Hydrogen Storage Capacity of Lead-Free Perovskite NaM T H 3 (M T =Sc, Ti, V): A DFT Study

Hydrogen is a promising clean energy carrier, but its storage is challenging. In this study, we investigate the potential of NaM T H 3 (M T =Sc, Ti, V) hydride perovskite as solid-state hydrogen storage material. Using density functional theory (DFT), we comprehensively analyze their structural, hydrogen storage, phonon, electronic, elastic


Introduction
Hydrogen is considered one of the most promising sustainable energy sources compared to conventional fossil fuels like oil, coal, and natural gas [1].Using solid-state materials as a hydrogen storage medium is one of the most desired solutions [2].There are two types of solid-state hydrogen storage: physisorption and chemisorption.Using weak van der Waals forces, hydrogen molecules are adsorbed onto the surface of a material, a process known as physisorption.In chemisorption, hydrogen molecules are chemically attached to the surface of a material.Compared to physisorption, chemisorption has a larger theoretical storage capacity but has higher activation energy needed for adsorption and desorption [3][4][5].Hydrogen is a possible alternative fuel with its high energy density and green attributes.Hydrogen storage utilizing hydrides, such as magnesium hydride, has been considered an alternate energy storage method.Due to the high gravimetric capacity of such materials, this approach is especially promising.The hydrogen storage capacities per unit mass of perovskite hydrides are typically lower than those of conventional metal hydrides.However, they also have other benefits that make them viable options in certain scenarios like high volumetric density, tunability, improved kinetics, and reversibility [6].Hydrogen storage, however, remains an important challenge due to higher desorption temperature and slow kinetics [7,8].Due to distinct crystal structures and adaptable characteristics, perovskite materials have been recognized as promising candidates among numerous materials.The general formula for perovskite materials is ABX 3 [9] where A denotes a large cation, B denotes a smaller cation, and X denotes an anion.As A, B, or X may be modified to accommodate additional elements, the cubic crystal symmetry of the perovskite structure makes it a flexible material for a wide range of applications.Substituting hydrogen for X, a hydride perovskite with the general formula ABH 3 is formed.The possible formulas for perovskite-type hydrides are A +1 B +2 H 3 and A +2 B +1 H 3 -1 .
Since A and B belong to alkali or alkaline earth metals, the possible hydride structures are limited [10][11][12].Hydride perovskites are gaining attention as a promising solution for hydrogen storage.Materials used for hydrogen storage usually have robust hydrogen bonding, wide interstitial spaces for substantial hydrogen storage, catalytic nature for improved hydrogen absorption, and adequate gravimetric hydrogen storage capacities [10,13].Such hydride perovskite has different gravimetric storage capacities with gravimetric and volumetric capacities of up to 6 wt% and 88 kg/m 3 depending on the choice of A and B up to six weight percent; therefore, single perovskite materials are the most investigated compounds for hydrogen storage applications [14,15].Due to their special qualities, Na-based perovskite materials have recently received a lot of interest; in this series of hydrides, NaMgH 3 became the prime focus of research [10,16].Surucu et al. studied the formation energy, mechanical stability, thermodynamics, and bulk modulus of NaM T H 3 (M T =Co, Fe, Mn) and found the gravimetric storage capacities 3.74 wt%, 3.70 wt%, and 3.57 wt% for NaMnH 3 , NaFeH 3 , and NaCoH 3 , respectively, using DFT [17,18].Gencer et al. calculated the electronic, elastic, thermal, and hydrogen storage properties of NaNiH 3 with a gravimetric storage capacity of 3.5 wt% [10,19].Inspired by these studies, NaM T H 3 (M T =Sc, Ti, V) first-principle calculations were carried out at atmospheric pressure and temperature.According to the author's knowledge, there have been no previous reports, and this work is the first detailed investigation of the crystal structure, phonon dispersion curves, gravimetric ratio, and electronic, mechanical, and thermal properties of NaM T H 3 (M T =Sc, Ti, V) lead-free hydride perovskites using DFT for hydrogen storage applications.The results of this study will be helpful for future works.The manuscript introduces the study, describes methods for calculations, discusses the results, and finally concludes the study in separate sections.

Computational Methodology
Computational work was performed in the domain of density functional theory (DFT) using plane wave pseudopotential approach of CASTEP and Quantum ESPRESSO code to investigate the properties of NaM T H 3 (M T =Sc, Ti, V) [20].

Result and Discussion
3.1.Structural Analysis.Figure 1 shows the cubic crystal structure of the NaM T H 3 (M T =Sc, Ti, V) hydride perovskite with the space group of (Pm-3m, 221).The crystal formation was within a cubic unit cell such that the Na atoms were positioned at the corners of the unit cell with coordinates at the origin (0, 0, 0).At the same time, M T atoms were accurately located at the body center of the unit cell at coordinates (0.5, 0.5, 0.5).The center of each face of the unit cell was clearly marked by the presence of H atoms, notably at the coordinates (0.5, 0, 0), (0, 0.5, 0), and (0, 0, 0.5).To seek the lattice parameters leading to the lowest energy of the structure, we changed the lattice constant a for the NaM T H 3 (M T =Sc, Ti, V) hydride perovskites.The unit's volume also changed with the modification of the lattice parameter.To investigate how the unit cell's total energy changes with changes in volume, we used density functional theory (DFT) to calculate forces.The Perdew-Burke-Ernzerhof (GGA-PBE) functional from the generalized gradient approximation was used to calculate the self-consistent field (SCF) energy, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was used for optimization.The system relaxed until the forces exerting pressure on the atoms were insignificant.
E-V curves, as shown in Figure 2, were drawn by plotting volume against the total energies represented by scattered points; curve fitting was done using the Birch-Murnaghan equation of state defined by Eq. ( 1), and continuous lines show how data fits using the B-M equation.The lowest 2 International Journal of Energy Research energy value on these curves indicates an equilibrium state that is the most stable.
In the context of this research, the term E V refers to the material's energy at a particular volume V. V o stands for the equilibrium volume, B stands for the bulk modulus, and E o is the total energy of the unit cell's ground state.B ′ is the pressure derivative of the bulk modulus.We determine the unit cell's volume value for which the total energy converges.These graphs show how the total energy varies for the unit cell volume and provide information about the material's bulk modulus and equilibrium volume.The order of relaxed unit cell volumes for NaVH 3 , NaTiH 3 , and NaScH 3 is as follows: NaScH 3 > NaTiH 3 > NaVH 3 , with corresponding volumes of 62.40 Å 3 , 52.30 Å 3 , and 46.79 Å 3 , respectively.This variation is primarily attributed to each compound's size of the M T atom.In NaScH 3 , the unit cell volume is the largest, mainly due to the relatively large size of the Sc atom.Conversely, in NaVH 3 , the unit cell volume is smaller, primarily owing to the smaller size of the V atom.These unit cells' total energies follow a different order: NaVH 3 > NaTiH 3 > NaScH 3 .It is concluded that these metal hydride perovskite crystal structures are considered energetically stable because the curve has been perfectly fitted using the Birch-Murnaghan equation of state to the calculated data points on the energy-volume curve [27][28][29].
3.2.Total Energies.Table 1 displays the computed lattice constants a (Å), volume V (Å 3 ), density (ρ), and total energies of the reactants and products of the reaction mentioned in Eq. ( 2).The formula for hydride perovskite formation for each compound is given below [30].
The formation of NaM T H 3 (M T =Sc, Ti, V) was confirmed using the total energies.The difference in the energies of E NaM T H 3 , E NaM T , and 3/2E(H 2 ) for M T =Sc, Ti, and V was negative, which confirms the favorable hydride formation, which is thermodynamically stable.For instance, NaScH 3 has the lowest total energy −2 64 × 10 +3 eV, indicating that it is the most stable, whereas NaSc −2 58 × 10 +3 eV has a less negative value, which shows it is less stable than NaScH 3 .Cwt% represents the gravimetric hydrogen storage capacity in Table 1.The sole purpose of this parameter is to access the compound for its potential use in hydrogen storage applications.If the calculated values of gravimetric hydrogen storage capacity are more than three, then the material is feasible for hydrogen storage.As shown in Table 1, the materials NaM T H 3 (M T =Sc, Ti, V) have storage capacities close to 4; hence, they are potential candidate for hydrogen storage applications.From the physical properties mentioned in Table 1, it can be seen that NaScH 3 has the largest value of lattice constant and, hence, the resulting unit cell volume, consequently, the lowest mass density.

International Journal of Energy Research
The study calculates the NaM T H 3 (M T =Sc, Ti, V) hydride perovskites for hydrogen storage applications.The gravimetric storage capacity of perovskite structures was calculated using Eq. ( 3) and represented by Cwt%.Gravimetric capacity measures the quantity of stored hydrogen per unit mass of the studied hydride perovskite [32,33].
In this expression, m H and m Host serve for the molar mass of H 2 and NaM T , whereas the H/M is the ratio of hydrogen atoms to transition metal atoms; in this case, it is three.The maximum storage capacity belongs to NaScH 3 with a 4.26% value.Remaining hydrides also have good storage capacities in the order Cwt%(NaScH 3 ) > Cwt%(Na-TiH 3 ) > Cwt%(NaVH 3 ), which follows the trend of unit cell volume.Figures 3 and 4 illustrate the gravimetric hydrogen storage capacity and volumetric hydrogen storage capacities of NaM T H 3 (M T =Sc, Ti, V) hydrides.Yet, it has the highest storage capacity values due to enough space available (due to the large lattice constant) within the unit cell for hydrogen to adsorb and desorb easily.

Phonon Dispersion and Phonon Density of State.
Figure 5 shows the phonon dispersion and vibrational density of state graphs of studied hydrides along high symmetry lines in the first Brillouin zone.The unit cell contains a total of 5 atoms; therefore, 3 N modes of vibrations are present, including three acoustic modes and the remaining optical modes.The branches seem to be less than the vibrational modes due to the degeneration of phonons along high symmetry directions X to G and R to G. The energy difference between the low-and high-frequency phonon modes is 30.38,34.45, and 33.05 THz for NaScH 3 , NaTiH 3 , and NaVH 3 , respectively.NaTiH 3 shows the highest energy gap.This huge difference in the energy of phonons is due to the different masses of elements Na, Sc, Ti, V, and H. Overall, phonon dispersion and DOS curve represent convergent behavior, which depicts the dynamic stability of the studied hydride perovskites [34].
Figure 6 shows the projected phonon DOS of the investigated materials.Additionally, we conducted computations to determine the partial density of phonon states for NaM T H 3 (M T =Sc, Ti, V) hydride perovskites, where M T represents Sc, Ti, or V.This analysis is aimed at explaining the atomic contributions within the overall density of phonon states of investigated material, as illustrated in Figures 6(a)-6(c).The black lines represent the total density of phonon states of compounds, while the red line indicates the density of phonon states for the element Na.The green line illustrates the density of phonon states for transition elements M T , where M T can be Sc, Ti, or V, and the blue lines represent the density of phonon states for hydrogen atoms (H).In Figure 6(a), it is evident that the Na atom exhibits the highest peak in the density of phonon states in the low-frequency range, extending up to 4.35 THz.Conversely, the H atoms display moderate peaks in the highfrequency range spanning from 4.5 to 14.9 THz.Additionally, the transition atom Sc demonstrates minor peaks between 2.1 and 9.7 THz, but beyond this, its density of phonon states becomes negligible.
In Figure 6(b), in the low-frequency range, spanning up to 6.7 THz, the Na atom exhibits the most prominent peak in phonon state density.Conversely, H atoms display  6(c).In the low-frequency range, spanning up to 6.4 THz, the Na atom exhibits the most prominent peak in phonon state density.Conversely, H atoms display moderate peaks in the high-frequency range from 6.4 to 15.9 THz.Transition atoms V also show minor peaks below 6.5 THz, but their phonon state density becomes negligible beyond this frequency.

Electronic Properties.
To analyze the electronic properties of a material, its band structure and density of states were calculated.The band structures show charge carriers' quantum mechanical behavior and their metallic or nonmetallic behavior.Figure 7 depicts the electronic band structure of NaM T H 3 (M T =Sc, Ti, V) in the first Brillouin zone along the high symmetry directions.The band structure of all three hydrides shows the overlapping valence and conduction bands across the Fermi level, which is the characteristic of metallic material.The red-dotted line shows the Fermi level; energy levels above this line are called conduction bands, and lower ones are called valence bands.The DOS N n E of n bands is defined as [35,36] N E n k indicates the band's dispersion across the first Brillouin zone.By summarizing all the bands, the overall DOS is determined, which is given as [35] The DOS and PDOS were calculated using the GGA-PBE functional technique.Figure 8 shows the DOS (a, e, and i) and PDOS (b-d, f-h, and j-l) of the studied materials NaM T H 3 (M T =Sc, Ti, V).In TDOS, the density of state at the Fermi level is not zero, which shows the metallic nature of all hydride materials under investigation.PDOS shows the contribution of different atomic orbitals to the compound orbitals.These results are calculated from -5 eV to 10 eV to evaluate the role of individual elemental states in constructing TDOS of material.Sc, V, and Ti played the most crucial role in the TDOS near the fermi level, and further analysis revealed that they originate from d orbitals of transition metals.The bands towards the left of E F are called valence bands, whereas the bands towards the right are called conduction bands [37].
In short, the metallic behavior and significant d orbital contributions of NaM T H 3 (M T =Sc, Ti, V) hydrides highlight their potential for potential as solid-state hydrogen storage materials.between molecules, depend crucially on the distribution of atomic charge.The distribution of electronic charge density within atoms is better understood using population analysis.The Mulliken population and charge analysis was executed for the compounds NaScH 3 , NaTiH 3 , and NaVH 3 .The Mulliken atomic population analysis represents a computational methodology employed for the examination of bonding characteristics, bond lengths, and electronic structures within both solids and molecules.The Mulliken analysis tells us how electrons are spread out in a material and can help us predict how it will react with hydrogen molecules.
Table 2 displays the atomic populations, charges, population averages, and average bond lengths of NaM T H 3 (M T =Sc, Ti, V).To evaluate the extent of covalent character in the given bonds, ionicity of population (f n ) was determined given by the following.
A positive population value indicates the covalent bonding nature, while a negative value signifies the ionic bond formation.In our investigation, the bonds formed between a hydrogen atom (H) and transition metal atoms represented by M T (where M T = V, Ti, and Sc) are characterized by positive bond populations, indicative of covalent bonding.In contrast, the negative populations observed in the H-H and H-Na bonds imply an ionic nature for these bonds.The computational results indicate a positive charge for sodium (Na) and transition metals M T (where M T = V, Ti, and Sc), while hydrogen (H) is consistently found to carry a negative charge.The data demonstrates electron transfer occurring within the unit cell of NaScH 3 , wherein both sodium and the transition metal element Sc transfer electrons to hydrogen.A similar trend is observed in the unit cells of NaTiH 3 and NaVH 3 , where electrons are also transferred from sodium and the respective transition metals (Ti and V) to hydrogen, forming strong bonds [13].Furthermore, in these compounds (NaScH 3 , NaTiH 3 , and NaVH 3 ), the bond length exhibits the following trend: H-Sc > H-Ti > H-V, with corresponding values of 1.98 Å, 1.87 Å, and 1.80 Å, respectively.Although all of these transition metal atoms (Sc, Ti, and V) make covalent bonds with hydrogen, the H-Sc bond length is the largest.Therefore, in this study, NaScH 3 is found to be the best material for hydrogen storage among NaM T H 3 (M T =Sc, Ti, V) hydrides, due to its favorable properties.The population analysis and electron density difference, along with charge density distribution, are vital in exploring the hydrogen storage capacities of lead-free perovskite NaM T H 3 as an efficient solid-state hydrogen storage material [38,39].

Charge Density Distribution.
To better understand the nature of bonds in our compounds and the distribution of charge around the atoms, we examined electronic charge    9 illustrates the findings from the investigation of the charge density distribution in NaM T H 3 compounds, where M T represents V, Ti, and Sc, within the (001) crystallographic plane.The graph's right side features a color scale indicating the charge density, with light grey indicating high density and dark grey representing low density.This implies the formation of strong covalent bonds between Na and elements such as V, Ti, and Sc, denoted as Na-M T .Conversely, H-H and H-Na bonds exhibit ionic interactions, affirming the findings indicated by population analysis.The characterization of bond nature and bond length between transition metals M T (V, Ti, Sc) and hydrogen atom H is of significant importance.Based on the information presented in Figure 9, it can be affirmed that the bonds between H and V, Ti, and Sc are covalent in nature, exhibiting considerable distortion and significant electron cloud overlap around these atoms.The bond lengths between a hydrogen atom (H) and transition metals (X) such as V, Ti, and Sc exhibit the following trend: H-Sc > H-Ti > H-V, with corresponding values of 1.98, 1.87, and 1.80, respectively.

Mechanical Properties.
The crystal energy, equilibrium volume, and the strain-dependent matrix having secondorder elastic constants (C ij ) are used to provide insight into the elastic behavior of the NaM T H 3 (M T =Sc, Ti, V) hydride perovskite.Mechanical characteristics are interlinked with the elastic constants denoted as C ij .The elastic constants of a material can define how it behaves under applied stress from the outside and offer important information about its bonding properties and structural stability [40].For the compounds NaVH 3 , NaTiH 3 , and NaScH 3 , the computed values of the elastic constants C 11 , C 12 , and C 44 for cubic single-crystalline structures are presented in Table 3.These values can be utilized for the evaluation of the mechanical stability of these compounds.The Born-Huang lattice dynamical theory has the ability to assess the mechanical stability of compounds.Given the cubic arrangement exhibited by these compounds, it is imperative to apply the Born stability criteria for the purpose of evaluating their stability.The requisite conditions are as follows [41]: Analyzing the data by Equations ( 3)-( 5) presented in Table 3, it is evident that all three compounds satisfy the established criteria for mechanical stability.Consequently, the structures of NaVH 3 , NaTiH 3 , and NaScH 3 demonstrate mechanical stability.The Cauchy pressure, C p , can be computed by deducting the C 44 value from C 12 .
The C p sign of the compound can be used to identify whether it would behave brittle or ductile.While a negative C p shows brittle behavior, a positive C p indicates ductile behavior.As in our case, all the studied compounds show brittle behavior as the C p for all compounds is negative.Furthermore, the utilization of elastic constants extends to the derivation of additional mechanical properties of a material, including the shear modulus, Young's modulus, and Poisson's ratio [42].Consequently, extensive inquiries have been conducted into the elastic constants of compounds involving NaM T H 3 (M T =Sc, Ti, V) hydride perovskites.The bulk modulus (B) Eq. ( 10), shear modulus (G) Eq. ( 12), Young modulus (E) Eq. ( 13), and Poisson's ratio (v) Eq. ( 14) were assessed utilizing the Voigt-Reuss-Hill  (VRH) approximations.The Voigt approximation establishes an upper limit for the modulus, whereas the Reuss approximation sets a lower limit.In comparison, the Hill approximation offers enhanced accuracy.The data presented in Table 4 was derived employing the Hill approximation method.
Table 3 shows the values of elastic constants of investigated hydrides.NaScH 3 has the lowest value of Cp.NaScH 3 expresses the strongest contraction under pressure with a Cauchy's pressure of -12.77, followed by NaTiH 3 (-8.60)and NaVH 3 (-2.99),indicating their decreasing response to contract under compression.The calculated values of bulk modulus are also mentioned in Table 4, which represents the ratio of a small increase in pressure to the decrease in volume (compression), or its resistance to compression; the highest value corresponds to NaVH 3 .The shear modulus (G) represents the ratio of shear stress to shear strain; the highest value of G belongs to NaVH 3 .Young's modulus reflects the degree of the stiffness of a solid material, defined as the ratio of stress to strain along the same axis.NaVH 3 has the highest Young's modulus of 118.49, followed by NaTiH 3 with 99.61 and NaScH 3 with 70.86, making NaVH 3 comparatively the stiffest material.
Another important parameter to determine the bonding strength of the material is Poisson's ratio.It is a measure of compression or expansion of studied perovskite hydride in the normal direction and is related to the bond type of the material.The ionic and covalent bonding criterion is v greater than 0.25 and v = 0 1, respectively.The respective Poisson ratios were calculated for investigated materials and can be found in Table 4.The results show that NaVH 3 has more covalent character, NaTiH 3 has a balanced covalent and ionic character, and NaScH 3 has more ionic character based on their Poisson ratios.The B/G (Pugh's ratio) was measured to predict the ductile or brittleness of the material.For ductile materials, the B/G ratio is greater than 1.75.The calculated values in Table 4 confirm that the studied materials are brittle.The elastic anisotropy index (A) values were calculated using the following [43].
If the material has isotropic elastic behavior, the anisotropic factor value will not be 1 (one).Therefore, all compounds exhibit elastic anisotropic behavior percentage anisotropy factor known as B k and G k using Eq. ( 17) [43].
These parameters indicate the level of elastic anisotropy in the bulk and shear modulus.For isotropic medium, the measured values of these parameters are zero; conversely, the medium is anisotropic.
The result mentioned in Table 3 indicates that all three hydrides are isotropic; only NaScH 3 shows a small value of 1.2% of anisotropy in shear modulus.The Vickers hardness of a material is given by Among the studied materials, NaVH 3 exhibits the highest HV value, indicating that it is a relatively hard material.Conversely, NaScH 3 has the lowest hardness due to its minimal HV value.Based on the hardness test, these compounds are soft materials because their calculated hardness values fall below the hardness threshold of 40 GPa.
The hydrogen adsorption and desorption processes are governed by thermal energy; therefore, it is essential to investigate the thermodynamic properties of the material.
The Debye temperature is used as an indicator to investigate the thermal conductivity and interatomic force.This temperature is interrelated to the elastic vibrations of the crystal and depends on the material's average sound velocity (V m ).The Debye temperature of crystals under investigation is calculated by Voigt-Reuss approximation as [44] In this context, h, k B ., N A , ρ, M, and n are, respectively, Planck's constant 1 381 × 10 −23 J/K , Boltzmann constant 6 626 × 10 −34 Js , Avogadro's number 6 022 × 10 23 , density of crystal, molar mass, and number of atoms per unit cell, and V m represents the average sound velocity.
The average sound velocities for NaScH 3 , NaTiH 3 , and NaVH 3 cubic crystals were around 4302 m/s, 4610 m/s, and 4673 m/s, respectively.These sound velocities are related to Debye temperature, reflecting the material's stiffness properties.For cubic crystals NaM T H 3 (M T =Sc, Ti, V), the respective Debye temperatures were calculated as 552.2 K, 626.3 K, and 660.1 K.The results depict that the Another important parameter is the melting temperature of these cubic crystals calculated by using the formula given as [44] T mel K = 5 911 × C 12 + 553 GPa ± 300K 20 The computed melting temperatures for hydrides NaM T H 3 (M T =Sc, Ti, V) were found to be 683.479K, 731.214K, and 778.858K, respectively.Results show that the melting temperature of these perovskites increases in the order of NaScH 3 < NaTiH 3 < NaVH 3 .NaScH 3 has the lowest melting temperature, whereas NaVH 3 has the highest melting temperature.The Debye temperature, average sound velocity, and other such quantities are mentioned in Table 5.
The mechanical properties, including elastic behavior and stability, of NaM T H 3 (MT=Sc, Ti, V) hydrides, as revealed by the analysis, underscore their potential as solid-state hydrogen storage materials, aligning with the study's objective.
3.8.Thermodynamical Properties.To investigate the thermal stability of the NaM T H 3 (M T =Sc, Ti, V) materials, quasiharmonic approximation was implemented using CASTEP code [45].This study was performed in the absence of any external pressure.The collective effect of heat absorption, elastic vibration (phonon), and electrons can be elaborated using the specific heat capacity of materials [46].The specific heat capacity at constant volume is equivalent to the amount of heat absorbed by the material as a result of a one-degree Kelvin change in temperature.Figure 10 shows the enthalpy, free energy, entropy, and specific heat capacities at constant volume of hydride materials under investigation.The specific heat capacities of hydride materials follow the order NaTiH 3 > NaVH 3 > NaScH 3 .Notably, the C v curves are steepest at the start, but after 200 K, the slope decreases.This may be due to the harmonic effects and coincides with the prediction of the Dulong-Petit law (C v = 3R for any solid material).Figures 10(a)-10(f) represents the thermodynamics of NaM T H 3 (M T =Sc, Ti, V) materials for the temperature range 0 K to 1000 K.In Figures 10(a), 10(c), and 10(e), before 110 K, the values of enthalpy (H), temperature times entropy (T * S), and Helmholtz free energy (F Helm ) were zero-afterward, the free energy decreases, whereas the enthalpy and entropy increase.Enthalpy itself is calculated by taking the sum of internal energy and the product of pressure volume H = E + PV .An enthalpy of 0.89 eV is noted for NaTiH 3 and NaVH 3 , which is greater than the corresponding values of enthalpy for NaScH 3 at 1000 K. To evaluate the thermal stability of these hydrides, free energies were noted.It is well established that negative values of free energies ensure the thermodynamic stability of a material.The maximum values of free energies of hydrides NaM T H 3 (M T =Sc, Ti, V) are -1.28 eV, -1.39 eV, and -131 eV, respectively, at 1000 K; these negative values show the thermodynamic stability of hydride under investigation.The entropy of the system represents the thermal energy of a system.From the graphs, the entropy or thermal energy of the system was zero at zero Kelvin.Still, after 110 K, the value started to increase due to an increase in thermal agitation of crystal atoms with the rise of temperature.
In Figures 10(b), 10(d), and 10(f), the heat capacity of NaM T H 3 (M T =Sc, Ti, V) and constituent elements are shown; graphs show that hydrogen has the minimum value of heat capacity at Dulong-Petit limit, followed by transition metal and sodium, and the highest value of heat capacity is achieved by perovskite hydrides.
For perovskite hydride material under investigation, ZPE represents the minimum energy at absolute zero temperature, E represents the system's internal energy, F shows the material's thermodynamic stability, and S reflects the degree of disorder.
From calculation of these thermodynamic properties, it can be inferred that the material properties of NaScH 3 , NaVH 3 , and NaTiH 3 are unique at a temperature of 298 K. NaScH 3 has the minimum value of internal energy (0.61 eV), Helmholtz free energy (0.30 eV), and heat capacity at constant volume (81.15 J/mol/K), but the maximum value of entropy (100.19J/mol/K), whereas NaVH 3 has the highest zero-point energy (0.54 eV) and internal energy (0.69 eV), but the lowest entropy (91.38 J/mol/K) and average value of heat capacity at constant volume (83.37 J/mol/K).Additionally, NaTiH 3 has an average value for all properties except the heat capacity (87.47 J/mol/K) which is highest among the three materials.To assess thermodynamic stability, we identified the lowest Helmholtz free energy (0.3 eV) value, which corresponds to NaScH 3 .Hence, NaScH 3 has the highest thermodynamic stability among the three materials at a temperature of 298 K.
Figure 11 shows the plots of variation of Debye temperature along with temperature for the respective hydrides.The highest values belong to NaScH 3 , which represents its comparatively high conductivity.The results show that NaScH 3 is thermodynamically stable and suitable for hydrogen storage.Out of all three hydrides, the heat capacity of NaScH 3 is the lowest, but the conductivity (Debye temperature) is high; therefore, it can conduct heat supplied for desorption.From Figure 11, it is obvious that the Debye temperature of NaScH 3 is the highest, and NaVH 3 is the lowest.
The mechanical and thermodynamic property analysis confirms the potential of lead-free perovskite NaM T H 3

Conclusion
NaM T H 3 (M T =Sc, Ti, V) hydride perovskites were investigated using DFT.Our results showed that the gravimetric hydrogen storage capacities of these hydrides NaScH 3 , NaVH 3 , and NaTiH 3 are 4.26 wt%, 4.09 wt%, and 3.83 wt%, respectively.As per Born's criterion, the studied perovskites express mechanical stability.The anisotropy index revealed the isotropic behavior for shear and bulk moduli.The Vickers hardness results showed that these perovskites are soft materials.The Poisson ratios revealed NaVH 3 being more covalent, NaTiH 3 being balanced, and NaScH 3 being more ionic.Electronic band structure and DOS analysis revealed the metallic nature, further supporting their potential for hydrogen storage.Population analysis revealed appropriate bond lengths, boosting the stable perovskite structure role as hydrogen storage medium.The dynamic stability of these hydrides was obvious from the absence of imaginary frequencies in phonon dispersion curves.As temperature rises, specific heat, entropy, and Debye temperature increase, while free energy turns negative.NaScH 3 shows exceptional

3 Figure 11 :
Figure 11: The calculated graphs of Debye temperature of NaM T H 3 (M T =Sc, Ti, V).

Table 2 :
The calculated values of the Mulliken population and charge analysis for NaM T H 3 (M T =V, Ti, Sc) hydride perovskites.

Table 3 :
The calculated elastic constants (C ij ) and C p for NaM T H 3 (M T =Sc, Ti, V).

Table 4 :
The calculated anisotropic factor (AU), bulk modulus (B in GPa), shear modulus (G in GPa), Young modulus (E in GPa), B/G, G/B, Poisson ratio (v), and Vickers hardness (H v in GPa) for NaM T H 3 (M T =Sc, Ti, V) hydrides.materials follows the order NaScH 3 < Na-TiH 3 < NaVH 3 .Notably, the highest value of Debye temperature belongs to NaVH 3 , which indicates the highest conductivity value.

Table 5 :
The calculated values of density (ρ in kg/m 3 ), molar mass (M in kg/mol), average sound velocity (V m in m/s), Debye temperature (θ D in K), and melting temperature (T m in K) for NaM T H 3 (M T =Sc, Ti, V) hydride perovskites.
T =Sc, Ti, V) as a solid-state hydrogen storage material, aligning with the focus on hydrogen storage capacity.