Table S9) and found that activated molecules reduce their stretching frequency to 4219 cm⁻¹, compared to 4439.86 cm⁻¹ for free H₂ calculated at the same level of theory, corresponding to a shift of Δν ≈ −221 cm⁻¹ for the most activated complex; the full series spans Δν = −183 to −221 cm⁻¹ (

Table 2. Adsorption descriptors for Li₆(H₂)_n global minima.

System (GM)	1 ΔGads (kcal/mol)	2 ΔEseq (kcal/mol)	3 dLi–H (Å)	4 νH–H (cm−1)	Status
Li6(H2)1	3.05	−2.25	1.99	4219	Favorable
Li6(H2)2	6.94	−1.85	2.01	4244, 4246	Favorable
Li6(H2)3	12.02	+0.32	2.59	4247, 4250	Saturated
Li6(H2)4	16.28	+0.06	2.91	4252, 4257	Saturated

¹ Gibbs free energies of adsorption. ² Sequential adsorption energies; this represents the sequential adsorption energy including ZPE, BSSE, and deformation corrections, excluding the thermal enthalpy correction ΔnRT. ³ Coordination distance. ⁴ Vibrational frequencies.

).

The electronic identity of the superatom remains stable throughout the series, as confirmed by the minimal variation in the Non-Nuclear Attractor (NNA) properties and their correlation with the NICS indices (Table S15). The HOMO–LUMO gap increases from 4.60 eV in the bare cluster to 4.89–5.05 eV across the complexes (Table 1), indicating progressive electronic stabilization upon ligand loading. We confirmed the presence of four NNAs distributed across the lateral triangular faces of the compressed octahedral (D₄h) core (Table S12), each carrying a negative charge of approximately −1.07 e and having volumes close to 212 Bohr³, an intrinsic feature of the superatom shell model. The NICS(0) indices range from −10.74 to −13.98 ppm across the global minima (Table 1 and Table S9), representing a substantial enhancement relative to the bare Li₆ cluster (−2.917 ppm) and confirming that σ-aromaticity is progressively reinforced upon hydrogen coordination.

To quantify the structural determinants of adsorption stability across the series, we performed a Pearson correlation analysis to identify the physical variables governing the stability of the system (Table S14). We found a very strong positive correlation (r = 0.98) between the total adsorption energy (ΔE_ads,Total, defined in Section 2.4) and the mean Li–Li distance (r_Li–Li), indicating that metallic core expansion is a necessary structural response to stabilize ligand loading. The electronic interaction energy (E_int) also shows a strong dependence on this core expansion (r = 0.95). We found a moderate inverse correlation (r = −0.69) between ΔE_ads,Total and the H–H bond distance (r_H–H), indicating that greater H–H bond weakening corresponds to a stronger interaction with the superatom. The RMSD parameter shows a correlation of 0.48 with the total adsorption energy, confirming that structural distortions, although minimal, are coupled to the energetic stabilization of the system.

3.2. Thermodynamics and Saturation Limits

We report the cumulative enthalpy (ΔH), entropy (−TΔS), and Gibbs free energy (ΔG) contributions at 298.15 K for the Li₆(H₂)_n series (n = 1–4) in Figure 2 and Table 2. In all cases, the entropic term outweighs the enthalpic stabilization, yielding positive ΔG values that increase monotonically along the series, ranging from 3.05 to 16.28 kcal/mol.

To decouple the contribution of each individual ligand, we analyzed the sequential thermodynamic parameters (ΔH_seq, ΔS_seq) and the equilibrium (crossover) temperature (T_eq), detailed in

Table 3. Sequential thermodynamics of H₂ adsorption on Li₆.

Step	1 ΔHseq (kcal/mol)	2 ΔSseq (cal/mol·K)	3 ΔG298K,seq (kcal/mol)	4 Teq (K)
0 → 1	−2.84	−19.76	3.05	143.76
1 → 2	−2.43	−21.20	3.89	114.64
2 → 3	0.46	−15.50	5.08	N/A
3 → 4	0.45	−12.78	4.26	N/A

¹ Sequential enthalpy. ² Sequential entropy. ³ Sequential Gibbs free energy. ⁴ Equilibrium temperature; this defines the thermodynamic threshold for spontaneity for each adsorption step.

. The first two adsorption steps are exothermic: ΔH_seq = −2.84 kcal/mol (0 → 1) and −2.43 kcal/mol (1 → 2). From the third step onward (2 → 3), the process becomes endothermic (ΔH_seq = +0.46 kcal/mol), marking the onset of saturation driven by electrostatic and steric repulsions.

Figure 3 illustrates the temperature dependence of ΔG. Spontaneous adsorption (ΔG < 0) is only achievable under cryogenic conditions. We identified crossover temperatures of T_eq = 143.76 K for the first step and T_eq = 114.64 K for the second; below these thresholds, the system enters the spontaneous regime, as shown by the zero-crossings in Figure 3. Steps n = 3 and n = 4 exhibit no defined T_eq and remain endergonic at all temperatures studied, due to the combination of an unfavorable adsorption enthalpy and a persistent entropic penalty. These results confirm that the effective thermodynamic loading capacity of Li₆ is restricted to n = 2 molecules.

3.3. Nature of the Interaction: Field-Activated Physisorption

We analyzed the physical nature of the binding interaction using the Ziegler–Rauk EDA scheme (

Table 4. Energy Decomposition Analysis (EDA) and binding energy for the Li₆(H₂)_n global minima. All values are reported in kcal/mol. Calculations were performed using the Ziegler–Rauk scheme.

System (GM)	1 Eint	2 Eprep	3 Ebind	4 Eelstat	5 EPauli	6 Eorb	7 Edisp	8 EXC
Li6(H2)1	−4.35	0.12	−4.23	−9.34	13.96	−4.03	−1.03	−3.91
Li6(H2)2	−7.95	0.31	−7.65	−17.06	25.25	−7.24	−1.72	−7.19
Li6(H2)3	−8.34	0.44	−7.90	−18.81	27.87	−7.54	−1.98	−7.88
Li6(H2)4	−8.62	0.51	−8.10	−20.01	29.62	−7.68	−2.16	−8.38

¹ Total interaction energy (E_elstat + E_Pauli + E_orb + E_XC + E_disp). ² Preparation (deformation) energy. ³ Binding energy (E_int + E_prep). ⁴ Electrostatic energy. ⁵ Pauli repulsion. ⁶ Orbital energy. ⁷ Dispersion energy. ⁸ Exchange–correlation energy.

; complete data for all isomers in Table S10), in which the exchange–correlation term (E_XC) is reported separately from the orbital term (E_orb), rather than being subsumed into it as in the classical formulation. For the n = 1 global minimum, the total interaction energy is E_int = −4.35 kcal/mol. The dominant attractive contribution is electrostatic (E_elstat = −9.34 kcal/mol), followed by the orbital term (E_orb = −4.03 kcal/mol), the exchange–correlation contribution (E_XC = −3.91 kcal/mol), and dispersion (E_disp = −1.03 kcal/mol). Pauli repulsion contributes a destabilizing term of +13.96 kcal/mol. Across all complexes, the E_elstat/E_orb ratio exceeds 2.0, confirming the predominantly electrostatic character of the interaction. The preparation energy (E_prep) remains minimal throughout the series, ranging from 0.12 to 0.51 kcal/mol (Table 4), representing the minor energetic penalty required to deform the fragments and confirming the structural rigidity of the metallic platform. Note that ΔE_ads,Total (Equation (5)) incorporates thermal and basis set corrections that are not part of the standard E_bind reported in EDA schemes. This distinction is crucial to differentiate between purely electronic coordination and effective thermodynamic adsorption.

The QTAIM topological parameters at the bond critical points (BCPs) are presented in

Table 5. Topological properties of the electron density (ρ(r)) and its Laplacian (∇²ρ(r)) at the Li⋯H bond critical points (BCPs) for the Li₆(H₂)_n global minima. All parameters are reported in atomic units (a.u.).

System (GM)	Interaction Type	ρ(r)	∇2ρ(r)	Characterization
Li6(H2)1	Li–H (Single)	0.0130	0.0821	Physisorption
Li6(H2)2	Li–H (Symmetric)	0.0122	0.0767	Physisorption
Li6(H2)3	Li–H (Relaxed)	0.0127	0.0765	Physisorption
	Li–H (Compressed)	0.0131	0.0827	Steric Confinement
Li6(H2)4	Li–H (Relaxed)	0.0125	0.0761	Physisorption
	Li–H (Compressed)	0.0130	0.0818	Steric Confinement

(extended comparison in Table S11). The electron density ρ(r) at the Li⋯H interactions ranges from 0.0122 to 0.0131 a.u. The Laplacian ∇²ρ(r) is positive in all cases, with values between 0.0761 and 0.0827 a.u., characteristic of closed-shell interactions. For n = 3 and n = 4, we detected a slight increase in the electron density at the BCPs associated with H₂ molecules in the first coordination sphere.

3.4. Electronic Characterization

We calculated Bader charges for all global minima (

Table 6. QTAIM charge analysis for Li₆(H₂)_n global minima.

System (GM)	1 Q(Li6) (e)	2 Q(H2)act (e)	3 Q(H2)spec (e)	4 ΔQ (e)
Li6(H2)1	+0.074	−0.074	—	+0.074
Li6(H2)2	+0.134	−0.067	—	+0.134
Li6(H2)3	+0.145	−0.065	−0.015	+0.145
Li6(H2)4	+0.154	−0.063	−0.014	+0.154

¹ Total charge of the lithium core (including NNAs). ² Average charges of activated. ³ Spectator hydrogen molecules. ⁴ Net charge transferred from the Li₆ core to the adsorbed H₂ molecules.

). The net charge transfer from the cluster to the H₂ molecules is small but differentiated across all complexes. Activated molecules carry negative charges between −0.074 and −0.063 e, while spectator molecules show a significantly smaller charge transfer (−0.015 to −0.014 e).

We identified Non-Nuclear Attractors (NNAs) at the center of the octahedral void of Li₆ (Tables S12 and S15). These NNAs concentrate a negative charge of approximately −1.07 e for n = 1, which decreases slightly to −1.04 e for n = 4. Their volumes range between 210 and 212 Bohr³ and remain virtually constant throughout the series. This simultaneous stability of the interstitial charge and the NNA volume confirms the closed-shell nature of the cluster and supports the superatom model.

The Molecular Electrostatic Potential (MESP) maps corroborate this charge distribution (Figure 4). The lithium nuclei act as the predominant electrophilic regions of the system. We detected moderate inductive polarization of the H–H bond exclusively at the primary adsorption sites.

We analyzed the spatial distribution of the frontier molecular orbitals (FMOs) in Figure 5 and Figure 6, as well as their energetic evolution in the orbital energy level diagram in Figure 7. The HOMO is localized exclusively on the metallic core throughout the series. The LUMO incorporates contributions from the σ* antibonding orbitals of the coordinated H₂ molecules. From these orbital energies, we derived the global reactivity descriptors (conceptual DFT, c–DFT) (

Table 7. Global reactivity descriptors for the Li₆(H₂)_n global minima. All parameters are reported in eV.

System (GM)	1 μ	2 η	3 χ	4 ω
Li6	−2.61	2.30	2.61	1.48
Li6(H2)1	−2.46	2.44	2.46	1.24
Li6(H2)2	−2.39	2.52	2.39	1.13
Li6(H2)3	−2.38	2.52	2.38	1.13
Li6(H2)4	−2.38	2.53	2.38	1.12

¹ Electronic chemical potential. ² Chemical hardness. ³ Electronegativity. ⁴ Electrophilicity index.

and Table S13) following the frontier molecular orbital framework for describing chemical reactivity [36]. Although absolute orbital eigenvalues are sensitive to the functional choice, the range-separated ωB97X-D4rev functional ensures a consistent description of the HOMO–LUMO gap and its relative trends across the series. The chemical hardness (η) and electronegativity (χ) show minimal variation relative to the isolated cluster. The chemical potential (μ) and electrophilicity index (ω) converge as ligand loading increases.

4. Discussion

4.1. Aromatic Superatom

Our results demonstrate that the Li₆ cluster functions as a structurally rigid platform, whose stability under ligand loading is confirmed by the preservation of D₄h symmetry and RMSD values below 0.032 Å (Table 1). We attribute this robustness to its intrinsic superatom character, whereby the four Non-Nuclear Attractors (NNAs) distributed across the lateral triangular faces of the compressed octahedral (D₄h) core act collectively as an electronic anchor that cohesively holds the metallic core against external perturbations, minimizing geometric deformation. A key finding is the near-perfect correlation (r = 0.98) between the adsorption energy and the Li–Li distance; the metallic core undergoes minimal expansion to optimize the interaction without compromising its octahedral topology. This resistance to perturbation is further reflected in the progressive increase in the HOMO–LUMO gap from 4.60 eV in bare Li₆ to ≈5.0 eV upon full loading (Table 1), Notably, starting from n = 2, the LUMO energy shifts into the positive regime (unbound), as shown in Table S13. This transition is physically consistent with the cluster’s high electronic hardness (η) and its character as a stable, closed-shell species that does not favor further electronic perturbation, confirming that hydrogen coordination reinforces rather than compromises the electronic stability of the cluster.

The integrity of the system is ensured by the ligand-induced strengthening of σ-aromaticity from −2.917 ppm in bare Li₆ to ≈−13.8 ppm under hydrogen loading, which acts as an “electronic glue” that minimizes deformation under external repulsions. This stability is a direct consequence of the superatom shell model: each NNA carries a substantial negative charge (≈−1.07 e) that remains nearly constant up to n = 4, with associated volumes ranging from 210 to 212 Bohr³ without significant variation despite increasing ligand loading (Tables S12 and S15), reinforcing their collective role as a stabilizing charge anchor.

By residing at the centroid of these four NNAs, the NICS(0) magnetic index probes the collective interstitial electron density and remains free from the local contamination typical of σ covalent bonds. The persistence of strong σ-aromaticity is therefore not a computational artifact but is intrinsically tied to the stability of this collective interstitial density. The parallel evolution of these descriptors (Table S15) confirms that the Li₆ core acts as an unaltered electronic anchor even under forced saturation regimes, thus supporting the concept of “NNA-supported aromaticity” that allows Li₆ to retain its electronic identity up to n = 4.

The isomeric distribution (Table S6) reveals a coverage-dependent behavior. For n = 1, the global minimum dominates with a Boltzmann population of 97.78%, consistent with a structurally pure species. However, for n = 2, the global minimum population drops to 71.2%, with three isomers competing with significant populations. For n = 3 and n = 4, the distribution broadens further, with global minimum populations of 42.8% and 31.12%, respectively. This progressive isomeric dispersion is consistent with the saturation regime identified in Section 3.2 and provides independent reinforcement of the effective thermodynamic loading limit at n = 2. Despite this broadening distribution, the descriptors obtained (HOMO–LUMO gap, NNA charge, NICS indices, and EDA components) constitute a contamination-free reference set that can be directly employed to assess how a given substrate modulates the adsorption capacity of the Li₆ motif in assembled materials. Deviations observed in supported systems can thus be unambiguously attributed to substrate-specific modulation rather than to the intrinsic superatom chemistry.

4.2. Nature of the Interaction

We interpret the binding in the Li₆(H₂)_n complexes as field-activated physisorption (consistent with the ion-quadrupole polarization-enhanced mechanism described by Lochan and Head-Gordon [37]) and rule out the formation of Kubas-type complexes. Our multi-physics evidence supports this assignment through three independent criteria: vibrational, energetic, and topological criteria.

We observe a redshift of Δν ≈ −221 cm⁻¹ and an H–H bond elongation of only 0.014 Å (Table 2). For context, classical Kubas complexes of transition metals such as W(CO)₃(PR₃)₂(H₂) exhibit ν_H–H frequencies in the range of 2600–3100 cm⁻¹, corresponding to redshifts of 1000–1500 cm⁻¹ relative to free H₂ (~4161 cm⁻¹) [38,39]. In contrast, Lochan and Head-Gordon report that Li⁺-doped light metal systems show typical shifts of only 130–160 cm⁻¹ [37,40], a range with which our value of Δν ≈ −221 cm⁻¹ is fully consistent. This behavior is characteristic of ion-quadrupole polarization-enhanced physisorption, lacking the d → σ back-donation required to form a Kubas-type σ-complex. The molecular integrity of H₂ is preserved nearly intact, favoring desorption without significant kinetic barriers.

The Ziegler–Rauk EDA scheme confirms that the stabilization is predominantly electrostatic, contributing between 65% and 67% of the primary attractive interaction (E_elstat, E_orb, E_disp); percentages are computed over these three physically interpretable terms, with E_XC reported separately as it represents the non-classical exchange–correlation correction of the hybrid functional rather than a classical interaction force [41], while the orbital component accounts for a secondary fraction of the stabilization (Table 4). The electric field generated by the superatom polarizes the H₂ molecules, but charge transfer is insufficient to establish a strong and stable three-center Li⋯H⋯H coordinative bond.

We find a positive Laplacian (∇²ρ > 0) and low electron densities (ρ ≈ 0.013 a.u.) at the Li⋯H bond critical points (Table 5). These values are diagnostic of closed-shell interactions and exclude the formation of covalent or strongly coordinative bonds. Bader charge analysis (Table 6) further confirms the non-covalent nature of the interaction: charge transfer toward H₂ is minimal, preventing the formation of stable hydrides and ensuring full process reversibility. The small dispersion contribution (E_disp = −1.03 kcal/mol, Table 4) indicates that storage does not rely on long-range van der Waals forces, but on a direct electrostatic response. The MESP maps (Figure 4) support this field-activated physisorption model.

4.3. The Entropic Bottleneck

The thermodynamics of the Li₆(H₂)_n system is governed by the competition between electronic stabilization and the loss of translational and rotational degrees of freedom. At 298.15 K, the −TΔS term consistently outweighs the enthalpic stabilization, yielding positive ΔG values that increase along the series (3.05 to 16.28 kcal/mol). Spontaneous adsorption under standard conditions is therefore thermodynamically unfeasible. Nevertheless, the equilibrium (crossover) temperature analysis (T_eq) allows us to precisely define the operational window of the system.

For the first two adsorption steps, the transition to the spontaneous regime (ΔG < 0) occurs under cryogenic conditions, with T_eq = 143.76 K for n = 1 and T_eq = 114.64 K for n = 2. While we recognize that the RRHO scheme tends to overestimate the entropic penalty in weakly bound systems by treating hindered translational and rotational modes as harmonic vibrations, we estimate that applying quasi-RRHO corrections would shift these Teq values upward by only 20–40 K. Since this adjustment does not alter our fundamental conclusion that spontaneous loading is restricted to the cryogenic regime (<200 K), we consider the RRHO results to be a robust upper bound that is physically representative of the system’s thermodynamic limits. For n ≥ 3, no defined T_eq exists: the endothermic character of the sequential adsorption (ΔH_seq > 0), combined with a persistent entropic penalty, keeps ΔG positive at all temperatures studied.

This effective loading limit at n = 2 has a structural origin: saturation of the first coordination sphere and the electrostatic repulsion generated by the high charge density at the central NNAs confine additional H₂ molecules to peripheral, energetically unfavorable positions. As a result, the lower symmetry observed for n = 4 is a direct consequence of this “forced saturation state,” where inter-ligand repulsions prevent the adoption of higher-symmetry configurations. It is important to distinguish that the structural ceiling at n = 4 represents a forced saturation state only, not spontaneously accessible under either standard or moderately cryogenic conditions.

That said, the cryogenic window defined by T_eq = 143.76 K and 114.64 K falls within a physically accessible temperature range, providing a quantitative thermodynamic reference rather than a practical storage prescription. In this context, Li₆ should not be regarded as a failed adsorbent, but as a well-characterized reversible physisorption platform whose operational window serves as a gas-phase theoretical reference: future Li₆-decorated materials in which the substrate favorably modulates the adsorption enthalpy could shift this window toward more practical temperatures.

5. Conclusions

Structural Stability: The Li₆ cluster acts as a rigid, non-fluxional platform (RMSD < 0.032 Å) throughout the adsorption series. The superatom shell model maintains geometric integrity, consistent with reversible adsorption–desorption cycles without symmetry breaking for n ≤ 2.

Electronic Identity: Ligand loading strengthens the metallic σ-aromaticity, with NICS(0) values shifting from −2.917 ppm to a range of −10.74 to −13.98 ppm. This is accompanied by an increased HOMO–LUMO gap (up to 5.05 eV) and enhanced interstitial electron delocalization.

Interaction Mechanism: Energy Decomposition Analysis (EDA) identifies the binding as field-activated physisorption, dominated by electrostatic contributions (65–67%). Kubas-type coordination is excluded based on moderate vibrational redshifts (Δv ≈ −183 to −221 cm⁻¹) and QTAIM parameters diagnostic of closed-shell interactions (∇²ρ > 0, ρ < 0.013 a.u.).

Thermodynamic Limits: The system exhibits negative cooperativity, establishing an effective loading capacity of n = 2 molecules under cryogenic conditions. An entropic bottleneck renders adsorption non-spontaneous at standard conditions, restricting the operational window to the cryogenic regime (T_eq = 143.76 K and 114.64 K). These values are subject to the intrinsic uncertainty of the double-hybrid functional (~0.5 kcal/mol). While we acknowledge these RRHO values as upper-bound estimates, the estimated 20–40 K increase from quasi-RRHO corrections does not alter the robust conclusion that spontaneous loading requires cryogenic conditions.

Gas-phase Theoretical Reference Utility: The substrate-free characterization of Li₆ yields a self-consistent set of contamination-free descriptors (Bader charges, NNA charge and volume, NICS(0) indices, EDA contributions, and crossover temperatures) that represent the intrinsic Li–H₂ interaction. While these conclusions are derived from high-level theoretical modeling, the consistency between the electronic descriptors (EDA, QTAIM) and the thermodynamic limits provides a robust physical picture. Consequently, the reported T_eq values should be interpreted as theoretical thresholds that define the operational window of the Li₆ superatom. This constitutes a theoretical baseline against which the performance of Li₆-decorated materials can be directly evaluated: deviations observed in supported systems can now be unambiguously attributed to substrate-specific modulation rather than to the superatom chemistry itself, enabling the rational, descriptor-guided design of lithium-decorated hydrogen storage materials.