Stability and hydrogen storage performance of Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides via DFT and AIMD

Abstract

This study aims to provide a comprehensive first-principles investigation, based on density functional theory (DFT) using the GGA-PBE functional, of Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides that crystallize in the Fm3̄m (225) space group. The structural, electronic, optical, and thermodynamic properties were systematically explored to evaluate their potential for advanced hydrogen storage and clean energy applications. Phonon dispersion and ab initio molecular dynamics (AIMD) simulations confirm the dynamical and thermal stability of the system at 300 K, without any structural distortion. Among the investigated compounds, Na₂LiVH₆ shows the highest gravimetric capacity (5.50 wt%) and an optimal desorption temperature (540.23 K), favorable for reversible hydrogen release. Electronic analysis reveals metallic conductivity, while thermodynamic parameters, including heat capacity, enthalpy, entropy, and free energy, exhibit stable temperature-dependent trends. Collectively, these DFT-GGA-PBE results demonstrate that Na₂LiVH₆ possesses superior mechanical endurance, lattice stability, and multifunctional potential for next-generation hydrogen storage and sustainable energy applications.

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Introduction

The increasing global energy needs and escalating environmental degradation necessitate the rapid advancement of sustainable energy technologies. Among the different alternative fuels, hydrogen has emerged as a highly promising energy carrier due to its exceptional energy density and clean combustion products.¹ Nevertheless, its large-scale utilization is still limited by difficulties associated with safe, efficient, and practical storage and transportation.² Traditional storage techniques, such as high-pressure gas compression, cryogenic liquid storage, and cryo-compressed systems, demand significant energy to maintain extreme pressure or low-temperature conditions and are further challenged by unavoidable boil-off losses.^3,4 Alternatively, solid-state hydrogen storage has emerged as a focal area of research owing to its superior volumetric density, enhanced safety, and reversible storage capability.^5,6 A wide range of solid materials has been investigated, including intermetallic alloys,^7,8 metal hydrides,^9–12 graphene-based nanostructures,¹³ metal–organic frameworks (MOFs),^14,15 MXenes,^16,17 and liquid organic hydrogen carriers (LOHCs).¹⁸ Among these, metal hydrides are particularly attractive because hydrogen is chemically bonded under moderate conditions, offering high volumetric capacities and intrinsic safety. Furthermore, their storage performance can be optimized through compositional engineering and catalytic modification.^19,20

Perovskite hydrides have recently emerged as a promising subclass of ternary hydrides with remarkable promise for hydrogen storage applications. These materials possess the ability to absorb and retain hydrogen both on their surfaces and within their crystal frameworks, offering an efficient approach to overcoming the persistent challenges of hydrogen storage and transport.²¹ Their high gravimetric capacity, excellent thermal and cyclic stability, and favorable reversibility make them highly attractive for next-generation solid-state hydrogen systems. Various perovskite hydrides such as NaMgH₃, KAlH₃, and NaAlH₃ have been synthesized and characterized, exhibiting orthorhombic or cubic crystal symmetries with remarkable thermodynamic stability.^22,23 Reported hydrogen storage capacities for AVH₃ (A = Na, K, Rb, Cs), XPtH₃ (X = Li, Na, K, Rb), and XFeH₃ (X = Ca, Sr, Ba) fall within 1–5 wt%, highlighting their competitive storage performance.^24–26 Double perovskite systems, including Na₂LiAlH₆ and K₂LiAlH₆, show dehydrogenation capacities of approximately 3.09 wt% and 5.2 wt%, respectively.^27–29 Li-doped K₂NaAlH₆ demonstrates an improved storage capacity of 4.91 wt% with increasing Li concentration,³⁰ while substitution of alkaline earth metals in A₂KNaH₆ (A = Be, Mg, Ca) further enhances capacity to 8.57, 5.19, and 4.1 wt%, respectively.³¹ Moreover, K₂XAlH₆ (X = Li, Na) hydrides exhibit wide band gaps and strong mechanical stability under external stress.³² Despite these advances, many double perovskite hydrides remain insufficiently examined, underscoring the need for detailed AIMD investigations to optimize their hydrogen storage characteristics and practical applicability.

In this study, Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides are investigated using DFT to explore their fundamental structural, electronic, mechanical, optical, and thermodynamic behaviors. The study aims to identify their feasibility and effectiveness as potential prospects for future solid-state H₂ storage technologies. Despite the growing interest in complex hydrides, these specific Zr, V, and Cr based Na₂LiXH₆ (X = Zr, V, Cr) compounds have not yet been systematically explored, either computationally or experimentally. The strategic substitution of X-site cations with varying electronegativities and ionic radii introduces tunable effects on formation energy, stability, electronic structure, and hydrogen release thermodynamics, an aspect that remains largely unreported in existing literature. To bridge this gap, DFT calculations were conducted to elucidate the atomic-level bonding interactions, energetic characteristics, and hydrogen storage potential of the compounds. Complementary AIMD simulations were employed to evaluate their thermal robustness. The results reveal the structural and energetic stability of these previously unexplored perovskite hydrides, providing a theoretical basis for the rational design of advanced materials for H₂ storage and energy conversion applications.

Computational methodology

All computational analyses were conducted using DFT within the CASTEP simulation package³³ to investigate the ground-state characteristics and optimized structural configurations of Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides. The Kohn–Sham equations were implemented with a plane-wave basis set, applying a kinetic energy cutoff of 600 eV,³⁴ and ultrasoft pseudopotentials were utilized to model the interactions between electrons and ions accurately. The exchange correlation interactions were described within the framework of the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) functional.³⁵ Structural optimization was performed through the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm, applying strict convergence thresholds of 2 × 10⁻⁵ eV per atom for total energy, 0.05 eV Å⁻¹ for atomic forces, 0.1 GPa for stress, and 0.002 Å for atomic displacements.³⁶ For Brillouin zone integration, a Monkhorst–Pack k-point mesh of 6 × 6 × 6 was utilized to ensure accurate sampling. After full relaxation, all Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides were verified to adopt a cubic symmetry within the Fm3̄m (No. 225) space group, characterized by lattice angles α = β = γ = 90°, thereby confirming their crystallographic and structural stability.³⁷ To evaluate the thermal stability, AIMD simulations were carried out using the pw.x module of Quantum ESPRESSO within the canonical (NVT) ensemble. The simulations employed the velocity Verlet integration algorithm coupled with a Berendsen thermostat fixed at 300 K to maintain temperature control. A time step of 0.96756 fs was adopted for 10 000 iterations, corresponding to a total simulation duration of approximately 9.7 ps. The calculations utilized ultrasoft pseudopotentials in combination with the PBEsol exchange-correlation functional, while the plane-wave and charge density cutoff energies were set to 60 Ry and 400 Ry, respectively. Convergence criteria were defined as 1.0 × 10⁻⁸ Ry for self-consistent field (SCF) cycles, 1.0 × 10⁻⁵ Ry for total energy, and 1.0 × 10⁻⁴ Ry bohr⁻¹ for forces, ensuring high numerical precision throughout the simulations.³⁸ These computational settings ensured robust convergence and high numerical precision across all investigated systems.

Results and discussion

Structural stability

The optimized crystal structure of Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides demonstrates a cubic symmetry classified under the Fm3̄m (225) space group, as illustrated in Fig. 1. The atomic positions within the unit cell are specified as follows: Na atoms reside at the 8c (1/4, 3/4, 3/4) Wyckoff sites, Li atoms are located at 4b (1/2, 0, 0), X = (Zr/V/Cr) atoms reside at 4a (0, 0, 0), and H atoms are positioned at 24e (1/2, 0.718326, 0). The structural motif consists of alternating LiH₆ and XH₆ (X = Zr, V, Cr) octahedra, forming a three-dimensional framework wherein Na⁺ ions occupy the cuboctahedral cavities, consistent with the characteristic rock-salt arrangement of perovskite-type hydrides.³⁹

Fig. 1 Optimized crystal structure of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides.

Table 1 shows the optimized lattice parameters (a = b = c) and unit cell volumes (V) of Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides. The results reveal a gradual contraction in lattice dimensions, decreasing from 7.93 Å (Zr) to 7.33 Å (Cr). This gradual reduction in lattice dimension results from the decreasing ionic radii of the X-site cations, which enhances the crystal compactness and strengthens the metal–hydrogen interactions. The bonding environment serves a crucial role in function in controlling hydrogen retention and desorption behavior, where an optimal balance of bond strength facilitates hydrogen release at moderately low temperatures, beneficial to efficient storage-release cycles. In this context, the Goldschmidt tolerance factor (τ_G), calculated using eqn (1),⁴⁰ exhibits a slight increase from 0.88 for Na₂LiZrH₆ to 0.93 for Na₂LiCrH₆, remaining well within the stability range (0.8–1.0) characteristic of cubic double perovskite frameworks. This gradual rise in τ_G reflects enhanced structural symmetry and lattice compatibility, which contribute to the observed thermodynamic stability and favorable hydrogen desorption characteristics of the Cr-based compound. All compounds studied have octahedral values (µ) within the optimal range (0.42–0.75). These values confirm that all three compounds possess geometrically stable frameworks without significant distortion.

Table 1 Calculated structural parameters of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides

Compound	Lattice constant a = b = c (Å)	Volume (Å)^3	τG	µ	ΔHf (eV per atom)	Ecoh (eV per atom)
Na2LiZrH6	7.93	499.35	0.88	0.53	−1.26	+1.26
Na2LiVH6	7.45	414.38	0.92	0.46	−1.47	+1.47
Na2LiCrH6	7.33	393.57	0.93	0.45	−1.61	+1.61

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The enthalpy of formation (ΔH_f) further provides insight into the thermodynamic stability of the hydrides, which can be calculated using eqn (2).⁴¹ The calculated ΔH_f values of −1.26, −1.47, and −1.61 eV per atom for Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆, respectively, indicate an increasing exothermic trend, and n represents the total number of atoms. The progressively more negative formation enthalpies imply that substitution of Zr with V and Cr enhances the overall thermodynamic stability of the lattice. Although this trend corresponds with the decrease in the effective ionic size of the X-site cation, it is important to emphasise that the enhanced stability cannot be solely attributed to ionic radii. Instead, the observed exothermic behaviour is primarily governed by changes in electronic structure, bonding characteristics, and metal-hydrogen hybridization induced by the transition-metal substitution. Likewise, the cohesive energy (E_coh) follows a similar sequence, increasing from +1.26 eV (Zr) to +1.61 eV (Cr), and was calculated using eqn (3),⁴² confirming stronger interatomic bonding as the X-site cation changes from Zr to Cr. The increase in cohesive strength is indicative of an augmentation in metal-hydrogen bonding interactions and an improvement in charge transfer. Rather than being a simple monotonic dependence on ionic radius alone, this phenomenon leads to enhanced lattice rigidity and a reduction in susceptibility to structural deformation. The combined structural, geometric, and energetic indicators reveal a consistent trend in the stability of the studied hydrides. Among the three, Na₂LiCrH₆ exhibits the highest structural and thermodynamic stability, followed by Na₂LiVH₆ and Na₂LiZrH₆. The enhanced compactness, favorable τ_G and higher E_coh collectively confirm the superior integrity of Na₂LiCrH₆ within the Na₂LiXH₆ series.

In addition to lattice parameters and formation enthalpies, the calculated M–H bond lengths provide significant insight into the bonding characteristics and hydrogen storage behavior of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides. The optimized structures exhibit short X–H bond lengths of 1.99 Å for Zr–H, 1.77 Å for V–H, and 1.71 Å for Cr–H, indicating strong M–H interactions within the [XH₆] octahedra. The continuing contraction of the X–H bond from Zr to Cr reflects increasing orbital overlap between transition-metal d-states and H-state, consistent with enhanced covalent character and improved framework stability. In contrast, the significantly longer Na–H separations of 2.81 Å in Na₂LiZrH₆, 2.64 Å in Na₂LiVH₆, and 2.59 Å in Na₂LiCrH₆ show weaker ionic interactions between hydrogen and the A-site cations. This pronounced bonding asymmetry suggests that while H atoms coordinated to the X-site ensure structural rigidity, those associated with the A site are more-weakly bound and thus more amenable to thermally activated release. Such coexistence of strong X–H bonding and weaker A–H interactions is a defining structural feature governing hydrogen uptake and release kinetics in perovskite hydrides and has been widely recognized as a key factor in optimizing reversible solid-state H₂ storage performance.^43,44

Dynamic stability

Dynamic stability is a key factor governing the structural integrity and long-term functionality of materials, particularly in applications such as hydrogen storage, where materials must endure repeated thermal and mechanical cycles without degradation.⁴⁵ To assess the dynamic stability of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides, phonon dispersion analysis was performed across the first Brillouin zone, as illustrated in Fig. 2a–c. Phonon dispersion relations serve as a fundamental tool for assessing the vibrational stability of crystalline materials, as they reveal how atomic vibrations propagate through the lattice. The appearance of imaginary (negative) phonon frequencies signifies dynamical instability, suggesting possible lattice distortions or phase transitions, whereas their absence confirms a stable crystal configuration suitable for practical applications. The lack of imaginary modes in the computed phonon band structures indicates that these compounds are dynamically stable under ambient conditions, ensuring their potential for practical applications.^46,47

Fig. 2 Phonon dispersions describing (a) Na₂LiZrH₆, (b) Na₂LiVH₆ and (c) Na₂LiCrH₆ perovskite hydrides.

As depicted in Fig. 2a–c, Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆ all exhibit stable phonon dispersions, with no imaginary phonon modes observed across the Brillouin zone. The acoustic branches smoothly converge to zero near the Γ-point, whereas the optical modes remain clearly divided, displaying well-defined oscillatory mode features that are indicative of a structurally stable lattice.⁴⁸ While the phonon frequencies vary slightly among the compounds, with Na₂LiCrH₆ showing higher frequencies indicative of a stiffer lattice, all three compounds maintain stable vibration profiles, suggesting that they are not susceptible to lattice distortions. The higher frequencies observed in Na₂LiCrH₆ indicate a stronger interatomic bond strength and more robust vibrational modes, potentially due to the higher atomic number and stronger bonding of Cr compared to Zr and V. This analysis confirms that the optimized structures of Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆ correspond to stable, minimum-energy states rather than saddle points, further supporting their dynamical stability and suitability for applications where long-term material performance is critical. The origin of the differences in phonon frequencies can be attributed to the differing electronic structures and bond strengths between the transition metals Zr, V, and Cr. The stronger metallic bonds in Na₂LiCrH₆ lead to higher phonon frequencies, reflecting a more rigid and robust lattice structure, while Na₂LiZrH₆ and Na₂LiVH₆ exhibit lower frequencies due to the weaker bonds in comparison. This gives rise to the subtle variation in dynamic behavior across these compounds.

The lack of low-frequency vibrational instabilities further verifies the vibrational robustness of these materials, indicating their ability to endure mechanical stresses while maintaining structural integrity. Additionally, the consistent stability observed across all three compounds reinforces their potential as reliable materials for high-temperature applications, such as H₂ storage and clean energy technologies.

Ab initio molecular dynamics (AIMD) calculations

The thermal and dynamic stability of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides were systematically evaluated through AIMD simulations conducted at room temperature, as depicted in Fig. 3a–c. The results reveal strikingly similar behaviors across all three compounds, highlighting their robust thermodynamic stability. Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆ exhibit stable total energy profiles, with fluctuations confined within tightly constrained ranges, underscoring a lack of significant lattice degradation or phase transitions throughout the simulation period. Specifically, Na₂LiZrH₆ fluctuates between −310.86 and −310.71 Ry, Na₂LiVH₆ between −356.67 and −356.52 Ry, and Na₂LiCrH₆ between −387.03 and −386.91 Ry. These confined energy oscillations suggest that the systems are thermally stable and do not undergo any energetic instabilities, ensuring that their structural integrity remains intact under thermal perturbations. The slight difference in energy profiles across the compounds can be attributed to the differing electronic states of the transition metals. The higher binding energies in Na₂LiCrH₆ are likely due to the stronger bonding associated with Cr, contributing to its higher thermodynamic stability. The temperature fluctuations, oscillating around the equilibrium values of 300 K for all compounds, further confirm the sustained thermodynamic equilibrium throughout the simulations, with temperatures fluctuating between 270 and 330 K for Na₂LiZrH₆, 280 and 340 K for Na₂LiVH₆, and 270 and 320 K for Na₂LiCrH₆. These results underscore the durability of Na₂LiXH₆ compounds under realistic, dynamic conditions, which is crucial for their potential applications in energy storage.

Fig. 3 AIMD total-energy traces versus time for (a) Na₂LiZrH₆, (b) Na₂LiVH_6, and (c) Na₂LiCrH₆ perovskite hydrides.

In addition to energy and temperature stability, the pressure fluctuations observed across all three compounds remain periodic and within expected bounds, further demonstrating their structural resilience.⁴⁹ The absence of any abrupt changes in pressure supports the notion that these perovskite hydrides can maintain their crystallographic framework under varying thermal and pressure conditions. Such stability is particularly important for materials intended for high-temperature applications, where they must endure fluctuating conditions without undergoing degradation or structural failure.^38,50 These results are of significant importance for the potential application of Na₂LiXH₆ hydrides in energy storage systems, such as H₂ storage, where materials must endure fluctuating temperatures and pressures without compromising their structural integrity. The differences in the pressure fluctuation behavior are minimal across the three compounds, with slight differences indicating the varying degrees of resistance to mechanical stress due to the electronic properties of Zr, V, and Cr. The consistent stability observed across Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆ positions these compounds as promising candidates for use in demanding high-temperature applications, reinforcing their potential for advancing clean energy technologies.

Hydrogen storage

Hydrogen is a dynamic, sustainable energy carrier with an unparalleled gravimetric energy density and flawlessly clean combustion, making it an essential part of the shift to carbon-neutral systems as the world looks for the next generation of clean power. However, a significant obstacle to its broad use is still the absence of safe, effective, and compact storage solutions. In this context, complex hydrides, particularly double perovskite-type hydrides, have attracted increasing theoretical interest due to their high hydrogen content and structural tunability. However, Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides remain largely unexplored in the literature, especially with respect to a systematic first-principles assessment of their stability and hydrogen-related properties.

Based on our first-principles calculations, Na₂LiXH₆ (X = Zr, V, Cr) exhibit favorable structural and dynamical stability, which motivates their consideration as potential candidates for solid-state hydrogen storage rather than established storage materials. As seen in Table 2, the H₂ storage capacity of Na₂LiXH₆ (X = Zr, V, Cr) was evaluated in terms of gravimetric capacity (C_wt%), volumetric capacity (ρ_vol), and desorption temperature (T_des). Eqn (4) was used to calculate the gravimetric H₂ capacity, which is defined as the ratio of hydrogen mass to the total mass of the hydride.⁵¹

where m_h and m_host represent the molar masses of hydrogen and the host material, respectively. Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆ have determined C_wt% of 4.03 weight percent, 5.50 weight percent, and 5.45 weight percent, respectively. Among these, Na₂LiVH₆ and Na₂LiCrH₆ show potential as high-capacity H₂ storage materials by getting close to the U.S. DOE 2025 objective of 5.5 weight percent. The decrease in molecular weight and increased hydrogen bonding strength within the lattice, which increases hydrogen uptake efficiency, are the main causes of the rise in C_wt% from Zr to V and Cr replacement. Another crucial performance parameter for real-world storage applications where compactness is necessary is the volumetric H₂ capacity (ρ_vol), which shows the hydrogen content per unit volume. It was evaluated using eqn (5).⁵²

Table 2 The ρ_vol, T_des and gravimetric ratios of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides

Compound	Cwt (%)	ΔHf (kJ per mol per H2)	ρvol (g per H2 per L)	Tdes (K)	References
K2NaAlH6	4.47	—	74.25	484.52	56
Cs2NaInH6	1.48	—	14.18	492.70	57
K2LiTiH6	4.38	−56.96	19.12	435.80	46
Ca2LiCuH6	3.86	—	15.68	717.20	39
Na2LiZrH6	4.03	−60.54	20.11	463.16	Present study
Na2LiVH6	5.50	−70.61	24.24	540.23	Present study
Na2LiCrH6	5.45	−77.69	25.52	594.44	Present study

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The ρ_vol were determined to be 20.11 g per H₂ per L for Na₂LiZrH₆, 24.24 g per H₂ per L for Na₂LiVH₆, and 25.52 g per H₂ per L for Na₂LiCrH₆, as indicated in Table 2. Denser atomic packing and stronger lattice cohesion increase hydrogen density per unit volume, as indicated by the increasing trend in ρ_vol from Zr to Cr replacement. This is beneficial for applications requiring both high energy density and compact storage. When evaluating a storage material's viability, especially for fuel cell integration, hydrogen desorption is represented by the desorption temperature (T_des) is crucial. Gibbs free energy governs the desorption process thermodynamically and is characterized by (eqn 6) and (7).^53,54

ΔG = ΔH − T_des × ΔS (6)

where the enthalpy and entropy changes during desorption are represented by ΔH and ΔS. According to the computed results, T_des of Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆ are 463.16, 540.23, and 594.44 K, respectively. Stronger metal-hydrogen interactions inside the lattice are correlated with a progressive increase in T_des, especially for Cr-substituted systems. These temperatures show that these materials are thermodynamically stable and capable of reversible hydrogen release under moderate heating conditions, even if they are somewhat higher than the optimal range for direct PEM fuel cell operation (233–333 K).⁵⁵ In brief, both gravimetric and volumetric measurements demonstrate that Na₂LiVH₆ and Na₂LiCrH₆ outperform Na₂LiZrH₆ in terms of H₂ storage capacity and stability. The higher desorption enthalpy and temperature values for Na₂LiCrH₆ confirm stronger M–H bonding and improved thermal resistance, which makes it particularly attractive for high-temperature H₂ storage and transport applications. The present results demonstrate that controlled transition-metal substitution can effectively tune the hydrogen-related properties of Na₂LiXH₆ perovskite hydrides. While the present analyses are based on thermodynamic stability, phonon dispersion, and AIMD simulations, they establish a theoretical foundation that motivates future experimental validation and kinetic studies.

Mechanical properties

The mechanical properties serve as a crucial indicator of the structure-dependent elastic response in perovskite hydrides. Accurate determination of second-order elastic constants (C_ij) is fundamental for evaluating mechanical stability and understanding a material's resistance to external deformation and applied stress. In the present study, the elastic constants of Na₂LiXH₆ (X = Zr, V, Cr) were systematically computed, from which essential mechanical moduli and anisotropic factors were derived to provide a comprehensive assessment of their stability and mechanical resilience. The obtained parameters not only validate the mechanical robustness of these compounds but further elucidate their deformability, rigidity, and bonding nature, which are key determinants of their reliability and efficiency under hydrogen storage conditions. The elastic response of Na₂LiXH₆ (X = Zr, V, Cr) was characterized by computing the distinct elastic coefficients C₁₁, C₁₂, and C₄₄ corresponding to the cubic symmetry; the full set of derived moduli is reported in Table 3. The calculated C₁₁, C₁₂, and C₄₄ values increase from Zr to Cr, indicating that substitution at the X site produces a progressive stiffening of the lattice. Physically, the rising C₁₁ and C₄₄ reflect enhanced resistance to longitudinal and shear deformations, respectively, as the X-site cation is varied from Zr to Cr.

Table 3 The computed elastic moduli of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides

Elastic moduli	Na2LiZrH6	Na2LiVH6	Na2LiCrH6
C11	47.74	71.24	93.23
C12	15.58	22.43	28.15
C44	18.45	26.74	30.15
Cp	−2.87	−4.31	−2.0
B(GPa)	26.29	38.69	49.85
G(GPa)	17.46	25.78	31.08
E(GPa)	42.89	63.28	77.20
B/G	1.51	1.50	1.60
υ	0.22	0.23	0.24
C′	16.08	24.41	32.54
A^U	1.15	1.10	0.93

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All three compounds satisfy the Born mechanical criteria for cubic symmetry, namely C₁₁ − C₁₂ > 0, C₁₁ + 2C₁₂ > 0, and C₄₄ > 0. In particular, the shear constant (C′ = (C₁₁ − C₁₂)/2) is positive in every case (16.08, 24.41, and 32.54 GPa), confirming mechanical stability against shear modes and indicating that the relaxed structures are mechanically robust (Table 3).

The bulk modulus (B) increases progressively from 26.29 to 38.69 to 49.85 GPa, while the shear modulus (G) rises from 17.46 to 25.78 to 31.08 GPa, and the Young's modulus (E) increases from 42.89 to 63.28 to 77.20 GPa. All elastic moduli were computed using eqn (8)–(12).^58–61 This consistent enhancement in elastic moduli demonstrates that Na₂LiCrH₆ exhibits the greatest stiffness and lowest compressibility within the series. The systematic improvement in elastic stiffness is closely associated with the observed reduction in lattice constants and the increase in cohesive energy, suggesting that the Cr-based compound possesses superior mechanical integrity under both hydrostatic and shear stresses.

Ductility/brittleness was assessed using Pugh's ratio (B/G) and Poisson's ratio (υ) as shown in Fig. 4. The computed B/G values (1.51, 1.50, and 1.60) are all below the empirical ductile threshold of ≈1.75, and υ (0.22, 0.23, and 0.24) remain below 0.26; both criteria therefore classify these materials as predominantly brittle with limited plasticity. Nonetheless, the slight increase of B/G and υ for Na₂LiCrH₆ indicates a modest shift toward less brittle behavior relative to the Zr- and V-variants. Cauchy pressure, defined as C_p = C₁₂ − C₄₄, is negative for all three compositions (−2.87, −4.31, and −2.00 GPa). A negative C_p generally signals significant directional (covalent-like) bonding contributions; thus, the elastic data imply that the bonding in these hydrides has a notable non-ionic/covalent component, consistent with strong X–H and M–H interactions in the framework. Elastic anisotropy was quantified using the universal anisotropy index A^U. The computed A^U values (1.15, 1.10, and 0.93) show that Na₂LiCrH₆ is the most nearly isotropic (A^U ≈ 1), whereas Na₂LiZrH₆ and Na₂LiVH₆ display modest anisotropy. The presence of finite anisotropy, particularly in the Zr and V compounds, suggests directional dependence of mechanical response that should be considered when designing single crystals, thin films, or polycrystalline components. The elastic data verify that all three Na₂LiXH₆ perovskites are mechanically stable in the cubic phase. Progressive substitution from Zr to V to Cr strengthens the lattice, as reflected by the higher values of C_ij, B, G, and E, and slightly decreases brittleness. Among them, Na₂LiCrH₆ exhibits the most favorable combination of stiffness and nearly isotropic elastic behavior. These mechanical characteristics, enhanced rigidity and structural stability for the Cr-based compound, are advantageous for practical handling and integration into devices. However, the inherently brittle nature of these materials may necessitate microstructural engineering strategies, such as the development of composites or protective coatings, to enable their use in applications requiring greater mechanical flexibility. Such elastic attributes significantly reinforce the structural stability and adaptability of these compounds, thereby emphasizing their strong potential for efficient hydrogen (H₂) storage applications.

Fig. 4 Graphs of (a) Poisson's ratio (υ), and (b) Pugh's ratio (B/G)of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides.

Electronic properties

Electronic characteristics, especially the energy bandgap (E_g) and density of states (DOS), serve a vital function in controlling the adsorption and desorption behavior, structure stability, as well as catalytic activity of hydrogen storage materials, thereby facilitating efficient hydrogen dissociation and recombination processes. A detailed understanding of the electronic band structure is fundamental to elucidating the electrical conductivity and optical behaviour of crystalline hydrides. It provides insights into energy distribution and bonding characteristics within the material. Fig. 5 presents the calculated band structures of Na₂LiXH₆ (X = Zr, V, Cr) along the high symmetry path X–R–M–Γ–R, obtained using the GGA-PBE functional. The Fermi level is indicated by a dashed olive line. All three compounds display an overlap at the Fermi level, with the valence band maximum (VBM) and conduction band minimum (CBM) located at distinct symmetry points, confirming the absence of a bandgap and indicating their metallic nature. The valence bands (VB) near the Fermi level appear relatively flat, suggesting localized bonding states, while the conduction bands (CB) show greater dispersion, implying enhanced electron mobility. With substitution from Zr to V to Cr, there is a gradual increase in the dispersion of the conduction bands and a narrowing of the valence bandwidth, reflecting subtle electronic restructuring across the series. These observations are key to understanding the DOS and optical response discussed in the subsequent section.

Fig. 5 Computed band structures of describing (a) Na₂LiZrH₆, (b) Na₂LiVH₆ and (c) Na₂LiCrH₆ perovskite hydrides.

Fig. 6 presents the total and partial density of states (PDOS) for Na₂LiXH₆ (X = Zr, V, Cr) hydrides, offering comprehensive insights into their electronic characteristics and bonding nature. In all three systems, the Fermi level (E_F) intersects prominent peaks, indicating metallic characteristics and significant electronic delocalization. For Na₂LiZrH₆, the VB region from about −5 eV up to the E_F is mainly formed by the hybridized Zr-d and H-s orbitals, with minor contributions from Na-s and Li-s states. The CB is primarily governed by Zr-d states, indicating pronounced metal-hydrogen interactions near the E_F. In the case of Na₂LiVH₆, the top of the VB primarily originates from V-d and H-s orbitals, while the CB reveals further hybridization between V-d and H-s states, pointing to strong covalent interactions within the VH₆ octahedra. Likewise, Na₂LiCrH₆ exhibits intense mixing between Cr-d and H-s orbitals near the E_F, where the partially filled Cr-d states play a significant role in verifying the electronic character. This pronounced d-s hybridization suggests a comparatively more covalent Cr–H bonding character than in the Zr- and V-based counterparts.

Fig. 6 TDOS and PDOS for Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides.

As the transition metal changes from Zr to V and then to Cr, the increasing covalent contribution reduces the purely ionic stabilization of H⁻, which may facilitate hydrogen mobility and lower the dehydrogenation barrier. Consequently, while Na₂LiCrH₆ exhibits strong metal–hydrogen interactions and high thermal stability, its hydrogen storage behavior should be viewed as a balance between enhanced bonding strength and potentially improved hydrogen release kinetics rather than exclusively superior high-temperature storage performance. Overall, the PDOS analysis confirms that Na₂LiXH₆ (X = Zr, V, Cr) exhibits metallic behavior at the E_F, enabling rapid charge transport and promoting efficient hydrogen storage-release dynamics.

Optical properties

Optical properties offer valuable information on the interaction between materials and incident electromagnetic waves, revealing the bandgap nature, electronic transition mechanisms, and hydrogen adsorption and desorption behavior. These attributes are fundamental for assessing the potential of hydride-based compounds in advanced functional and energy-related applications. In the context of hydrogen storage, the metallic or insulating nature of a material critically influences thermal and electronic transport during hydrogen uptake and release.^62–65 Metallic behavior, associated with enhanced electronic and thermal conductivity, can promote efficient heat management and improved cycling kinetics, whereas insulating or wide-bandgap characteristics may reduce electronic losses and contribute to structural stability.⁶⁶ The optical response discussed herein is fully consistent with this electronic classification, where metallic systems exhibit pronounced low-energy absorption without a distinct optical band edge, while insulating or semiconducting compounds show a clear absorption onset arising from interband transitions. This correlation establishes a coherent link between optical behavior, electronic structure, and hydrogen storage performance. Fig. 7 shows the computed optical properties of Na₂LiXH₆ (X = Zr, V, Cr) in the photon energy range 0–10 eV, including the real [ε₁(ω)] and imaginary [ε₂(ω)] parts of the dielectric function, absorption coefficient [α(ω)], and reflectivity R(ω). The trends in these spectra are consistent with the previously discussed electronic structures, highlighting the influence of the transition-metal cation on light–matter interaction. Fig. 7a illustrates the real part of the dielectric function, ε₁(ω), where Na₂LiVH₆ displays the highest static dielectric constant, signifying stronger polarization and enhanced electronic screening. Na₂LiZrH₆ exhibits a comparatively moderate response, whereas Na₂LiCrH₆ shows the lowest value, indicating reduced polarizability. Hence, the pronounced dielectric response of Na₂LiVH₆ enhances its suitability for hydrogen storage systems, as efficient polarization promotes effective charge transfer and strengthens the overall adsorption–desorption dynamics. The imaginary part ε₂(ω) (Fig. 7b) demonstrates a similar order, where Na₂LiVH₆ and Na₂LiZrH₆ possess pronounced low-energy peaks arising from interband transitions near the Fermi level, whereas Na₂LiCrH₆ displays weaker intensity and a shift of major optical activity toward higher energies. Hence, the distinct interband transitions observed in Na₂LiCrH₆ enhanced its potential for hydrogen storage applications by promoting light-induced electronic excitation and facilitating efficient hydrogen desorption processes.

Fig. 7 Optical response of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides.

The absorption spectra (Fig. 7c) confirm that Na₂LiVH₆ and Na₂LiZrH₆ absorb significantly in the visible and near-UV regions, with multiple absorption edges starting near their band gaps, whereas Na₂LiCrH₆ exhibits a sharp increase only in the ultraviolet domain, consistent with its wider gap. Reflectivity (Fig. 7d) trends complement this observation: Na₂LiVH₆ and Na₂LiZrH₆ exhibit high low-energy reflectance (>0.5), indicating reduced transmission, whereas Na₂LiCrH₆ maintains relatively low reflectivity and a smoother spectral variation. Hence, Na₂LiVH₆ and Na₂LiZrH₆ demonstrate strong optical screening and intense interband transitions, making them promising for broadband absorption-based devices, whereas Na₂LiCrH₆, with its moderate dielectric response and dominant UV absorption, stands out as a potential candidate for ultraviolet optoelectronic or protective coating applications.

Thermal properties

The thermodynamic properties of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides are crucial in assessing their capabilities for hydrogen storage and release mechanisms. Favorable thermodynamic characteristics in H₂ storage materials facilitate effective cycles of hydrogen absorption and desorption, which are essential for preserving reversibility and long-term stability. The structural response of these hydrides to temperature changes and their capacity to efficiently store and release hydrogen can be better understood by examining characteristics like heat capacity, enthalpy, entropy, and free energy. If the hydrides' thermodynamic profile remains constant, they can endure multiple cycles of hydrogenation and dehydrogenation without experiencing noticeable degradation in structure.⁶⁷ Examining thermodynamic parameters like longitudinal sound velocity (v_l), transverse sound velocity (v_t), average sound velocity (v_m), Debye temperature (θ_D), and melting temperature (T_m) is essential for understanding the vibrational and lattice dynamics that affect hydrogen diffusion and lattice resilience at high temperatures. Eqn (13)–(15) were used to derive these parameters:^68,69

T_m = [553 + 5.911(C₁₁)] ± 300 (15)

where ρ is the material density, M is the molecular weight, C₁₁ is the elastic constant, and B and G are the bulk and shear moduli. These equations connect the material's thermal and dynamic stability, which is essential for H₂ storage devices that operate in temperature-varying environments, to mechanical stiffness and atomic vibrations.

All three compounds (Na₂LiZrH₆, Na₂LiVH₆, and Na₂LiCrH₆) show identical thermodynamic trends with increasing temperature, as shown in Fig. 8a–d and detailed in Table 4. At increasing temperatures, the heat capacity (Fig. 8a) and enthalpy (Fig. 8b) increase monotonically, indicating increased phonon contributions and lattice vibration amplitudes. Notably, Na₂LiCrH₆ exhibits the highest enthalpy and heat capacity values, indicating improved energy storage and stronger interatomic bonding. The vibrational and configurational entropy contributions grow with temperature, supporting the desorption process by lowering the Gibbs free energy barrier, as the total entropy (Fig. 8c) rises linearly. Fig. 8d shows that all compounds exhibit a consistent decline in free energy as the temperature rises, a typical behavior of thermodynamically stable hydrides that facilitates hydrogen release at higher temperatures.

Fig. 8 Thermal parameters of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides.

Table 4 Calculated thermodynamic parameters of Na₂LiXH₆ (X = Zr, V, Cr) perovskite hydrides

Parameters	Na2LiZrH6	Na2LiVH6	Na2LiCrH6
vl (km s^−1)	3.08	3.74	4.20
vt (km s^−1)	2.09	2.54	2.79
vm (km s^−1)	2.28	2.77	3.05
θD (K)	369	497	546
Tm (K)	835	974	1104

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The most reliable thermodynamic performance is shown quantitatively by Na₂LiCrH₆ with v_l = 4.20 km s⁻¹, v_t = 2.79 km s⁻¹, v_m = 3.05 km s⁻¹, θ_D = 546 K, and T_m = 1104 K, signifying higher lattice stiffness and strong atomic interactions. Na₂LiZrH₆ has slightly lower values (θ_D = 369 K, T_m = 835 K), suggesting a relatively softer lattice, while Na₂LiVH₆ follows closely (θ_D = 497 K, T_m = 974 K). By increasing vibrational frequency and phonon transport efficiency, Cr inclusion strengthens lattice rigidity and improves thermal stability, as demonstrated by the observed rise in Debye temperature from Zr to Cr substitution. Na₂LiCrH₆'s mechanical and thermal endurance during hydrogen cycling is improved by its higher Debye temperature, which indicates a higher phonon frequency and a smaller atomic vibration amplitude. These findings demonstrate that Na₂LiCrH₆ has the strongest lattice among the compounds under investigation, which makes it a viable option for H₂ storage applications where effective absorption–desorption performance depends on high thermal and structural stability.

Conclusion

In summary, this in-depth first-principles investigation elucidates the intrinsic correlation between the structural, mechanical, electronic, optical, and thermal characteristics of Na₂LiXH₆ (X = Zr, V, Cr) double perovskite hydrides. The compounds crystallize to form a cubic Fm3̄m structure, showing negative formation energies and favourable tolerance factors. These factors indicate that these compounds are structurally and thermodynamically stable. All compounds maintain mechanical and thermodynamic stability within the cubic phase, exhibiting progressive improvements in elastic stiffness, cohesive strength, and lattice integrity from Zr to Cr substitution. Among the studied systems, Na₂LiVH₆ demonstrates the most resilient configuration, supported by its bulk modulus, Debye temperature (597 K), melting point (974 K), and sound velocities, confirming strong atomic bonding and high thermal endurance. Its favorable desorption enthalpy (−70.61 kJ per mol per H₂) and volumetric hydrogen density (24.24 g per H₂ per L) further verify its superior hydrogen storage capability. In addition, the compound's broad optical absorption and pronounced dielectric response suggest potential for dual hydrogen-optoelectronic functionality. Collectively, the synergistic combination of mechanical rigidity, thermal stability, and optical activity designates Na₂LiVH₆ as a potential multifunctional material for sustainable hydrogen storage and advanced optoelectronic technologies.

Ethical statement

Approval all authors affirm that this submission is entirely original, has not been published elsewhere, and fully adheres to the established ethical standards for scholarly research and publication.

Author contributions

All authors affirm their substantial contributions to this research and collectively assume full responsibility for its content, encompassing conceptualization, computational analysis, manuscript preparation, and revision processes.

Conflicts of interest

The authors declare that there are no financial or personal relationships that could have influenced the outcomes, interpretation, or conclusions presented in this study.

Data availability

The data will be provided on request by the corresponding author.

Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d5ra08708b.

Acknowledgements

The authors sincerely acknowledge the International Islamic University, Islamabad, Pakistan, for extending essential computational facilities and research infrastructure that enabled the successful execution of this study.

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