DFT and AIMD predictions of Li₂XBH₆ (X = Li, Na, K) double perovskites for efficient hydrogen storage and photocatalytic applications

Abstract

Energy demand in today's world is increasing. So, to meet this energy demand and reduce reliance on fossil fuels, we need to look to renewable energy sources, and hydrogen storage can be one of them. This study conducted a magnificent investigation into the thermo-dynamical stability, optoelectronic properties, hydrogen storage capability, and photocatalytic activity of Li₂XBH₆ (X = Li, Na, K) Double perovskite hydrides (DPH) using a density functional theory (DFT)-based approach, with ab initio molecular dynamics (AIMD) simulations, which specifies that the compounds are thermally stable. The electronic structure analysis indicated that all compounds exhibit semiconductor behavior, with band gap energies of 1.79 eV for Li₂LiBH₆, 1.66 eV for Li₂NaBH₆, and 1.48 eV for Li₂KBH₆, suggesting their potential for efficient photonic and optoelectronic applications such as solar cell absorbers. This study shows a prominent initial theoretical GHSC, with Li₂LiBH₆ demonstrating the highest storage potential at 16.05 wt%, Li₂KBH₆ the lowest at 8.66 wt%, and Li₂NaBH₆ 11.25 wt% – all considerably surpassing the US-DOE target of ≥5.5 wt%. The study shows that these substances are promising theoretical candidates for advanced optoelectronic devices, hydrogen storage materials, and photocatalytic applications.

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1. Introduction

Rising environmental concerns and the ongoing global energy demand have heightened the demand for energy-efficient alternatives to conventional fossil fuels.^1,2 Hydrogen has emerged as a key energy carrier due to its high energy density and potential for direct conversion into mechanical, electrical, and thermal energy.^3,4 However, practical utilization of hydrogen requires the development of a safe, lightweight, efficient, and reversible storage system.⁵ There are many hydrogen storage methods, but solid-state storage in complex hydrides has obtained notable recognition due to their high gravimetric hydrogen storage capacity (GHSC), thermodynamic stability, and mechanical reliability.^6,7 Perovskite-type hydrides (ABH₃) represent an encouraging class of materials for solid-state hydrogen storage.⁸ Their extensive surface areas enable substantial hydrogen adsorption, while their excellent thermal and chemical stability allows operation under elevated temperatures and harsh chemical environments.⁹ There is another advanced subclass of double perovskites, A₂BB′H₆, which increases lattice flexibility and enhances mechanical, electronic, and thermodynamic characteristics. Such structural advantages make DPHs particularly suitable for reversible hydrogen storage at moderate temperatures.¹⁰ Releasing hydrogen from a substance is termed desorption.¹¹ The thermal stability of perovskites makes them well-suited for hydrogen desorption at elevated temperatures. Their outstanding chemical stability makes them suitable for use in corrosive or demanding chemical environments.¹² The ease of manufacture and economic feasibility of these materials increase their suitability for large-scale hydrogen storage applications.^13,14 Though perovskites are seen as a promising solution for hydrogen storage, multiple difficulties must be solved before real-world application.¹⁵ Key obsessions include increasing the kinetics of hydrogen adsorption and desorption, improving the stability and long-term durability of substances under operational conditions, and boosting storage capacity. Because of their magnificent gravimetric hydrogen densities, usually between 1.2 and 6.0 wt%, light metal perovskite hydrides demonstrate strong potential for use in hydrogen storage technologies.¹⁶

A significant amount of research is currently dedicated to investigating materials for hydrogen storage applications. Researchers are conducting many studies and investigations on materials to improve H₂ storage capacity. Recent density functional theory (DFT)-based investigations have found that Mn-based hydride single perovskites KMnH₃ and LiMnH₃ exhibit GHSC of 3.12 and 4.67 wt%, respectively.¹⁷ Similarly, other studies reported GHSC of 7.946, 4.306, 5.588, and 5.456 wt% for LiSiH₃, KSiH₃, NaSiH₃, and MgSiH₃.¹⁸ In addition, Cu-based hydrides exhibited relatively lower capacities, namely 3.23 wt% for AlCuH₃, 2.22 wt% for GaCuH₃, and 1.67 wt% for InCuH₃.¹⁹ Extensive investigations into single-perovskite hydrides have provided valuable insights into their hydrogen-storage performance. Specifically, the NaXH₃ (X = Be, Mg, Ca, Sr) family exhibits capacities between 2.6 and 8.6 wt%,²⁰ while XNiH₃ (X = Li, Na, K) compounds show values of 3.30–4.40 wt%.²¹ Among chalcogen-based hydrides, LiSH₃ and LiSeH₃ achieve capacities of 7.19 wt% and 3.40 wt%,²² respectively. Transition-metal-based cesium hydrides demonstrate moderate performance, with CsCoH₃ and CsZnH₃ offering 2.82 wt% and 3.09 wt%,²³ respectively. But nowadays, DPHs are attracting the attention of researchers because of their greater structural flexibility and improved functionality, as we mentioned before. So, a large number of investigations on DPH are ongoing. In recent times, DFT-based research has found that Cs₂CaCdH₆, Rb₂CaCdH₆, and DPH show GHSC of 1.39 wt% and 1.69 wt%, respectively.²⁴ DPHs that have already investigated and their GHSCs are 1.78 to 5.39 wt% for X₂CaAlH₆ (X = K, Rb, and Cs),²⁵ 2.956 wt% for Na₂CaCdH₆,²⁶ 1.14 and 1.86 wt% for Rb₂AlXH₆ (X = In, Tl),²⁷ 5.94 and 3.11 wt% for X₂LiBH₆ (X = K, Rb)²⁸ and 2.40 to 6.39 wt% for A₂LiCuH₆ (A = Be/Mg/Ca/Sr).²⁹ However, recent DFT investigations on single perovskite hydride and DPHs report GHSCs typically under 8.6 wt%, with most DPHs remaining under 7.05 wt%. These limitations encourage the research of lightweight boron-based DPHs, which remain largely uninvestigated and may enable substantially enhanced hydrogen storage performance.

Most reported studies focus on the individual physical characteristics or hydrogen storage properties of related DPHs. In contrast, a unified evaluation that combines hydrogen storage, optoelectronic, and photocatalytic performance for the Li₂XBH₆ system has not been explored. In this research, we present a comprehensive first-principles investigation of Li₂XBH₆ (X = Li, Na, K) DPHs, highlighting their structural stability, electronic properties, hydrogen storage behavior, and photocatalytic potential using DFT. This provides new information on their structural, optoelectronic, and hydrogen storage characteristics.

In this study, we theoretically present a comprehensive first-principles investigation of Li₂XBH₆ DPHs using DFT coupled with AIMD simulations. We examine their structural stability, electronic properties, hydrogen storage performances, and photocatalytic potential. While experimental synthesis is influenced by precursor choice, chemical purity, and stoichiometric control,^30–32 this theoretical work provides an initial framework for guiding future experimental studies under controlled conditions. Remarkably, these compounds exhibit theoretical GHSCs that exceed the U.S. DOE target and favorable band gap alignments for photocatalytic applications. This work, therefore, provides novel insights into the multifunctional energy potential of Li₂XBH₆ DPHs and establishes a theoretical roadmap for future experimental efforts.

2. Methodology

The first-principles calculations are executed within the DFT framework using the full-potential linearized augmented plane wave (FP-LAPW) method as applied in the WIEN2k computational package,^33,34 which is well-established for accurate treatment of electron interactions in solid-state systems. The exchange–correlation effects for structural and mechanical optimization are treated using the GGA-PBE_sol functional,³⁵ selected for its improved reliability in predicting equilibrium lattice parameters and elastic behavior of perovskite-type hydrides. Recognizing the common tendency of semi-local GGA functionals to underestimate band gaps, the Tran–Blaha modified Becke–Johnson (TB-mBJ) potential is applied to achieve a high-accuracy evaluation of electronic and optical properties with reduced computational cost.^36–38 Brillouin-zone integrations were performed using a Monkhorst–Pack k-point mesh of 12 × 12 × 12 to ensure reliable total-energy convergence. The basis size (R_MT × K_max = 8), Fourier expansion cutoff charge density (G_max = 13), charge convergence = 10⁻⁵ Ry, and energy convergence = 0.0001 eV were set for optimization. To assess the thermal stability of the Li₂XBH₆ (X = Li, Na, K) DPH, AIMD simulations were carried out under constant-temperature conditions, allowing evaluation of structural robustness at elevated temperatures. Highly dense k-point meshes of 5000 and 10 000 points were used to accurately calculate the electronic and optical properties.

The optical parameters were computed from the real ε₁(ω) and imaginary part ε₂(ω) of dielectric tensor ε(ω) and follows:^39–41

3. Results and discussion

3.1 Structural, mechanical stability, and AIMD simulations

The Double perovskite's general structure is A₂BB′X₆,⁴² where A, B, and B′ are cations and X is an anion. The structural characteristics of DPH Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ were examined in detail using first-principles DFT-based computational techniques. This compounds forms crystals with a double perovskite structure that belong to the Fm3̄m space group (no. 225). These structures contain 40 atoms per cubic cell, where Li atoms occupy the Wyckoff position at 8c , X atoms are at 4b , B atoms occupy 4a (0, 0, 0), and H atoms are at the positions 24e . The compounds' lattice properties were optimized by GGA-PBE_sol, demonstrating a remarkably stable crystal structure in all three materials (Fig. 1). The lattice constants for Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ were 6.2136 Å, 6.8034 Å, and 7.4815 Å, respectively, demonstrating minimal change with increasing cation size, except for potassium, where we observe a noticeable change. From Fig. 2, we found that Li₂LiBH₆ minimizes its overall energy (E₀) at −101.0542 Ry, with a volume (V₀) of 404.7378 bohr³. For Li₂NaBH₆, at a volume of 531.2702 bohr³, it minimizes its overall energy at 409.9652 Ry, whereas at a volume of 706.489 bohr³, Li₂KBH₆ minimizes its overall energy at −1288.2530 Ry. These values are consistent with the expected behavior of perovskite structures, in which greater cations result in greater unit cell diameters. A crystalline material's formation energy, (ΔE_f) offers important information about its primary stability and chemical makeup.⁴³

Fig. 1 Unit cell of Li₂XBH₆ (X = Li, Na, K): 2D view shown is on the left side, and 3D view is on the right side.

Each one's formation energy was determined using eqn (7):⁴⁴

ΔE_f = E_total(Li₂XBH₆) − (2E_tot(Li) + E_tot(X) + E_tot(B) + 6E_tot(H)) (7)

An element's atomic or molecular arrangement and bonding inside the crystal structure are specified by its formation energy.^43,45 The physical or structural stability factors are:Here, t refers to the tolerance factor and μ refers to the octahedral factor, and R_A, R_B, R_B′, and R_H are the corresponding ionic radius of the structures’ materials. The tolerance factors, which lie within the favorable range of 0.81–1.1,⁴⁶ were calculated to be 0.953 for Li₂LiBH₆, 0.892 for Li₂NaBH₆, and 0.825 for Li₂KBH₆. These values imply that the structures are stable and won't distort under typical circumstances.

Fig. 2 Energy vs. volume graphs of (a) Li₂LiBH₆, (b) Li₂NaBH₆, and (c) Li₂KBH₆ after optimization.

The bonding environment of the BX₆ octahedra remains favorable for all three compounds, as indicated by the octahedral factors, which were calculated to be 0.37, 0.46, and 0.58, respectively. These values fall within the acceptable range of 0.377 to 0.895⁴⁷ (Table 1).

Table 1 The structural properties evaluated for the studied materials

Hydrides	a0 (Å)	V0 (bohr^3)	E0 (Ry)	t	μ	ΔEf (eV per atom)
Li2LiBH6	6.22	404.74	−101.05	0.953	0.37	−2.12
Li2NaBH6	6.80	531.27	−409.97	0.892	0.46	−1.96
Li2KBH6	7.48	706.48	−1288.25	0.825	0.58	−1.83

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Several numerical experiments on the elastic constants (C₁₁, C₁₂, C₄₄), bulk (B), shear (G), and Young moduli (E), Poisson's ratio (ν), and Pugh's ratio (B/G) are used to confirm their mechanical stability. Stable elastic constants, high stiffness, and resistance to deformation under applied stress are among the materials' advantageous mechanical characteristics. A key consideration for the use of materials in hydrogen storage systems is their mechanical stability, especially for those that are subjected to multiple hydrogen absorption and desorption cycles, as well as for transportation implementation.^48,49 A well-known checklist (Born stability) for determining a material's mechanical soundness uses its elastic constants.^50–53

(C₁₁ − C₁₂) > 0, (C₁₁+ 2C₁₂) > 0, C₁₁ > 0, C₄₄ > 0, C₁₂ < B < C₁₁ (10)

Here, B denotes the bulk modulus, and the values of C₁₁, C₁₂, and C₄₄ are given in Table 2.

Table 2 Mechanical properties evaluated for Li₂XBH₆ (X = Li, Na, K) hydrides

Hydrides	C11	C12	C44	B	G	E	ν	B/G	DT
Li2LiBH6	75.546	61.403	57.954	66.117	26.272	69.598	0.325	2.517	921.139
Li2NaBH6	31.186	59.507	16.261	50.067	59.844	128.382	0.073	0.837	1186.029
Li2KBH6	−17.302	60.787	8.572	34.757	3.131	9.119	0.456	11.101	260.910

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The encouraging outcome of this work is that Li₂LiBH₆ hydride stands out as the one that fully satisfies all the specified criteria. This material is mechanically stable and strong enough to withstand deformation, as demonstrated by its easy compliance with all the requirements. This indicates that it has a robust structure capable of withstanding mechanical stress. In contrast, Li₂NaBH₆ and Li₂KBH₆ do not satisfy the required stability conditions. Therefore, both materials are mechanically unstable, as Li₂NaBH₆ fails to satisfy the conditions (C₁₁ − C₁₂) > 0, and C₁₂ < B < C₁₁, while Li₂KBH₆ has a negative C₁₁ violating C₁₁ > 0.

The bulk modulus measures how much a material retains stability under pressure and is used to evaluate both its incompressibility and elasticity.⁵⁴ From Table 2, we found that the bulk moduli of Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ are 66.117, 50.067, and 34.757 GPa, respectively, with Li₂LiBH₆ showing the highest bulk modulus. Conversely, the stiffness of a material is characterized by Young's modulus, with higher values indicating increased rigidity.⁵⁵ In this study, we found Young's modulus 69.598 GPa for Li₂LiBH₆, 128.382 GPa for Li₂NaBH₆, and 9.119 GPa for Li₂KBH₆. Shear modulus measures a material's resistance to shear, i.e., to shape changes while keeping its volume constant. In relation to other properties, the shear modulus offers a more accurate representation of a material's stiffness.⁵⁶ The shear modulus was found to be 26.272 GPa for Li₂LiBH₆, 59.844 GPa for Li₂NaBH₆, and 3.131 GPa for Li₂KBH₆. Poisson's ratio, which indicates a material's degree of ductility, is another important consideration when evaluating its mechanical qualities. The Poisson's ratio for the hydrides under analysis is less than 0.3, which often shows brittle behavior. This indicates that under stress, the materials are more likely to crack than to flex or stretch. This is further confirmed by Pugh's ratio, which shows that the materials are brittle with values below 1.75.⁵⁷ From Table 2, we can see that Poisson's ratio (ν) of Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ are 0.325, 0.073, and 0.456, as well as Pugh's ratio (B/G) are 2.517, 0.837, and 11.101, respectively. By analyzing this information, we found that Li₂LiBH₆ and Li₂KBH₆ exhibit ductile behavior, whereas Li₂NaBH₆ is brittle. The Debye temperature (DT) shows how much heat metals can tolerate before melting. By calculating the DT, we found that the DTs of Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ are 921.139 K, 1186.029 K, and 260.910 K, respectively.

The AIMD simulation was performed to evaluate the thermal stability of these compounds.⁵⁸ The AIMD simulations were performed at 300, 500, and 800 K, with the smallest variance in fluctuations for the perovskite-type hydrides Li₂KBH₆, Li₂LiBH₆, and Li₂NaBH₆ under the NVE ensemble over a total simulation period of 30 000 fs (Fig. 3). The structural stability of the materials was assessed by analyzing the time-dependent behavior of the total energy and temperature. The energy trends for the three systems show a gradual increase with minor oscillations, indicating that the materials undergo dynamic relaxation in the simulation. Among the studied compounds, Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ show minor energy fluctuations, indicating that their structures are more stable under thermal stress. Throughout the simulation, the temperature profiles for all systems remain stable, fluctuating around 300, 500, and 800 K without notable deviations. The observed consistency demonstrates that the systems effectively maintain thermal equilibrium throughout the NVE ensemble simulation. In summary, the AIMD simulations show that Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ maintain their structural integrity at 300, 500, and 800 K. This stability is comparable to other perovskite hydrides such as LiZnH₃, NaZnH₃, and KZnH₃,⁵⁹ where ionic size and lattice flexibility play important roles in determining thermal stability.⁶⁰ Therefore, the AIMD results confirm the thermal stability of Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ at 300, 500, and 800 K. However, additional structural analyses, such as structural snapshots, radial distribution functions, and mean-square displacement, would be desirable to provide a more comprehensive assessment of long-term thermal behavior.

Fig. 3 The AIMD simulation of energy and temperature as a function of time for (a) Li₂LiBH₆, (b) Li₂NaBH₆, and (c) Li₂KBH₆ materials.

3.2 Electronic properties

The electronic properties of Li₂XBH₆ (X = Li, Na, K) are analyzed in this work through a comprehensive analysis of the band structure, TDOS, and PDOS. Band structure analysis plays a crucial role in determining a material's electronic properties. We used two methods, GGA-PBE_sol and TB-mBJ, to study the band structures and check the accuracy of our results. The band structure of Li₂XBH₆ (X = Li, Na, K) DPH along the k-path (W–L–Γ–X–W–K) in the energy range −6 to 6 eV, is shown in Fig. 4, where the left side of the figure shows the approach of the GGA-PBE_sol method and the right side shows the TB-mBJ method. The TB-mBJ approximation was employed in this study because of its superior accuracy in estimating band gaps compared to the GGA-PBE_sol approach. The TB-mBJ approach produced band gap values that showed close agreement with those reported experimentally.⁶¹ By the GGA-PBE_sol method, all three materials exhibit a metallic nature because the valence band and the conduction band overlap. But in the TB-mBJ approach, the compounds exhibit band gap energies of 1.79 eV for Li₂LiBH₆, 1.66 eV for Li₂NaBH₆, and 1.48 eV for Li₂KBH₆. From this observation, we notice that the band gap energy is decreasing with the increase in the size of the X-site atom. In addition, we also observe that the conduction band maximum of all these three compounds is placed at the Γ point, where the red dotted line represents the Fermi energy, and the valence band maximum of these compounds is placed at the X point, except for Li₂KBH₆. Because we found that the valence band maximum of Li₂KBH₆ components is actually placed at the L point, these observations indicate that all of the materials are indirect band gap semiconductors (CBM and VBM at different symmetry points).⁶² Therefore, the favorable band gap positions of Li₂XBH₆ (X = Li, Na, K) DPH make them attractive materials for optoelectronic applications such as solar cell absorbers and UV-responsive devices.^63,64

Fig. 4 The electronic band diagram of (a) Li₂LiBH₆, (b) Li₂NaBH₆, and (c) Li₂KBH₆, shown using GGA-PBE_sol (on the left side) and TB-mBJ (on the right side) techniques.

By studying the DOS and PDOS, we can better understand how electronic states are distributed near the Fermi level, which directly influences conduction and overall electronic properties.⁶⁵ So, the density of states (DOS) and partial density of states (PDOS) were also studied for Li₂XBH₆ (X = Li, Na, K) DPH to understand the distribution of electronic states near the Fermi level. Fig. 5 shows the TDOS and PDOS of titled materials, explaining the distribution of electronic states across the conduction and valence bands of the considered compounds. It is found that the total DOS below the Fermi level is higher than that in the conduction band, confirming that hole conductivity is major. In the case of the valence band, hydrogen atoms were found to contribute remarkably to the partial DOS of the Li₂XBH₆ compounds. Therefore, the findings reveal that the electronic states of H⁻ ions are the major contributors to bonding interactions in these compounds. The strong peaks in H-s within the valence band also refer to the dominance of H₂ states and H⁻ ions, which act as a significant contributor toward the overall permanence of these compounds. Conversely, the Li atom dominates the conduction band. The contribution of Li-s states in the conduction band is major in all three materials, but B-s states also have a significant role in PDOS. Here, we also found some minor contribution of Na-p in the Li₂NaBH₆ compound, but the contribution of H₂ is minimal in the conduction band, which was the major contributing state in the valence band. Furthermore, we observe that an increase in atomic radius leads to a reduction in the band gap, as the interaction between the nucleus and valence-shell electrons weakens.

Fig. 5 Total DOS and partial DOS of (a) Li₂LiBH₆, (b) Li₂NaBH₆, and (c) Li₂KBH₆.

In conclusion, the findings indicate that Li-s, B-s, and H-s states dominate the electronic transitions, enabling efficient light absorption and making them suitable for hydrogen storage and optoelectronic devices.

The effective mass of charge carriers significantly influences their mobility and response to external fields, thereby playing a critical role in photovoltaic performance.

The electron and hole effective masses were determined from the TB-mBJ band dispersion at the conduction band minimum (Γ point) and valence band maximum (X point and L point for Li₂BH₆) using the parabolic relation:⁶⁶

where ℏ is the reduced Planck constant (1.05 × 10⁻³⁴ J s⁻¹) and d²E/dK² is the second derivative of the band energy E with respect to the wave vector k.

The exciton binding energy E^ex_b is a fundamental parameter governing the efficiency of a material for photovoltaic energy-conversion applications. For high-performance solar cell absorber candidates, this energy must be sufficiently low. A low E^ex_b enables excitons (bound electron–hole pairs generated by photo-excitation) to dissociate into free charge carriers (electrons and holes) more efficiently using only the available thermal energy at room temperature. This directly improves charge transport, enhances carrier mobility, and suppresses recombination loss. Conversely, excessively high exciton binding energy hinders dissociation, leading to reduced charge-carrier mobility and elevated recombination rates, which critically degrade solar-cell efficiency and device performance. Excitonic effects originate from the coulombic attraction between photo-excited electron–hole pairs inside the crystal lattice.

In this work, the exciton binding energies of the Li₂XBH₆ family (X = Li, Na, K) were estimated using the Wannier–Mott exciton model, involving the carrier reduced mass and static dielectric response of the respective compounds. The model applied for E^ex_b evaluation follows the hydrogenic formalism:⁶⁶

where the reduced effective mass (μ_r) was calculated asThe density-of-states effective masses and the corresponding effective density of states in the valence band (N_v) and conduction band (N_c).

These are key parameters that determine the intrinsic carrier concentration and thereby influence the electrical transport and optoelectronic performance of semiconductors. Using the calculated electron and hole effective masses, the density-of-states effective masses were evaluated, and the effective densities of states at 300 K were computed using the well-established expressions:⁶⁷

where k is Boltzmann's constant, h is Planck's constant, and T is the absolute temperature.

The electron and hole carrier mobilities (μ_h and μ_e) were estimated using the effective mass approximation by assuming a relaxation time of τ = 10⁻¹⁴ s. The mobilities were calculated using the following expressions:⁶⁸

where q denotes the elementary charge 1.6 × 10⁻¹⁹ C. The calculated carrier mobility from Table 3 show that electron transport is more favorable than hole transport in all three components, as μ_e is consistently higher than μ_h. Among them, Li₂KBH₆ shows the highest electron mobility, whereas Li₂LiBH₆ shows the highest hole mobility, suggesting promising charge-transport characteristics for optoelectronic applications, as well as Li₂LiBH₆ exhibits a flat band-edge profile, indicating a lower degree of band dispersion near the valence and conduction band extrema. In contrast, Li₂NaBH₆ and Li₂KBH₆ show comparatively strong curved bands, reflecting higher dispersion.

Table 3 Electronic parameters of Li₂XBH₆ (X = Li, Na, K) compounds, including TB-mBJ band gaps, effective masses, density-of-states effective densities (N_c, N_v), and exciton binding energies

Compound	TB-mBJ (eV)	Nature			Nv	Nc	μh (cm^2 V^−1 s^−1)	μe (cm^2 V^−1 s^−1)	E^exb
Li2LiBH6	1.79	Indirect	0.23	0.19	2.77 × 10^18	2.08 × 10^18	76.37	92.45	0.08
Li2NaBH6	1.66	Indirect	0.33	0.13	4.76 × 10^18	1.18 × 10^18	53.23	135.12	0.11
Li2KBH6	1.48	Indirect	0.27	0.11	3.52 × 10^18	9.16 × 10^17	65.06	159.68	0.06

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3.3 Optical properties

A comprehensive understanding of a material's optical properties is essential for its interaction with electromagnetic radiation and for its usefulness in optoelectronic and photovoltaic devices. Some of the important optical parameters that are crucial for understanding the role of Li₂XBH₆ (X = Li, Na, K) with respect to incident photon flux and their use in various electronic devices, coatings, solar cells, as well as hydrogen storage applications include the dielectric functions, refractive indices, absorption coefficients, extinction coefficients, reflectivity, and optical conductivities.⁶⁹ These are determined by the complex dielectric function given by:^70,71

ε(ω) = ε₁(ω) + iε₂(ω) (16)

The real part, ε₁(ω), indicates the material's ability to get polarized in the presence of the external electromagnetic field, thereby allowing it to get information about the refractive properties of the material. On the other hand, the imaginary part ε₂(ω), is indicative of the material's extinction properties.⁷² From Fig. 6(a) and (b), the static dielectric constants for Li₂XBH₆ are ∼4.00, which are recorded well with cations (Li, K) providing strong polarization response, and 3.3 for Li₂NaBH₆, ensuring intermediate polarization collapse. In the visible region, all compounds feature relatively small but distinct peak features at around 2.6 eV for Li₂KBH₆, and some further fall-off ε₁(ω) reaching maxima between about 5 and 7.5 eV in the ultraviolet (UV). For ε₂(ω), negligible contributions are found at zero photon energy, whereas a small peak is again visible for Li₂KBH₆ at 3.2 eV in the visible range, with the maximum absorption occurring in the UV region for all compounds.⁷³ These results are consistent with the Penn model, which empirically correlates the static dielectric constant with the optical band gap, expressed as:⁶⁶Here, the reasonable values of ε₁(0) align with the relatively wide band gaps of Li₂XBH₆ compounds and reflect the reduced polarizability expected for materials with high-energy interband transitions. This behavior also suggests that the dielectric response is slightly increased as the effective band gap decreases (as in Li₂KBH₆), which agrees with Penn's model. The real part n(ω) and the imaginary part k(ω) in Fig. 6(c) and (d) have similar trends to those of the dielectric function, according to the optical relation n²(ω) = ε₁(ω).^74,75 At zero photon energy, n(0) lies between 1.8 for Li₂NaBH₆ and 2.0 for Li₂LiBH₆ and Li₂KBH₆ compositions. Both n(ω) and k(ω) increase steadily into the UV region, where they reach their maxima. The visible region shows a distinct peak for Li₂KBH₆, indicating a stronger optical response to the Li- and Na-based materials.

Fig. 6 (a) Real dielectric function, ε₁(ω), (b) imaginary dielectric function, ε₂(ω), (c) refractive index, n(ω), and (d) extinction coefficient, k(ω) for Li₂XBH₆ (X = Li, Na, K) materials.

Fig. 7(a) depicts that the maximum absorption coefficients for Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ are approximately 1.99 × 10⁶ cm⁻¹, 1.57 × 10⁶ cm⁻¹, and 1.37 × 10⁶ cm⁻¹ occurring at photon energies of 12 eV, 9.8 eV, and 6 eV, respectively, which measures how efficiently incoming photons drive electronic transitions from the valence to the conduction band.⁷⁶ All the compounds exhibit strong absorption in the UV region, consistent with their characteristic as wide-band-gap semiconductors. Li₂KBH₆ is the only compound that exhibits an additional, but smaller, absorption coefficient of 2.61 × 10⁵ cm⁻¹ at 3.2 eV in the visible spectrum, indicating better light-harvesting properties compared to the other compounds. The other Li and Na-based compounds also show an absorption peak (10³ to 10⁴ orders) in the visible range of 1.77 to 3.26 eV. Therefore, the studied materials may be used as the absorbing layer of a solar cell, as a silicon solar cell captures the absorption coefficients of 10³ to 10⁵ orders.⁷⁷ The optical conductivity σ(ω) in Fig. 7(b), showing the shape typical of their absorption process, exactly like the real space coupling between the photon absorption and the charge carriers’ excitation. In the low-energy region, all three materials exhibit nearly no conductance, which becomes appreciable around 4–5 eV where interband transitions set in. The main conductivity peaks are concentrated in the ultraviolet (UV) region, which presents a great number of open electronic states. Among the studied systems, Li₂NaBH₆ achieves both the highest peak and the smallest half-width, whereas a similarly strong but broader response is seen for Li₂LiBH₆. In contrast, a significant feature is observed for Li₂KBH₆, which does reach into the visible region, in agreement with its previously observed weak absorption in this range. The general tendency also indicates that the higher absorption of the larger population of photo-generated charge carriers causes an increase. The increased optical conductivity value that diminishes the reflection capabilities and enables efficient energy transfer within optoelectronic devices further emphasizes Li₂XBH₆ compounds as a potential system for light-driven applications.⁷⁸

Fig. 7 (a) Absorption, α(ω), (b) conductivity, σ(ω), (c) reflectivity, R(ω), (d) energy loss function, L(ω) for Li₂XBH₆ (X = Li, Na, K) materials.

Fig. 7(c) depicts the reflectivity spectra R(ω) of the Li₂XBH₆ (X = Li, Na, K) compounds in percent to evaluate their surface photon-reflection response. All are characterized by low reflectivity in both low energy and visible (typically less than 15 percent), meaning not all incident light is reflected from the surface.⁷⁹ The low reflectance facilitates enhanced light absorption, which is beneficial for optoelectronics and photovoltaics. As photon energy increases into the UV, reflectivity is more significantly enhanced across all compositions. The highest responses are observed at Li₂KBH₆, with peaks around 35–45%, while for Li₂LiBH₆ and Li₂NaBH₆, they are in the 20–30% range. This behavior indicates that Li₂XBH₆ (X = Li, Na, K) has low reflectivity in the visible but strong reflectivity in the UV, due to high-energy interband transitions. Fig. 7(d) shows the energy loss function, L(ω) (e-loss), which represents the energy lost by fast electrons traversing the material and is closely related to plasmonic excitations. The collective electronic excitations in the Li₂XBH₆ compounds exhibit well-defined trends in their e-loss spectra. Li₂KBH₆ shows a much-enhanced e-loss peak, approximately an order of magnitude higher, reaching about 1.4, compared to Li₂LiBH₆ and Li₂NaBH₆, with maximum values around 0.4 and ∼0.3, respectively. Such a strong peak in Li₂KBH₆ implies enhanced plasmon resonance and a higher probability of inelastic electron scattering, leading to more energy dissipation inside the lattice. The progressively increasing extremum e-loss values at higher energies also demonstrate the increasing density of accessible unoccupied states involved in plasmonic excitations.⁸⁰ Therefore, the studied materials may serve as the absorbing layer of a solar cell due to their low reflectivity, appreciable absorption in the visible range, and bandgap values close to the S.Q. limit,⁸¹ although their optical response remains more pronounced in the UV region.

3.4 Hydrogen storage properties

The primary challenge in utilizing hydrogen is the absence of efficient storage materials, but researchers have developed structures capable of storing high hydrogen concentrations reversibly.^82,83 In the near future, the demand for hydrogen storage as a renewable energy source will increase significantly. Volumetric hydrogen storage capacity is an important consideration for hydrogen storage. To determine whether these materials can be used in hydrogen storage technology, the following equations are used:⁸⁴where, m_host is the molar mass of the host material, H/M is the atomic ratio of hydrogen to the host material, and m_H is the molar mass of hydrogen. The initial theoretical gravimetric hydrogen storage capacity (GHSC) of Li₂LiBH₆ is 16.05 wt%, where Li₂NaBH₆ is 11.25 wt%, and Li₂KBH₆ is 8.66 wt%. Hydrogen desorption is the mechanism by which hydrogen is liberated from a substance. Since lower desorption temperatures are more practical for real-world applications, they are considered highly desirable. Favorable hydrogen desorption temperatures (T_des) typically lie in the range of 450–650 K.⁸⁵ For determining the desorption temperature, which equation must be used:The T_des can be calculated by the Gibbs reaction^86,87

ΔG = ΔH − T_desΔS (20)

Here, ΔH denotes the formation enthalpy, whereas ΔS indicates the enthalpy change of hydrogen, which is −130 J mol⁻¹ K⁻¹.⁸⁸ The calculated T_des of Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ are 521.67, 482.30, and 451.18 K, respectively, which lie within an optimal range, supporting their potential for controlled hydrogen release in an energy system.

Hydrogen storage performance can’t be evaluated solely based on theoretical gravimetric capacity, because practical hydrogen release is also governed by thermodynamic stability, decomposition pathway, intermediate phase formation, desorption kinetics, and reversibility. In complex hydrides, hydrogen desorption generally occurs via thermally activated bond breaking and often proceeds through multistep reactions rather than a single ideal step. For comparison, MgH₂ is a well-known hydride with a theoretical hydrogen capacity of about 7.6%, but its practical deployment is limited by relatively high desorption temperature and slow reaction kinetics. Likewise, NaAlH₄ exhibits a theoretical capacity of about 5.6% and releases hydrogen through multistep decomposition involving intermediate phases such as Na₃AlH₆,⁸⁹ while LiBH₄, despite its very high theoretical hydrogen capacity, remains challenged by strong thermodynamic stability and sluggish dehydrogenation behavior.⁹⁰ Compared with these experimentally established hydrides, the Li₂XBH₆ compound investigated here shows encouraging theoretical gravimetric capacities and favorable desorption characteristics. Nevertheless, their practical hydrogen-storage applicability can’t yet be confirmed, and further experimental studies on decomposition pathways, kinetic barriers, and cycling reversibility are required.

A comparison of our materials in Table 4 with other reported DPHs shows that Li₂XBH₆ compounds exhibit significantly higher GHSC. Notably, the Li₂LiBH₆ compound exhibits an exceptionally high hydrogen storage capacity of 16.05%, exceeding both previously reported materials and the U.S.DOE target, while maintaining a practical T_des below 650 K. Conclusively, these properties make this material convincing for future hydrogen storage applications.

Table 4 Band gap, GHSC, (Cwt%), and desorption temperature, (T_des) of present and reported work

Hydrides	Band gap (eV)	C wt%	Tdes (K)	Ref.
Li2LiBH6	1.79	16.05	521.67	This work
Li2NaBH6	1.66	11.25	482.30	This work
Li2KBH6	1.48	8.66	451.18	This work
MgH2	—	7.6	∼673–723	91
NaAlH4	—	5.6	∼463–498	92
Na2LiAlH6	5.20	7.05	373.9	93
K2LiAlH6	2.904	5.08	360.2	94
Na2LiGaH6	3.41	4.71	337.1	93
K2LiGaH6	1.451	3.71	241.3	94
Na2CaCdH6	2.08	2.95	—	95
Rb2NaAlH6	3.88	2.30	215.78	96
Rb2NaInH6	2.81	2.01	175.63	96
Rb2CaCdH6	2.30	1.69	—	97
Ba2LiNbH6	0.95	1.59	—	98

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3.5 Photo-catalytic properties

The photocatalytic activities of Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ are primarily assessed by their ability to absorb sunlight, the appropriateness of their band gaps, the positions of their band edge levels, and their optical absorption features. According to the thermodynamic theory, the alignment of the potential distribution of a photocatalyst and band gap, that is, a band gap of at least 1.23 eV, or more practically, of ∼1.5 eV, is both a necessary and sufficient prerequisite to water splitting in the sun.^99,100 On the other hand, H₂O oxidation and reduction are multi-step processes, with the former a four-electron reaction and the latter a two-electron reaction.

2H₂O → O₂ + 4H⁺ + 4e⁻

4H⁺ + 4e⁻ → 2H₂

During the electron transfer process, an activation barrier separates the water molecules from the photocatalyst. This resistance causes the water oxidation and reduction reactions to require a significant overpotential, meaning that incident photons must have energies above the photocatalyst's band gap to effectively drive the reactions. As a result, the lower band gap should be significantly larger than 1.23 eV to fully split water. To utilize the visible light and maximize solar energy, the band gap must be smaller than 3.0 eV.¹⁰¹ To analyze the potential of perovskites as photocatalysts, the Mulliken electronegativity approach was employed to investigate their photocatalytic activity. Mulliken defined the electronegativity of an atom as the arithmetic mean of its atomic electron affinity (A_f) and first ionization energy (I₁).¹⁰²For a compound with 4 elements, a, b, c, and d, with numbers of atoms p, q, r, and s respectively, for each.^103,104 The electron negativity for a compound can be calculated using eqn (23) as:Mulliken's electronegativity approach was used in calculating the band edge potentials (E_CBM and E_VBM), and they are expressed as:¹⁰⁵where χ_GM is the absolute electronegativity of the material (the geometrical mean of the Mulliken electro-negativities),¹⁰⁶ E_g is the band gap (TB-mBJ), E₀ energy of free electrons relative to NHE (normal hydrogen electrode) is approximately 4.5 eV, E_CBM is the potential of the conduction band minima (CBM), and E_VBM is the potential of the valence band maxima (VBM).

To be successful in photocatalytic water splitting, a semiconductor must fulfill two conditions: (1) the valence band maximum of the semiconductor must be more positive than the potential edge of H₂O/O₂, and (2) the conduction band minimum of the semiconductor must be more negative than the redox potential of H⁺/H₂ (0 V vs. NHE).¹⁰⁷ It is observed that the compounds have band gaps ranging from 1.23 to 3 eV in the visible light spectrum. Indirect band gaps reduce the rate of recombination and carrier lifetimes, which are beneficial in photocatalytic surface reactions.¹⁰⁸ The VBM values of all three compounds are more positive than that of H₂O/O₂ (greater than +1.23 V vs. NHE), while their CBM values are more negative than the redox potential of H⁺/H₂ (0 V vs. NHE). This indicates that the compounds possess strong oxidative and reductive properties, making them highly effective for water oxidation. For the Li₂XBH₆ (X = Li, Na, K) compounds, Li₂LiBH₆ exhibits edge potentials of +1.65 V for the VBM and −0.15 V for the CBM. Similarly, Li₂NaBH₆ shows edge potentials of +1.55 V (VBM) and −0.11 V (CBM), while Li₂KBH₆ presents +1.38 V (VBM) and −0.10 V (CBM). The indirect band gaps of Li₂NaBH₆ and Li₂KBH₆ are relatively narrow, and their conduction band edges lie close to the CO₂ reduction potential for CO formation in the light spectrum. Moreover, the conduction bands of all three compounds are positioned near the CO₂ reduction potential for HCOOH production, indicating a high potential of visible-light-driven CO₂ photo reduction, and in particular when combined with co-catalysts to enhance the efficiency of charge separation.^109,110 For better context, the predicted photocatalytic behavior of Li₂XBH₆ may be compared with well-known photocatalysts such as TiO₂ and g-C₃N₄.^111,112 TiO₂ is a classical photocatalyst with strong redox capability, but its activity is mainly restricted to the UV region because of its wide band gap. In contrast, g-C₃N₄ has a narrower band gap and is widely investigated for visible-light photocatalysis. Compared with these benchmark systems, the Li₂XBH₆ compounds (Fig. 8) show theoretically favorable band-edge positions for water splitting, indicating their potential as promising photocatalytic materials. On the other hand, the VBM values lie slightly below the ˙OH/OH⁻ potential (1.90 V vs. NHE), indicating that they are not sufficiently positive to drive oxidation, and their VBMs can drive strong water oxidation under visible light, while their CBMs are promising candidates for water reduction in the visible light spectrum. This study is entirely theoretical, and further experimental investigations will be conducted in the future to validate these results.

Fig. 8 Calculated edge potentials vs. NHE of Li₂XBH₆ (X = Li, Na, K) with respect to the reduction and oxidation potentials of H₂O at pH = 0.

4. Conclusion

Li₂LiBH₆, Li₂NaBH₆, and Li₂KBH₆ exhibit remarkable solid-state structural and Thermodynamic stability for multifunctional applications such as significant theoretical hydrogen storage capabilities and photocatalytic applications. However, in the case of mechanical stability, only Li₂LiBH₆ satisfies the required criteria, whereas Li₂NaBH₆ and Li₂KBH₆ fail to meet those conditions. Among these, Li₂LiBH₆ demonstrates the highest initial effectiveness, exhibiting a remarkable theoretical GHSC of 16.05 wt% and a T_des of 521.67 K. Li₂NaBH₆ and Li₂KBH₆ also meet the U.S. DOE hydrogen storage benchmarks, exhibit considerable storage capacities of 11.25 and 8.66 wt%, and desorption temperatures of 482.30 and 451.18 K that lie within the operational range. They also display semiconducting nature, creating opportunities for advanced optoelectronic systems. Additionally, these hydrides' optical absorption characteristics also point to possible applications in photovoltaics and other optoelectronic domains. Moreover, CBM of all three compounds is more negative than 0 V vs. NHE, while VBM is more positive than +1.23 V vs. NHE. This alignment indicates strong oxidation and reduction potentials, respectively, rendering these materials suitable for photocatalytic applications. In conclusion, this study shows that the materials are promising for hydrogen storage and photocatalytic applications.

Author contributions

Partha Podder: writing – original draft, methodology, formal analysis, data calculations, validation. Joy Kumar Sharkar: writing – original draft, methodology, formal analysis, data calculations, validation. Md. Al-Amin: writing – review & editing, investigation, validation. R. M. Tanvir: writing – review & editing, investigation, validation. Siyam Sheikh: writing – review & editing, validation. Md Tarikal Nasir: writing – review & editing, validation. A. Rayhan: writing – review & editing, validation. S. Mahmud: conceptualization, formal analysis, methodology, validation, writing – review & editing, software, supervision, project administration.

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Acknowledgements

The authors gratefully acknowledge the Smart Computing Research Laboratory (SCRL), Department of Electrical and Electronic Engineering, Jatiya Kabi Kazi Nazrul Islam University (JKKNIU), Mymensingh-2224, Bangladesh, for providing the computational facilities and support essential to the completion of this work.

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