Modeling of Hydrogen Atom Adsorption and Diffusion in Ti₃Sb Intermetallic Crystal with A15 Cubic Structure

Abstract

For the first time, the adsorption of hydrogen on the (110) surface of the A15 Ti₃Sb compound with a cubic structure (Cr₃Si type; space group $Pm\overline{3}n$) for the accumulation of hydrogen H was calculated using the density functional theory method (DFT SGGA-PBE). Taking into account the relaxation of the Ti₃Sb–H system, the equilibrium positions of hydrogen on the Ti₃Sb (110) surface were determined depending on the supercell size. Hydrogen adsorption on the Ti₃Sb (110) surface of supercells is preferable in pit sites. All DFT calculations of the Ti₃Sb–H system were performed on relaxed and optimized supercells (2 × 1 × 1, 3 × 3 × 3, and 5 × 5 × 5). Relaxation of the supercell reduced the calculated adsorption energy compared with the non-relaxed supercell. The calculated band structure and curves of local and partial densities of states of Ti₃Sb–H were used to explain the interaction of hydrogen with the Ti₃Sb (110) surface. The activation energy of H diffusion along the coordinates tetrahedral interstitial site → tetrahedral interstitial site (TIS–TIS) and tetrahedral interstitial site → octahedral interstitial site (TIS–OIS), along with the diffusion coefficient of H in the cubic lattice of Ti₃Sb, were calculated.

1. Introduction

Intermetallic alloys and compounds of metal (metalloid)–hydrogen (H) systems are used in various fields, including hydrogen energy for storing and transporting hydrogen [1,2,3,4]. Hydrogen absorption in intermetallic compounds can change the crystal structure [5] and physical properties [6]. These issues are of great importance for the practical use of hydrogen-absorbing materials such as H storage [7]. Intermetallic alloys and compounds based on titanium are also used for hydrogen storage. In such alloys, the hydrogen capacity is not high, for example, TiFe-1.50, TiMn_1.5–1.86 TiCr_1.8–2.43 wt.% [8]. Various methods (e.g., machine learning methods, addition or substitution of components [9,10], surface and heat treatment, microstructure improvement) are used to optimize the material properties, in particular to improve hydrogen storage performance. In addition, improving the properties requires obtaining information on the relationship between the composition, structure, and activity of the efficient hydrogen storage material, in particular, based on Ti.

Thus, intermetallic hydride compounds (IHCs) are formed by introducing hydrogen atoms H into the voids of the metal crystal lattice, which expands the lattice (for different materials from 10 to 30%) [11,12,13]. IHCs are suitable for creating hydrogen storage systems due to their hydrogen capacity (1.5–3 wt.%) and kinetic and thermodynamic properties (high rates of hydrogen adsorption/desorption at moderate pressures and temperatures).

A class of intermetallics with stoichiometric composition A₃B is known, which can crystallize in cubic, tetragonal, and hexagonal structures [13,14,15,16]. In recent years, the properties of compounds such as Ti₃Sb [14,15,16,17,18], Ti₃Ir [19], Ta₃Sb, Ta₃Pb, and Ta₃Sn [20] with cubic syngony (phase A15, prototype Cr₃Si) have been intensively studied. The absorption and diffusion of hydrogen in intermetallic compounds of the A15 Ti₃Sb type have been poorly studied. Therefore, the study of the features of the interaction of hydrogen with compounds based on Ti₃Sb is relevant, in particular, for the creation of new hydrogen storage systems [21,22,23,24].

However, experimental determination of the characteristics and coefficients of hydrogen adsorption and diffusion in a solid is difficult. Calculations based on density functional theory are used to study hydrogen adsorption and various adsorption sites on the surface of the material [25]. Reliable diffusion coefficients of atoms are determined both theoretically, for example, using Monte Carlo (MC) methods and diffusion rate calculations [26,27], and by experimental methods [28].

However, reliable interpretation of experimental data on the adsorption and diffusion of atoms in a solid is difficult. This is due to the need to take into account the features of the calculation method, the dependence of the material properties, and the determined coefficients of various factors, such as the structure, composition, chemical bond, and shape of the sample [6,7].

The analysis shows that adsorption and subsequent diffusion of introduced elements into metallic and intermetallic systems are complex processes for experimental study. Taking this into account, in recent years, ab initio methods for calculating the migration coefficients of gases and light atoms in solids have been developed [29,30,31,32,33], in particular, in Ti₃Sb [14]. The mechanisms of diffusion of implanted hydrogen atoms in metallic systems are also known [28]. In these works, in particular, it is shown that interstitial atoms in fcc systems are not necessarily located in the usual interstitial sites. The diffusion of interstitial hydrogen atoms in metals is modeled, taking into account phonon properties. In addition, it is proposed to also correct the formulas used for the diffusion coefficient.

The task that we set in this article, the study of the mechanism of hydrogen transfer in intermetallic structures of Al5 for storing H, has apparently not been studied to date. Such studies are needed, in particular, for the development of adsorption and chemical methods for dissolving and storing hydrogen, taking into account the properties and capacity characteristics of the hydrogen absorbent material [34].

It should be noted that the preparation of the methodology, including clean surfaces for experiments, in particular, in H-intermetallic systems, is complex and time-consuming. From the analysis of the above literature, it follows that theoretical calculations related to the behavior of H on the surfaces of intermetallic compounds are limited. However, the interaction between H atoms or molecules and the surface of solids is of interest to a number of industries, for example, for the accumulation and storage of H₂. Therefore, in order to provide further understanding of the mechanism of interaction of H with intermetallic surfaces (the processes of H penetration into the crystal lattice), we studied supercells based on Ti₃Sb–H systems.

Taking into account the above and the development of solutions to the problems of hydrogen storage and the use of hydrides by accumulating hydrogen, the task considered in this article is relevant. Since the processes of H adsorption and diffusion in Al5 Ti₃Sb occur at the atomic scale, the exact mechanisms leading to the change in properties and accumulation of H are not easy to experimentally determine.

Modeling the interaction of particles at the atomic level can yield practical important results for describing adsorption and diffusion. Taking this into account, the task was set to calculate the parameters of adsorption and diffusion of H on the surface and in the crystal lattice of the previously unstudied Ti₃Sb–H system. This paper does not consider issues related to the evaluation of the interaction of H with defects in the Al5 Ti₃Sb compound with a cubic lattice.

The aim of this work is DFT GGA-PBE modeling of the processes of adsorption and diffusion of H atoms on the surface and in the bulk of the lattice of cubic supercells based on the A15 Ti₃Sb–H system.

2. Methodology and Calculation Models

2.1. Methodology

Possible stable positions, diffusion barriers, and electronic properties of H atoms adsorbed on the surface of the compound A15 Ti₃Sb (110) were calculated, along with the binding energy and diffusion barrier of H in the bulk of Ti₃Sb. We also considered the results of calculations of the adsorption properties from low (Θ = 0.25 ML–monolayer) to high (1 ML) hydrogen coverage of the surface Θ.

The calculations of the electronic structure and structural properties of Ti₃Sb–H were performed from first principles based on the density functional theory (DFT) [35,36,37,38,39]. To perform density functional theory calculations, Vienna Ab initio Simulation Package VASP6.1.2 software packages (https://www.vasp.at/) accessed on 1 January 2022 and the Atomistic Toolkit (ATK) were used [34]. The interaction between valence electrons and core electrons was considered within the pseudopotential (PAW-PBE) approximation and the plane wave approach. Exchange–correlation energy was estimated using the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) [36] for solids and surfaces. To construct molecular orbitals, a plane wave basis set with an energy cutoff of 325–500 eV was used for supercells with different sizes (2 × 2 × 1, 2 × 2 × 2, 3 × 3 × 3, and 5 × 5 × 5) of k-points [14,17,20].

The magnetic structure of condensed matter in quantum chemistry is known to be determined by interactions between the orbital and spin moments of the electrons of atoms: exchange energy, dipole–dipole energy, and spin–orbit interaction energy. Taking into account the electron spin in the Ti₃Sb–H structure is important because the interaction of the magnetic moment associated with the spin with the magnetic field created by the orbital rotation of the electron makes a certain contribution to the total energy of the system. Taking this into account, the spin-polarized generalized gradient approximation SGGA was used to calculate the energy $E_{\mathrm{G}\mathrm{G}\mathrm{A}}^{\mathrm{X}\mathrm{C}}$, which depends, in particular, on the charge and spin density d of titanium electrons. That is, the calculations of the energy $E_{\mathrm{G}\mathrm{G}\mathrm{A}}^{\mathrm{X}\mathrm{C}}$ of Ti₃Sb–H structures included spin-dependent exchange correlation functionals. The spin–orbit interaction (SOC parameter) is taken into account by the so-called internal quantum number J, equal to the sum of the quantum numbers l and s: $\left|J=\left|l\pm s\right|=\left|l\pm 1/2\right|\right|$.

In the calculations, it was assumed that the exchange correlation enhancement factor, consisting of the sum of the exchange part and the correlation part, also depends on the spin density. The spin polarization is equal to zero for non-magnetic systems. The spin-dependent part of the exchange–correlation functional allows taking into account the effective interaction of atomic orbitals (J_H, eV) in calculations within the framework of Hund’s rules acting on the d-orbitals of the transition metal. Typical values of J_H for the energy state of transition metal atoms are as follows: 3d (0.7–1 eV), 4d (0.5–0.8 eV), and 5d (0.3–0.5 eV). Thus, the SGGA PBE density functional encodes the exchange splitting of the spin states of the d-orbitals of metal atoms (mainly titanium) in the studied Ti₃Sb-based structures. The calculated J_H value for d-metals is larger when the spin-dependent SGGA PBE functional is used compared with using the spin-local density functional (LSDA).

Integration over the Brillouin zone was carried out in accordance with the Monkhorst–Pack scheme using a k-point grid separation of 0.03 Å⁻¹. Convergence tests of the total energy per atom showed that the limiting kinetic energy for the wave functions converged within a convergence threshold of less than 0.1 meV/atom. The following valence states of orbitals were used for the atomic configurations: Ti–[Ar] 3d²4s², Sb–[Kr] 4d¹⁰5s²5p³, and H–1s¹. The geometry optimization of the crystal structures based on A15 was carried out within the framework of the BFGS minimization scheme [40]. To determine the convergence of the geometric parameters of the crystal, a force limitation of 0.02 eV/Å was applied. The following threshold values were used for convergent Ti₃Sb–H supercells: energy change per atom, maximum residual force, maximum atomic displacement, and maximum stress less than 1 × 10⁻⁷ eV, 1 × 10⁻⁴ eV/Å, 1 × 10⁻⁴ Å, and 0.001 GPa, respectively. For comparison, the experimental lattice constant (a = 5.2228 Å) [14,41] along with the A15 Ti₃Sb structure, which have cubic symmetry with the space group $Pm\overline{3}n$, No 223, were used (Figure 1). The optimized lattice constant of Ti₃Sb is a = 5.215 Å by DFT GGA, which is in good agreement with the experimental value [41].

Our preliminary DFT calculations of Ti₃Sb–H showed that the difference in the calculated adsorption energy values between different surfaces was insignificant. A noticeable difference in the calculated values from the direction is usually observed in the calculated mechanical properties in metals. In our DFT calculations, for ease of visualization, for comparison of the calculated adsorption energies and activation energies, only the (110) surface of the A15 cubic cell of the Ti₃Sb compound was considered (Figure 1).

In the important planes (110; 111; 100) of an ideal cubic lattice, the number of particles should be almost the same, since there is no anisotropy of crystals. It is known that the surface energy, for example, of metals and crystalline materials with an ideal cubic lattice mainly depends on the valence, electron charge, lattice constant, ion radius, and electron work function. It is also known that DFT-calculated values of surface energy for different orientations (110; 111; 100) of the surface of cubic materials depending on the number of layers in the plate also do not differ much from each other and there is no anisotropy of surface energy.

When calculating adsorption within the framework of density functional theory (DFT), atomic hydrogen (H) is usually used rather than molecular hydrogen (H₂) for several reasons as follows:

• Simplicity of modeling. Atomic hydrogen is easier to model at the DFT level because its electronic structure is simpler. The H₂ molecule has two hydrogen atoms with additional electronic interactions that are more difficult to describe, especially in the context of adsorption.
• Reaction kinetics. In real conditions, hydrogen adsorption on a surface often occurs in atomic form, since H₂ molecules dissociate on the surface, turning into atomic hydrogen. This is especially true for processes such as diffusion, catalytic reactions, or hydrogenation reactions, where atomic hydrogen participates in the reaction.
• Energetic considerations. Adsorption of atomic hydrogen is often more stable than that of molecular hydrogen, since hydrogen molecules generally tend to dissociate when adsorbed onto active sites of catalysts or other surfaces.
• Simplification of calculations. In DFT calculations, it is usually assumed that the adsorbed atom or molecule has already gone through all intermediate stages (e.g., dissociation), and only the atomic form of hydrogen actively interacts with the surface, which simplifies the modeling.

Thus, the use of atomic hydrogen in our calculations reflects the real physical situation in Ti₃Sb–H supercells and allows us to more accurately model the interaction of H with Ti₃Sb. This allows us to model with materials in various applications, such as hydrogen accumulation and storage, catalysis, or fuel cells.

To optimize and improve the accuracy of DFT calculations, we constructed a model of a layer plate [42] of the Ti₃Sb (110) surface with a vacuum region of 15 Å in supercells. In other words, the surface properties of the cubic structure of A15 Ti₃Sb (110) (Figure 1) were simulated using a plate model from one layer to three layers.

The atoms in the upper two layers of Ti₃Sb were allowed to relax. And the atoms in the lower three layers of the cell were fixed in their bulk positions. For the adsorption calculation, the H atoms were placed in the top site (TS), bridge site (BS), and hollow site (HS) on the Ti₃Sb surface. Tetrahedral (TIS), octahedral (OIS), and diagonal (DIS) interstitial sites were considered for the diffusion of H atoms in the subsurface layer and bulk position in the constructed Ti₃Sb (100) supercells.

The interlayer distance between the plate surface and the volume models was determined as $∆d=\left(d_{i-j}-d_0\right)/d_0$, where $d_0$ and $d_{i-j}$ are the interlayer distances between the i-th and j-th layers of the plate model before and after relaxation, respectively.

Thus, in the H adsorption calculations, the surface unit cell was doubled along the lateral directions of the cubic crystal. Such a cell was laid by identical neutral planes corresponding to the Ti₃Sb stoichiometry. The vacuum gap between the layers was 15 Å. In the calculations, the atoms of the upper one and two layers of the crystal together with the H adsorbate were allowed to relax. In this case, the atoms in the other layers were frozen and were in optimized bulk positions in the Ti₃Sb crystal.

The adsorption energy of H ($E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$) was calculated as the difference between the total energy of the supercell with one adsorbed H and the clean Ti₃Sb (110) surface and the H atom in vacuum:

$$E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}={\displaystyle \frac{1}{N}}E\left({\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{b} \mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}+N\left(\mathrm{H}\right)\right)-E\left({\mathrm{s}\mathrm{l}\mathrm{a}\mathrm{b} \mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\right)-NE\left(\mathrm{H}\right),$$

(1)

where E(slab Ti₃Sb + N(H)) is the total energy of the Ti₃Sb–H system with an adsorbed H atom, E(slab Ti₃Sb) is the total energy of the Ti₃Sb slab, N is the number of adsorbed H atoms, and E(H) is the total energy of free H atoms, respectively. Lower adsorption energies of H atoms corresponded to more stable adsorption positions. The spontaneous and exothermic process of H adsorption on the Ti₃Sb (110) surface corresponded to the calculated negative value of $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$. The effects of relaxation and increasing the supercell size to 5 × 5 k-points reduced the adsorption energy. Compared with the unrelaxed 2 × 1 × 1 supercell, the value of $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ decreased by ∼0.15 eV on average.

Probable diffusion paths of hydrogen atoms on the Ti₃Sb surface were studied using the molecular dynamics (MD) method. Hydrogen atoms were introduced into supercells with Ti₃Sb cubic structures in tetrahedral interstitial sites. The calculations were performed on both 2 × 2 × 2 (96 atoms) and 3 × 3 × 3 (324 atoms) supercells. The results of data convergence for these supercells differed slightly from each other. The formation energies of interstitial H₀ (charge states of hydrogen q = 0) were 1.82 eV (2 × 2 × 2 supercell) and 1.85 eV (3 × 3 × 3 supercell), respectively, in calculations with the GGA-PBE functional. The sample of the k-point of the supercell was limited to the Γ point in the calculations. The positions of hydrogen in the Ti₃Sb crystal lattice were considered in a grid of regularly located possible sites for H introduction. The distance between positions of two nearest hydrogen atoms was chosen to be on average 0.2 Å. The calculations were performed at nonequivalent possible insertion sites for hydrogen in the Ti₃Sb cubic cell. One hydrogen atom was introduced into each of these sites in a weakened atomic structure. By limiting the hopping distances to a maximum of up to 4 Å, a saddle point in the DFT GGA calculations for three-dimensional H migration in Ti₃Sb was obtained.

In principle, one could consider all atomic hops between any pair of stable H species positions (q = –1, 0, +1 charge states) in Ti₃Sb. However, taking into account the charge state of H was not part of the task and was not considered by us in this article. The possible charge states of hydrogen (hydride ion, H^–, atomic hydrogen, H⁰, and proton, H⁺) for all sites can be considered by appropriately changing the number of electrons in the cell. The corresponding migration energies of the different hydrogen species (q = –1, 0, +1) can be estimated by calculating the two positive vibrational frequencies at each saddle point of the associated hopping frequencies of these hydrogen species.MD simulation of the hydrogen atom diffusion coefficient ($D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$) in the Ti₃Sb crystal was performed using the LAMMPS 2022 version accessed on 1 January 2022. (https://docs.lammps.org/Manual) software package. The calculation used the grand canonical ensemble for isothermal–isochoric (NVT) from 1110 to 1375 K (melting point of Ti₃Sb) for 50–100 ps with a time step of 1 fs [43]. To calculate $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$, the mean square displacement (MSD) method was used. In LAMMPS, the mean square displacement of hydrogen atoms was tracked to calculate the MSD over time, and then the Einstein equation was used to obtain the diffusion coefficient. Postprocessing and diffusion calculations were performed in the standard way, taking into account the equilibrium of the system and the step interval (up to the run 50,000 command) in time. For hydrogen interactions, it is usually sufficient to use a pair potential, such as Lennard–Jones. However, we modeled hydrogen in a cubic Ti₃Sb crystal, and therefore a more complex potential was required.

When describing the nature of interactions between atoms in metals and alloys, the interatomic interaction potential based on the embedded atom method (EAM) is often used. Unlike the pair potential, the EAM method takes into account not only interactions between atoms but also interactions of electron clouds with atoms and, conversely, atoms with clouds. Therefore, when numerically integrating the equations of motion and when choosing the integration step in the microcanonical ensemble for NVT diffusion processes, EAM potentials were used.

After calculating the diffusion coefficient, we compared it to experimental values of similar systems and our theoretical predictions for verification. This showed that the data on the mobility of hydrogen atoms in the Ti₃Sb crystal lattice were noticeably “scattered” relative to the average experimental value $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ for cubic materials.

Taking into account the above, the diffusion coefficient H was also calculated using the Arrhenius equation:

$$D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}\left(T\right)=D_{0 }\exp \left(-{\displaystyle \frac{E_{\mathrm{a}}^{\mathrm{H}}}{k_{\mathrm{B}}T}}\right)$$

(2)

where $D_{0 }$ is the pre-exponential factor, $E_{\mathrm{a}}^{\mathrm{H}}$ is the activation energy of atomic diffusion of H, $k_{\mathrm{B}}$ is the Boltzmann constant, and T is the temperature.

The value of $E_{\mathrm{a}}^{\mathrm{H}}$ was calculated using the classical transition state theory. Then, Equation (2) can be rewritten as:

$$D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}\left(T\right)=D_{0 }\exp \left(-{\displaystyle \frac{E_{\mathrm{a}}^{\mathrm{H}}}{k_{\mathrm{B}}T}}\right)=l^2\nu \exp \left(-{\displaystyle \frac{E_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}}{k_{\mathrm{B}}T}}\right)$$

(2a)

where l is the jumping distance between two nearest sites, ν is the frequency of jumping attempts, and $E_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ is the diffusion energy barrier (or activation energy of atomic diffusion of H).

The distance of the H atom from one stable position to another stable position in the Ti₃Sb–H supercell varied from 0 to 3 Å. And the frequency of jumps of the interstitial hydrogen atom per unit time was v = 10¹²–10¹⁴ s⁻¹.

The DFT method strategy allows all calculations of the total energy of the ground state of the system at 0 K “for a static crystal”, where the atoms are fixed in their crystallographic positions. In this case, at 0 K, the quantum zero-point motion of the atomic nucleus was very small and was not taken into account.

The activation energy $E_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ of H diffusion in Ti₃Sb was calculated using the DFT method. When H migrates along the diffusion path from one site to another site, the H atom must overcome the corresponding potential barrier $E_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$. The maximum point of the potential barrier is defined as the saddle point, which is located in the middle of the migration path or reaction paths (coordinate, Å) of the H atom in the Ti₃Sb supercell. The activation energy of diffusion $E_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ in Ti₃Sb was calculated as the energy difference:

$$E_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}=E_{\mathrm{s}\mathrm{a}\mathrm{d}}-E_0$$

(3)

where $E_{\mathrm{s}\mathrm{a}\mathrm{d}}$ and $E_0$ are the total energies of the supercells for the saddle point and for the equilibrium region, respectively, for the dependence of energy (eV) on the reaction coordinate (Å). The activation energy of H $E_{\mathrm{a}}^{\mathrm{H}}$ in Ti₃Sb(110)–H was determined in the following two ways: (1) by using the Arrhenius equation and (2) by searching for a saddle point using the DFT calculation. In the first method, statistical data on the diffusion coefficient of hydrogen through bcc metals [28] and alloys [34] were processed using the MC method [43] at several temperature values. The pre-exponential factor and the activation energy for the considered bcc metals and alloys were averaged. Then, taking into account the value of $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}\left(T\right)$ and the temperatures that we specified in the Arrhenius coordinates (ln$D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ = f(1/T)) for bcc materials containing hydrogen, a straight line was constructed. Such a straight line cut off a segment equal to ln$D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^0$ on the ordinate axis, and the activation energy $E_{\mathrm{a}}^{\mathrm{H}}$ was found from the slope tangent (tana = −$E_{\mathrm{a}}^{\mathrm{H}}$/R) of the resulting line. However, this calculation method did not provide sufficient accuracy of $E_{\mathrm{a}}^{\mathrm{H}}$ for Ti₃Sb(110)–H compared to $E_{\mathrm{a}}^{\mathrm{H}}$ of similar materials with a cubic structure. Therefore, to control the accuracy of $E_{\mathrm{a}}^{\mathrm{H}}$, a second method was also used, namely the activation energy of H was found from a DFT calculation using the NEB model. These DFT data for $E_{\mathrm{a}}^{\mathrm{H}}$ were consistent with the data of similar materials, in particular, as can be seen in Section 3.8. To calculate the diffusion coefficient of hydrogen in a cubic Ti₃Sb(110) crystal using the MD molecular dynamics with LAMMPS, activation energy $E_{\mathrm{a}}^{\mathrm{H}}$ was obtained using the NEB model.

Comparison of the results of calculations of the activation energy of diffusion of supercells 2 × 2 × 2 and 5 × 5 × 5 showed that they did not differ much from each other. The convergence of these values corresponded to an accuracy of 0.001 eV.

Magnetic configurations of Ti₃Sb, with a spin magnetic moment of Ti, were set taking into account two eigenvectors of the spin state (spin up and spin down) along the z-axis, providing a magnetic moment equal to zero [14].

The H coverage (Θ) of the surface was defined as the ratio of the number of adsorbed H atoms to the number of metal atoms in each surface layer, taking into account different configurations of the Ti₃Sb–H supercells. The calculated average adsorption energies of H atoms on different sites at 0.25 ≤ Θ ≤ 1 showed that, with the same coverage of H atoms, the adsorption energy on the same adsorption sites did not differ significantly. The value of $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ for identical sites depended on the size of the supercell.

Our calculations of the spin-polarized band structure of Ti₃Sb–H (Θ = 0.25 ML) showed that increasing the supercell size from 3 × 3 × 3 to 5 × 5 × 5 did not lead to a significant change in the minimum value of the total energy of Ti₃Sb–H. The 3 × 3 × 3 and 5 × 5 × 5 Ti₃Sb–H supercells with hydrogen coverage Θ = 0.25 ML had almost the same minimum total energy as the 2 × 2 × 2 supercell, when one Ti atom in the supercell was substituted by one H atom. With this in mind, as an illustrative material, in particular for the density of states (DOS), the DOS of the 2 × 2 × 2 supercells with Θ = 0.25 ML is presented below.

2.2. DFT SGGA PBE Method

As is known, intermetallic compounds of the A₃B type with the A15 structure are effective hydrogen adsorbents and are promising for storing H. However, to study the adsorption equilibrium and thermodynamic and kinetic parameters of H adsorption, it is necessary to use various synthesis methods, measuring instruments, and physicochemical analysis methods. To determine, in particular, the adsorption energy of H, it is necessary to study the Langmuir adsorption isotherms, obtain data on equilibrium adsorption, and use various adsorption models, for example, the Freundlich and Redlich–Peterson models. In addition, it is necessary to determine the thermodynamic properties of hydrogen adsorption on the surface and evaluate the spontaneity and endothermicity of the adsorption process in the Ti₃Sb–H system. It is also necessary to select a model for the adsorption kinetics in the Ti₃Sb–H system. For example, if a binary system is better described by a pseudo-second-order model, this indicates that chemisorption occurs.

To synthesize the Ti₃Sb material, it is necessary to use materials science methods, for example, to obtain SEM images of the material and Ti₃Sb–H, to conduct X-ray structural analysis to identify the production of high-quality crystals, and other methods. However, these experimental methods are labor-intensive, and the speed and accuracy of the estimated characteristics are low. The DFT SGGA PBE method for studying the Ti₃Sb–H system used in this work eliminated these difficulties and disadvantages.

If we set a goal, we could analyze the natural bond orbitals (NBOs) in the Ti₃Sb–H system. NBOs are known to allow us to study intramolecular hydrogen bonds, charge transfer, and/or conjugated interactions between bonds in molecular, complex, and non-molecular systems. In this work, the Ti₃Sb–H supercell was considered a non-molecular system, and the charges of chemical interactions between hydrogen (donor) and Ti₃Sb (acceptor) were not taken into account. Therefore, NBO analysis was not included in the problem we were solving or the goal of the work.

3. Results and Discussion

3.1. Structural Features of Ti₃Sb–H

The interaction of hydrogen with metals (M), as is known from physical chemistry surfaces [28,34], can be accompanied by the following processes: on the metal surface: physical adsorption and chemical adsorption (activated adsorption or chemisorption); in the volume of the metal: diffusion, migration, absorption, and formation of hydrides.

Physical adsorption is caused by intermolecular, mainly van der Waals forces, when the energy of interaction of the gas with the adsorbate has a value of about 10–30 kJ/mol. In this case, the molecules of the adsorbed gas do not dissociate into atoms. This is the first stage of sorption processes during the interaction of metals with hydrogen at low temperatures.

An increase in temperature facilitates the transition of physical adsorption to chemisorption, when the H₂ molecule dissociates into atoms that chemically interact with the surface atoms of the metal. In this case, hydrogen atoms saturate the uncompensated bonds of the metal surface. The interaction energy during chemical adsorption ≥30 kJ/mol.

Correct description of the results of interaction of H with crystals (metals, intermetals, alloys, complex compounds, etc.) is a complex task. In the first approximation, we will consider a simple mechanism of adsorption of H atoms on the metal surface, i.e., we will consider the M–H system.

The electron gas model describes the behavior of electrons in bodies with electron conductivity. In this model, the Coulomb interaction between electrons in metals is neglected, and the electrons themselves are weakly bound to the ions of the crystal lattice. In this case, the adsorbed H atoms on the metal surface will be outside the “electron surface” of the metal. We assume that the temperature of the M–H system is high and contact of the metal surface with hydrogen is possible. Then, due to the H₂→2H process, the transition of hydrogen atoms into the metal crystal lattice will be possible. The flow of hydrogen atoms in the surface layer of the metal will create a concentration gradient, and then the hydrogen atoms will diffuse into the metal lattice.

Analysis of the results of studies of metal–hydrogen systems (in particular, DFT of H adsorption [44,45,46,47,48,49,50,51,52,53,54,55,56] and H diffusion [57,58,59]) shows that the interaction of metals with hydrogen can be divided into three classes (A, B, C). Class A metals (Li, Na, K, Ca, etc.) form salt-like hydrides with hydrogen. Class B metals (Mg, Al, Zn, Cd, Hg, etc.) form hydrides with a covalent bond when interacting with hydrogen. Class C metals can interact with hydrogen via both endothermic (Fe, Ni, Cu, Ag, Pt, etc.) and exothermic (Ti, Zr, Ru, Rd, etc.) reactions.

If the adsorption of H is purely physical, diffusion in the metal does not occur. The necessary preliminary stage of H diffusion in metals is activated adsorption, caused by chemical interaction. Diffusion of hydrogen into the lattice of an intermetallic compound precedes the formation of interstitial or substitutional solid solutions. It is known from experimental and calculation data that the dissolution and diffusion of H is enhanced in those metals whose crystal lattice has defects. In this case, the captured H can be in the form of a proton in the lattice defects [60,61,62].

Intermetallics occupy an intermediate position between metals and ceramics both in the type of chemical bond and in the properties. They have chemical bonds of metallic and covalent types, and intermediate bond types are also possible. The range of problems to be solved related to atoms’ adsorption and diffusion in intermetals has expanded over the last decade [14,17,28,34]. However, there are few new observations, methods, and theoretical models in this area that simultaneously relate surface properties and diffusion based on intermetallic compound A15 A₃B–H systems.

The A15 structure of A₃B compounds has a cubic edge with an average length of 5.22–5.29 Å (lattice parameter “$a_0$”) (Figure 1). In A₃B, the atoms of component B are located at the vertices of the cube and form a body-centered cube. The A atoms form one-dimensional chains in the orthogonal directions x, y, z. The interatomic distances along the chains are 1/2$a_0$. The interatomic distance A–A in A₃B is smaller than the interatomic distance in pure metal A. This structure of A₃B affects its physical properties.

Some intermetallic compounds of the A₃B type demonstrate a phase transformation from a cubic structure to a tetragonal one. In such a structure (space group $Pm\overline{3}n$) A₃B, in particular, Ti_3−xSbH_x (Ti₃Sb_1−xH_x), the H atoms can be located in the following crystallographic sites: 6(d), 16(i), and 24(k) [14,17] (Figure 2).

Site 6(d) corresponds to the nodes of the metallic sublattice Ti and the non-metallic sublattice Sb, and the centers of tetrahedral interstices (site 16(i)) and tetrahedral interstices 24(k) formed by four nodes of the metallic sublattice Ti (or four nodes of the Sb sublattice) (Figure 2).

Hydrogen atoms can be located in the vacant site of the Ti₃Sb lattice at different sites, for example, in the 6(d) sublattices of Ti and/or Sb. It is also possible that some of the H atoms can occupy the vacant site 16(i) of the Ti and Sb sublattices, and the remaining H atoms are statistically located in the tetrahedral positions 24(k) (Figure 2). Thus, the position of hydrogen on the surface and its location in the volume can change the properties of the Ti₃Sb–H system, for example, increasing the hydrogen capacity of Ti₃Sb.

In accordance with the stated objective of this work, the calculated reference state of the binding energy $E_{\mathrm{b}}$ between the components and/or objects of the Ti₃Sb–H system corresponded to the energy of a supercell without defects. That is, the properties of an ideal system based on a Ti₃Sb crystal were calculated. All ab initio calculated quasi-binary Ti₃Sb–H supercells in our case contained the same number of intermetallic sites and had the same size. In such a binary system, the binding energy between two components is calculated by the formula:

$$E_{\mathrm{b}}\left(A_1,A_2\right)=\left[E\left(A_1\right)+E\left(A_2\right)\right]-\left[E\left(A_1+A_2\right)+E_{\mathrm{r}\mathrm{e}\mathrm{f}}\right]$$

(4)

where $E\left(A_j\right)$ is the energy of the supercell based on $A_j$ only, $E\left(A_1+A_2\right)$ is the energy of the supercell based on both $A_1$ and $A_2$, which interact with each other, and E_ref is the energy of the supercell without $A_1$ or $A_2$.

If the system contains more than two objects (components, particles, sites, defects, etc.), Equation (4) must be extended and the influence of defects on the change in the total energy of the system must be taken into account. It is necessary to additionally calculate the energy of the most stable configurations, the energy of vacancy formation, and the energy of migration of intrinsic point defects, divacancies, and vacancy clusters in A15 Ti₃Sb–H. The solution of the listed issues related to point defects was not included in the objectives of this article. Let us consider the adsorption of hydrogen on the surface of A15 Ti₃Sb, where the interaction of hydrogen occurred both in the Ti and Sb sites of the Ti₃Sb lattice.

3.2. DFT SGGA-PBE Calculation of the Ti₃Sb–H Structure

In the cubic unit cell of A15 A₃B, space group $Pm\overline{3}n$, the A atoms occupy the 6c sites (¼, 0, ½), with point symmetry D_2d. The B atoms occupy the 2a sites (0, 0, 0), with point symmetry T_h. The structure of A15 A₃B is characterized by a sequence of internuclear separations of atoms: A−A (2×; 0.5a₀), A−B (12×; 0.559a₀), A−A (8×; 0.612a₀), and B–B (8×; 0.866a₀, where a₀ = cubic lattice parameter) [14,17].

Figure 3a–d show the atomic structures of Ti₃Sb–H cells, where the hydrogen atom is adsorbed on the Ti–Ti and Ti–Sb sites in the crystal lattice. The calculated lattice parameter both in the primitive unit cell and in different configurations of Ti₃Sb–H supercells coincide with each other and on average is a₀ = 5.22 Å.

3.3. Adsorption Energy of H on Metal

Let us first take a brief look at the adsorption of hydrogen on a metal surface. The bond energy M–H depends on the degree of overlap of the electron orbitals of hydrogen and metal. It is known that the relation between the M–H bond energy and the energy of the metal crystal lattice is almost linear [44,45,46,47,48,49,50]. In this case, it is important to determine the contribution of hydrogen to the M–H bond energy. The M–H bond energy affects the mechanisms of hydrogen adsorption, diffusion, and emission. Materials and the mechanism of H adsorption (H_ads) for M–H systems, taking into account experimental data [50,51,52,53,54,55,56], can be divided into three groups: low, medium, and high adsorption energy of atomic hydrogen.

• With increasing H_ads energy ($E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$), the rate of hydrogen emission in M–H systems increases. This mechanism is typical for metals with low H_ads energy (Ag, Au, Zn, Cd, In, Hg, Pb, Tl). A similar mechanism of H adsorption also takes place in some binary refractory compounds (e.g., borides, carbides, nitrides, silicides). In such compounds, the contribution of the covalent chemical bond is significant.
• With an increase in the density of the layer of adsorbed H atoms, $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ increases and the process of surface recombination begins. With a further increase in the adsorption energy, the recombination rate remains constant in the case of a defect-free metal surface. The rate of H_ads gradually slows down if adsorption takes place mainly on the defects of the crystal lattice and the boundaries of crystal grains. For metals such as Pt, Pd, Ru, Rh, Os, Fe, Co and Ni, the adsorbed hydrogen atoms are still far from the metal surface; therefore, they cannot tear an electron from the metal, and chemical desorption does not occur.
• At high energy $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$, a relatively dense layer of H_ads is formed on the metal surface. Hydrogen release from the surface can occur by different mechanisms, for example, electrochemical desorption. The electrochemical desorption mechanism is typical for metals with high adsorption energy, for example, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Mn, and Re. Similar data on H_ads for binary and complex alloy intermetallics have not yet been generalized.

3.4. DFT Calculation of the H Adsorption Energy

The DFT calculation of the adsorption energy of H on solid surfaces has its own characteristics (see, for example, [57,58,59,60,61,62,63,64,65,66,67]). Below, we consider the results of our DFT SGGA calculation adsorption for the Ti₃Sb–H system. The adsorption energy of the hydrogen atom $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ on the Ti₃Sb (110) (Figure 1c) surface was calculated using this formula:

$$E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}=E\left({\mathrm{H}/\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\right)\left(110\right)-E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\right)\left(110\right)-E\left(\mathrm{H}\right),$$

(5)

where E(H/Ti₃Sb)(110) is the total energy of the surface layer (Ti₃Sb)(110) with an adsorbed H atom, E(Ti₃Sb)(110) is the total energy of the surface layer (Ti₃Sb)(110), and E(H) is the total energy of the H atom, respectively.

The adsorption energy between an H adatom and a Ti₃Sb surface depends on various factors, in particular, the bond length between the component atoms and the adatom position. When the bond length between the H adatom and the Ti₃Sb surface component atoms, in particular, the titanium atoms, is shorter, the adsorption energy of H is relatively greater. When the bond length between the adatom and the surface component atoms is long, physical adsorption may take place. Conversely, when the bond length is short, the adsorption energy increases due to chemical adsorption.

The following possible adsorption sites on the Ti₃Sb (110) surface were included in the scheme for calculating the adsorption energy of the H atom (Figure 4).

We considered the hydrogen atom site on top of Ti or Sb (TS), the H bridging site of the Ti–Ti or Ti–Sb bond (BS), and the H site in the lattice voids (H hollow site (HS)). The adsorption energy of H was calculated for the first subsurface of Ti₃Sb, which contained tetrahedral (TISs) and octahedral interstices (OISs). Thus, the most probable adsorption sites of one H atom on the first and second subsurface regions of TS, BS, and HS were estimated by us by a DFT calculation of the adsorption energy of H. The H adsorption sites with the minimum H adsorption energy were taken as stable adsorption sites for H adatoms on the Ti₃Sb (110) surface.

The study of the effect of coverage $\Theta$ on $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ showed that the adsorption of H atoms was stronger with vertical adsorption on the hollow site HS of the Ti₃Sb surface. Comparison of the calculated $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ on different regions of the Ti₃Sb (110) surface showed that, with the same coverage (0.25 < $\Theta$ < 1), the adsorption energies were close to each other in the same regions and equivalent configurations of the supercells.

The calculated absorption energies of H atoms in these regions for different supercells and similar structures with different sizes differed slightly from each other (

Table 1. DFT SGGA PBE calculated hydrogen adsorption energies $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}} \left(\mathrm{e}\mathrm{V}\right)$ on the surface of a cubic crystal A15 Ti₃Sb (110) with a hydrogen atom coverage of a monolayer ML of the surface Θ = 0.25 ML of a 2 × 1 × 1 supercell.

Ti3Sb (110)		Surface H
Site	${\mathit{E}}_{\mathbf{a}\mathbf{d}\mathbf{s}}^{\mathbf{H}}$ (eV)	dH−Ti (Å)	dH−surf (Å)
TS	−0.55	1.77	1.72
BS	-0.77	1.84	1.33
HS	−0.92	1.85	0.57

,

Table 2. Dependence of $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ of hydrogen on the hollow site (HS) of the surface on the bond length and the height of the adsorbate from the Ti3Sb (110) surface (Θ = 0.25 ML) in supercells of different sizes.

Adatom H on HS Site	Ti3Sb (110)	${\mathit{E}}_{\mathbf{a}\mathbf{d}\mathbf{s}}^{\mathbf{H}}$ (eV)	dH−Ti(Sb) (Å)	dH−surf (Å)
Ti–Ti bond supercell	2 × 1 × 1	−0.55	1.882 (H–Ti)	0.357
Ti–Sb bond supercell	2 × 1 × 1	−0.47	1.844 (H–Sb)	0.273
Ti–Ti bond supercell	5 × 1 × 1	−0.93	1.951 (H–Ti)	0.351
Ti–Sb bond supercell	5 × 1 × 1	−0.97	1.975 (H–Sb)	0.272

and

Table 3. DFT SGGA PBE calculated adsorption energy of H on Ti₃Sb(110) surface compared to $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ of similar systems.

System	${\mathit{E}}_{\mathbf{a}\mathbf{d}\mathbf{s}}^{\mathbf{H}}$(eV)
H–Ti3Sb (110) this work	−0.55
H–Fe3Si (110) [63]	−0.87
H–TiO2 [64]	−0.61
H–Ti/TiFe (001) [65]	−0.52

).

The adsorption energies of H atoms in three regions, taking into account the distance between the H atom and the Ti₃Sb atom ($d_{\mathrm{H}-\mathrm{T}\mathrm{i}\left(\mathrm{S}\mathrm{b}\right)}$) and the height of the adsorbate ($d_{\mathrm{H}-\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}$), are presented in Table 1.

Let us briefly consider how association occurs during gas adsorption on the surface of a solid. Ti₃Sb crystals absorb hydrogen. This ability is due to the fact that the opposite atoms of the surface of the Ti₃Sb–H system are under the influence of unbalanced attractive forces. In this case, the resulting force is directed inside the Ti₃Sb crystal. When the adsorbed H gas diffuses inside the crystal lattice, it enters the force field of the crystal, which exists between the atoms, ions, or molecules of the solid. In this case, chemical absorption of the adatom occurs.

As can be seen in Table 1, the adsorption energy of H $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ depends on the distance between hydrogen and the metal atom (d_H−Ti) on the Ti₃Sb (110) surface, the perpendicular distance of H to the surface plane (d_H−surf), and the preferred arrangement of H atoms (TS site, BS site, and HS site) on the surface. A similar dependence was also found for the calculated $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ on the number of Ti₃Sb (110) surface layers. More precisely, the dependences of $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ on the perpendicular distance of hydrogen to the first subsurface plane (d_H−1surf) and the perpendicular distance of hydrogen to the second and third subsurface planes of Ti₃Sb (110) that we estimated were qualitatively similar.

For equilibrium states, the adsorption energy of gases on metal surfaces is approximately 0 to −20 kJ/mol (−0.21 eV/H atom) for physical adsorption and −80 kJ/mol (−0.83 eV/H atom) to −400 kJ/mol (−4.1 eV/H atom) for chemical adsorption.

From the comparison of the calculated data $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$, eV for important sites (TS, BS, and HS sites) in Table 1, Table 2 and Table 3, it follows that the adsorption energy of hydrogen on the Ti₃Sb (110) surface (Θ = 0.25 ML) was −0.55 eV/H atom (for the TS site), −0.77 eV/H atom (for the BS site), and −0.92 eV/H atom (for the HS site), respectively. For the first two sites (TS, BS), the adsorption energy H corresponded to physicochemical adsorption, and for the HS site, the energy $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ corresponded to chemical adsorption.

Increasing the degree of surface coverage from 0.25 to 0.5 H on ML Ti₃Sb, for example, in a 2 × 1 × 1 supercell, where d_H−Ti = 1.88 Å and d_H−surf = 1.35 Å, increased the absolute value of the adsorption energy. This may be due to an increase in the degree of overlap of the orbital states of the H, Ti, and Sb atoms and an increase in the attractive energy between the atom Ti₃Sb. The calculated adsorption energy of H was lower at the hydrogen position on the HS site of the Ti₃Sb lattice surface with Ti–Ti bonds than with Ti–Sb bonds, regardless of the supercell size.

Thus, the adsorption energies calculated by the DFT GGA method after relaxation of the Ti₃Sb–H system showed the following: H atoms were adsorbed predominantly over the Ti and Sb positions in the first atomic layer of Ti₃Sb 2 × 1 × 1 configurations. The strongest adsorption energy ($E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ = −0.55 eV) was observed for the 2 × 1 × 1 (H–Ti) configuration.

The H atom tended to move to the nearest Ti atom. The Ti–H distance was, on average, 1.86 Å, which is typical for titanium hydrides (>1.80) [66]. The adsorption energies of the H atom in other regions of the Ti₃Sb (110) surface varied significantly, indicating that the adsorption of the H atom on the Ti₃Sb (110) surface with corresponding relaxed configuration belongs to chemical adsorption (Table 1, Table 2 and Table 3). Therefore, relatively strong forces act on the H atom at these positions on the Ti₃Sb (110) surface.

The most stable adsorption site for H atoms adsorbed on the surface was the initial site of the 5 × 1 × 1 (H–Ti) configuration after relaxation with an adsorption energy of –1.23 eV. The (H–Ti) distance was 1.87 Å with the H atom in the first atomic layer. A chemical bond was formed between the H atom and the nearest Ti and Sb atoms, similar to the 2 × 1 × 1 (H–Ti) configuration. The adsorption energy for 5 × 1 × 1 (H–Sb) was –1.17 eV, which was less energetically favorable than the 5 × 1 × 1 (H–Ti) configuration.

3.5. Electronic Properties of H Adatoms on the Ti₃Sb (110) Surface

It is known from physical chemistry that chemically bonded non-polar H₂ molecules can interact with the adsorbent surface via van der Waals potential forces.

3.5.1. DOS ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$

The physicochemical interaction of the adatom ${\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ with the surface was estimated by calculating the band electron structure of ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$. Similarly, to the above, it can be assumed that the potential function of the van der Waals interaction of the hydrogen atom with the ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ surface included three non-valent forces: repulsion of electron shells, dispersion forces of van der Waals attraction, and electrostatic interactions.

In the DFT SGGA calculations of ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$, these three non-bonded forces were combined into one van der Waals potential function. The model of electrostatic interaction in the calculations was simplified. It was assumed that the contacts of atoms in ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ were formed only between the nearest neighbors of atoms, and the electrostatic interactions of atoms were averaged. It was assumed that the resulting contribution to the electrostatic interaction of atoms depended on the position of the adatom ${\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ and the distance between the contacting atoms. The effective charges $q_i{q}_j$ of the contacting atoms were not taken into account in the calculations.

In the DFT SGGA calculations of the interaction of the hydrogen atom with the surface of the ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$ system, non-bonded forces were considered as a single function. The resulting contribution to the van der Waals interaction was taken only between the nearest atoms. Our preliminary calculations, carried out with several functionals and basis sets without taking into account the spin based on GGA, were compared with the calculated data, including valence low-spin and high-spin states. It was found that the use of corrections of the DFT-D3 can affect the total energy of the system calculated by DFT SGGA but not the properties of the orbitals of the atoms in the calculations.

The results of the DFT SGGA calculations showed the following. In the band structure of ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\left(110\right)–{\mathrm{H}}_{\mathrm{a}\mathrm{d}\mathrm{s}}$, there was a metallic type of bond near the Fermi energy, strengthened by electrons of the Ti d-orbital. From the analysis of the band structure (Figure 5) and density of electron states (DOS), it can be assumed that the Fermi energy was located in a narrow pseudo-gap in the electron density of states in Ti₃SbH_x. The pseudo-gap occurred in Ti₃SbH_x containing hydrogen due to some shift of the energy bands in the energy spectrum. Within the framework of the p-d-s orbital interaction model, it can be assumed that, in Ti₃SbH_x, the interaction of the 5p states of antimony, the 3d states of titanium, and the 1s states of hydrogen led to a change in the number of states at the Fermi level and its position in the pseudo-gap.

The partial density of states (PDOS) of hydrogen was noticeably smaller than that of Ti (Figure 6a,b). The PDOS of the final state of Ti–H indicate that the valence states of H (1s orbital) and Ti (3d orbital) shifted to a lower level (by about 0.5 and 8 eV, respectively). Compared with the DOS curve of pure Ti₃Sb (110) surface, the change for Ti 3d states in the PDOS of Ti–H can be observed from 5 to –2.5 eV. The peak heights indicate that several electrons of Ti 3d shifted to lower levels when H was adsorbed on the Ti₃Sb (110) surface, which shows the strength of the interactions. In Figure 6b, the s-electrons of H and the s and p-electrons of Ti are shown with a clear peak at around –15 eV. A clear peak is also found in the interaction of the s-electrons of H and the s, p, and d-electrons of Ti at around 8 eV. These peaks indicate that a chemical bond was formed between the H and Ti atoms. The PDOS results are consistent with the calculated adsorption energies (Table 1 and Table 2).

It follows from Figure 6b that the PDOS of 1s orbitals of H adatoms interacting with s-p-d orbitals of Ti atoms on the Ti₃Sb(110) surface was close to zero. This shows that the charge localization between the orbitals of H adatoms and the orbitals of Ti atoms on the surface was very weak. As a consequence, there was no covalent bond in this region of the configuration. There was a slight charge polarization in the H atom and in the neighboring atoms of the Ti₃Sb compound due to the electrostatic repulsion. The left and right regions of the PDOS showed that there was electron localization between the H atom and the nearest metal atoms. These features of the electronic structures suggest that the H–M bonds formed in the right region were much stronger than that of the left region of the surface. Therefore, such a stronger chemical bond would result in a lower adsorption energy of H, for example, in 2 × 1 × 1 Ti₃Sb–H with Ti–Ti bond (Table 2). In general, for the adsorption configurations of H atoms on the Ti₃Sb (110) surface, there were both strong chemisorption and weak physical adsorption depending on the adsorption energies and electronic structure.

3.5.2. PDOS in 2 × 1 × 1 Supercells Ti₃Sb–H

The DOS of Ti₃Sb–H supercells is shown in Figure 7. It follows from the DOS spectra that the s- and p-orbitals in Ti₃Sb–H made an insignificant contribution to the electron density of the valence band. That is, the electrons in Ti₃Sb–H were strongly localized in the s- and p-orbitals. The low-energy bands in the valence band were generated by the s-orbitals, and the high-energy bands arose from the 3d orbitals of Ti. The interaction between the 3d orbitals of Ti and the 3p orbitals of Sb was observed near the Fermi level. The energy gap of Ti₃Sb–H was ≤0.05 eV/atom. DOS indicates the presence of small peaks of Ti–Sb bond in the valence band of Ti₃Sb–H and a weak electron interaction.

The calculated density of states of Ti₃Sb (Figure 7a,d; from –6 to 4 eV) enabled to evaluate the nature of chemical bonds in which the 3d and 4s orbitals of Ti atoms participated. As can be seen, the overlap region of the Ti density of states with the Sb density of states was small. This was due to the highly localized profile of the Ti density of states. Based on this, it can be assumed that Ti atoms participated in the formation of ionic bonds with Sb atoms and, to a lesser extent, in the formation of covalent bonds.

The above states for Ti₃Sb–H were consistent with the PDOS of the components of pure Ti₃Sb in energy [14]. However, the PDOS of Ti₃Sb–H (Figure 7) showed different regions in the distributions of the electron density of states of Ti and Sb compared with pure Ti₃Sb. From the comparison of the PDOS spectra, it followed that the density of states of Sb was small near the Fermi energy E_F. Therefore, it can be assumed that the main part of the metallic properties of Ti₃Sb–H was due to the presence of d- orbitals of Ti near the Fermi level.

3.5.3. DOS in Ti₃Sb_1−xH_x and Ti_3−xSbH_x

The atomic structure of Ti₃Sb_1−xH_x and the PDOS of Ti₃Sb_1−xH_x and Ti_3−xSbH_x solid solutions are shown in Figure 8 and Figure 9, respectively. An increase in the number of H adatoms on the surface did not significantly complicate the DOS in the Ti₃Sb–H system. More precisely, the presence of a larger number of H atoms per unit cell did not significantly activate the participation of Ti 3d orbitals in determining the electronic properties.

3.6. Magnetic Moment in Ti₃Sb–H

DFT SGGA calculations of optimized Ti₃Sb–H structures indicated a uniform spatial distribution of Ti and Sb component atoms in the lattice. As for pure A15 Ti₃Sb, in solid solutions Ti₃Sb_1−xH_x and Ti_3−xSbH_x, the titanium and antimony atoms were stoichiometrically located at different sites of the crystal lattice. Partial substitution of Ti and/or Sb atoms by hydrogen atoms did not lead to relaxation of the Ti₃Sb–H structure. Therefore, the nonequivalent positions of the component atoms and the symmetry of the Ti and Sb sites formed a partial magnetic moment. The atomic magnetic moment consisted of spin and orbital moment contributions. Spin-orbit coupling (SOC) influenced the formation of the magnetic moment in Ti₃Sb–H. The size of the SOC splitting was of the order of the Ti 3d sublevel bandwidth.

From the DOS spectra of Ti₃Sb–H supercells, it followed that the contribution of titanium 3d electrons to the magnetic interaction between the crystal atoms was noticeable. Since the 3d states of Ti were only partially occupied by electrons, the empty levels of the components would be filled due to the hybridization of the s- and p-electrons of Ti and Sb. In the case of Ti₃Sb_1−xH_x and Ti_3−xSbH_x, we observed a very small magnetic moment. For the partial magnetic moment of Ti in Ti₃Sb_0.98H_0.02, the following values were obtained: 0.23 ${\mu }_{\mathrm{B}}$(Ti) at the Ti–Ti sites and 0.28 ${\mu }_{\mathrm{B}}$(Ti) at the Ti–Sb sites of the lattice.

3.7. Diffusion of H Atoms on the Ti₃Sb (110) Surface

We have shown that the most stable configuration of the H atom on the Ti₃Sb (110) surface corresponded to adsorption on the 2 × 2 × 1 supercell (Ti–Ti bond). Therefore, we further considered the diffusion of atomic hydrogen on the Ti₃Sb (110) surface between two sections of the 2 × 2 × 2 supercell. This allowed to characterize the surface diffusion. The minimum energy path for H diffusion from the terminal interstitial site of Ti to the nearest terminal interstitial site of Ti on the T_i3Sb (110) surface ranged from 0 to 7 Å (reaction coordinates). The activation energy of the transition state was determined for the saddle points and the path of the minimum reaction energy between two sites of the Ti₃Sb lattice. $\delta E_{\mathrm{Z}\mathrm{P}\mathrm{E}}$ (zero-point energy) [67] correction due to the motion of hydrogen at the zero point was not taken into account. The (negative) value of $\delta E_{\mathrm{Z}\mathrm{P}\mathrm{E}}$ can decrease $E_{\mathrm{a}}$ according to $E_{\mathrm{a}}=E_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}}+\delta E_{\mathrm{Z}\mathrm{P}\mathrm{E}}$. Thus, the energy of migration and diffusion paths of atomic hydrogen on the Ti₃Sb (110) surface between two sites of the supercell were determined. In other words, the energies of hydrogen migration ($E_{\mathrm{a}}\approx E_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}}$) along interstitial paths in the Ti₃Sb lattice were calculated.

The contact of the components of the Ti₃Sb–H system required a certain amount of energy to activate the interphase surface. This energy can be communicated to the system in the form of heat, elastic–plastic deformation energy, and electron and ion irradiation. The interphase energy of interaction of two condensed phases usually decreases exponentially, i.e., the driving force of diffusion is the difference in thermodynamic potentials at the interface between the components. In the absence of the appearance of new phases in the diffusion zone, equilibrium of thermodynamic potentials is established. Accordingly, a certain energy balance was established in the ${\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}-{\mathrm{H}}_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}$ system.

The activation energy of hydrogen diffusion in Ti₃Sb was determined using Equation (2) for the transition states of the H atom. It was assumed that the initial H atom occupied the TIS interstitial site, which was the most stable. The diffusion path of H in the Ti₃Sb lattice mainly went from one TIS interstitial site to another identical TIS (TIS–TIS) and from a TIS interstitial site to the second nearest neighboring non-identical OIS interstitial site (TIS–OIS). From the dependence of the energy barrier of H diffusion on the reaction coordinate between two neighboring identical TIS interstitials (i.e., TIS–TIS), it was found that $E_{\mathrm{a}}^{\mathrm{H}(\mathrm{T}\mathrm{I}\mathrm{S}-\mathrm{T}\mathrm{I}\mathrm{S})}$ = 1.57 eV. This energy was lower than $E_{\mathrm{a}}^{\mathrm{H}(\mathrm{T}\mathrm{I}\mathrm{S}-\mathrm{O}\mathrm{I}\mathrm{S})}$ for diffusion between two neighboring non-identical interstitials from TIS to OIS, that is, $E_{\mathrm{a}}^{\mathrm{H}(\mathrm{T}\mathrm{I}\mathrm{S}-\mathrm{O}\mathrm{I}\mathrm{S})}$ = 2.33 eV for the TIS–OIS interstitials. It can therefore be concluded that the diffusion of H in Ti₃Sb took place predominantly along the identical TIS–TIS interstitials.

Let us consider the results of calculating the diffusion of H in a 2 × 2 × 2 supercell and a 5 × 5 × 3 grid of Ti₃Sb k-point mesh. We assumed that H migrated only between relatively stable interstitials from A1 (initial position) to A2 (final position for diffusion paths) through selected transition sites E1 and E2 in T_i3Sb. Interstitials A in the Ti₃Sb–H supercell can be located on the surface, in the body, and at the edge sites of the lattice (Figure 10). Calculations showed that the migration paths of H through the transition sites (e.g., E1 and E2 in Figure 10) had higher energy barriers compared with $E_{\mathrm{a}}^{\mathrm{H}(\mathrm{T}\mathrm{I}\mathrm{S}-\mathrm{T}\mathrm{I}\mathrm{S})}$ for surface diffusion.

Thus, the activation energy of H atom diffusion into the Ti₃Sb lattice depth, determined within the framework of the transition state model (A1→E1→E2→A2 in Figure 10), showed the following: $E_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}}$ for identical tetrahedral Ti–Ti interstitials (TIS–TIS) and for non-identical Ti–Sb interstitials (TIS–OIS) differed greatly from each other. The maximum value of $E_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}}$ at the saddle point on the dependence of the energy on the migration path of H for identical tetrahedral Ti–Ti interstitials (TIS–TIS) was almost two times smaller than $E_{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{r}}$ for interstitial jumps of H atoms between two adjacent non-identical positions (TIS–OIS).

The stability of the H pair in the Ti₃Sb matrix was estimated by calculating the binding energy E^b of the H pair. In this case, E^b characterized the decrease in the total free energy of the Ti₃Sb–H system when two H atoms approached each other from afar.

$$E_{\mathrm{i}\mathrm{n}\mathrm{t}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}=2E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\mathrm{H}\right)-2E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}{\mathrm{H}}_2\right)-E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\right)$$

(6)

$$E_{\mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{s}\mathrm{u}\mathrm{b}}^{\mathrm{b}}=2E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\mathrm{H}\right)-2E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}{\mathrm{H}}_2\right)-E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\right)$$

(7)

$$E_{\mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}=E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\mathrm{H}\right)+E\left({\mathrm{T}\mathrm{i}}_{3-n}\mathrm{S}\mathrm{b}{\mathrm{H}}_n\right)-E\left({\mathrm{T}\mathrm{i}}_{3-n}\mathrm{S}\mathrm{b}{\mathrm{H}}_{2n}\right)-E\left({\mathrm{T}\mathrm{i}}_3\mathrm{S}\mathrm{b}\right)$$

(8)

where $E_{\mathrm{i}\mathrm{n}\mathrm{t}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}$ is the total free energy of the supercell containing interstitial–interstitial H pairs, $E_{\mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{s}\mathrm{u}\mathrm{b}}^{\mathrm{b}}$ is the total free energy of the supercell containing substituted–substituted H pairs, $E_{\mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}$ is the total free energy of the supercell containing substituted–interstitial H pairs. Similar equations for the binding energy E^b of an H pair can be written for Ti₃Sb_1−xH_x.

For the substituted–interstitial H pair, the binding energy (attraction) was $E_{\mathrm{i}\mathrm{n}\mathrm{t}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}$= –0.11 eV in Ti₃SbH₁. Increasing the hydrogen concentration (number of H atoms from 1 to 4) in the Ti₃Sb lattice reduced the binding energy to $E_{\mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}$= –0.14 eV between these non-identical sites.

The binding strength for the interstitial–interstitial and substituted–substituted H pairs was negligible in the Ti₃Sb lattice. This indicates that it was more difficult for H atoms to move in these identical positions of the Ti₃Sb lattice. That is, the driving force of attraction of H was weak between like sites (TIS–TIS) for the formation of H clusters in the Ti₃Sb matrix.

3.8. Calculation of the Diffusion Coefficient

The diffusion coefficient H was calculated for an ideal Ti₃Sb supercell without taking point defects into account. Surface effects, hydrogen capture by impurities, dislocations, grain boundaries or precipitates, and the formation of molecular hydrogen in Ti₃Sb micropores were not taken into account in the calculations. The interstitial diffusion mechanism was considered. It was assumed that the hydrogen atom diffused in the cubic Ti₃Sb crystal by migrating from a tetrahedral site to its nearest tetrahedral site.

Using the molecular dynamics MD and MC methods [43], taking into account the activation energy of the migration barrier $E_{\mathrm{a}}^{\mathrm{H}(\mathrm{T}\mathrm{I}\mathrm{S}-\mathrm{T}\mathrm{I}\mathrm{S})}$ = 1.57 eV along the interstices of the Ti–Ti lattice of Ti₃Sb (110), the diffusion coefficient $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ was calculated.

Table 4. DFT SGGA PBE-calculated activation energy of H diffusion in Ti₃Sb(110) compared with $E_{\mathrm{a}}^{\mathrm{H}}$ of different compounds.

System	${\mathit{E}}_{\mathbf{a}}^{\mathbf{H}}$ (eV)
H–Ti3Sb(110) this work	1.57
H–Fe3Si (110) [63]	1.91
H–TiO2 [64]	2.69

shows the calculated activation energies of the hydrogen atom of Ti₃Sb–H in comparison with similar compounds.

Once again, it should be noted that the activation energy for H $E_{\mathrm{a}}^{\mathrm{H}}$ diffusion was calculated for the paths from the Ti₃Sb(110) surface to the subsurface on Ti₃Sb(110) using the non-elastic plate model (NEB) [63,67], which allowed the saddle point to be determined. For the studied Ti₃Sb(110)–H supercell configurations, the transition state was determined from the dependence of the energy barrier for H diffusion on the H diffusion coordinate between two adjacent identical interstitials. As indicated above, for the TIS–TIS pair, $E_{\mathrm{a}}^{\mathrm{H}}$ (TIS–TIS) was relatively smaller.

As noted above, the diffusion coefficient of H through bcc metals [28] and alloys [68] was processed by the MC method [43]. In calculating $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$, the $E_{\mathrm{a}}^{\mathrm{H}}$ value was taken from the NEB data (Table 4), since the $E_{\mathrm{a}}^{\mathrm{H}}$ obtained from the Arrhenius equation differed significantly from the EA of similar materials at high temperatures. Thus, we obtained an estimated value of $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ = 1.4 × 10^–8 exp(–1.57/$k_{\mathrm{B}}T$) m²/s ($k_{\mathrm{B}}$ is the Boltzmann constant) at a temperature of 1110–1350 K.

The results obtained for the A15 Ti₃Sb–H system were compared to similar systems based on metals and alloys with a cubic structure for the purpose of using it as a hydrogen storage material. It was found that Ti₃Sb–H had some advantages (e.g., it was more stable, had a lower activation energy) compared with the materials already described above. In particular, these results are given in Table 3 and Table 4.

Taking into account the applied significance of materials based on intermetallic crystalline compounds A₃B with a cubic structure A15 (Cr₃Si type), we can also note some important properties of other similar materials [46,69,70,71,72,73,74,75,76,77]. Intermetallic compounds A₃B exhibited high superconducting transition temperatures and hydrogen-absorbing properties. For example, hydrogen H or deuterium (D) retaining materials with the compositions Nb₃SnH_n [69], Ti₃SbD_n [1], and Ti₃IrDn [70] were known. These A₃B crystals absorbed significant amounts of H or D without changing their structure. The lattice parameter expanded linearly as the concentration of H or D in the lattice increased.

The structural properties of A₃B crystals studied by neutron diffraction allowed us to determine the diffusion characteristics of the H- or D-doped Ti₃IrD_X compound. In particular, taking into account the absorption of D atoms by interstitial positions of the cubic crystal Ti₃IrD_X with x = 0.70 and 3.63, the arrangement of D atoms at the lattice sites was determined.

The compounds Nb₃Au, Nb₃Pt, Nb₃Ir, and Nb₃0s retain their crystal structure type and remain superconducting when saturated with hydrogen to H/M = l. It has been established that the maximum hydrogen content in the Nb₃Sn–H and Nb₃Pt–H systems varies from Nb₃SnH_1.0 to Nb₃PtH_5.1 [71].

Hydrogen absorption isotherms in the intermetallic phase of Ti₃Sb were measured in the range of 400–800 °C and hydrogen partial pressure of 0.001–0.925 atm [72]. It was shown that the number of accessible interstitial sites per metal atom (z) in cubic Ti₃Sb was higher (z = 0.75) compared to similar materials (Ti₃Al hexagonal z = 0.25; Ti₃Sn hexagonal z = 0.25; Ti₃P tetragonal z = 0.5).

The measurement of hydrogen absorption in A15 V₃Ga and Ti₃Ir compounds and X-ray analysis of samples showed that hydrogen absorption in them was high. The presence of hydrogen did not change the structure of these hydrogen-containing solid solutions of A15. The lattice parameter and partial enthalpy and entropy depend monotonically on the hydrogen concentration in solid solutions based on A15 [73].

Other similar compositions of A15 compounds, such as Nb₃Sn [74,75] and Nb₃Sb [76,77], have also been studied theoretically and experimentally. They have been shown to have anomalous properties and other applications. For example, Shubnikov–de Haas and magnetoresistance effects in a strong field have been discovered in the A15 Nb₃Sb compound [76]. In addition, Nb₃Sb wires have been used for thermonuclear magnets [77].

We also noted the following feature of theoretical studies. Comparison of our calculated DFT results for the Ti₃Sb–H system with experimental physicochemical data for similar systems showed the following advantages of the theoretical method. For example, using the DFT method, the adsorption energy of hydrogen was determined by location on specified areas of the Ti₃Sb surface, taking into account the H concentration, the degree of H surface coverage, crystallographic directions of the lattice, the distance between the adatom and the Ti₃Sb components, and the number of crystal monolayers. For the experimental determination of these characteristics, in particular, it was necessary to synthesize the corresponding compositions, prepare the surfaces, and conduct studies in specially controlled and created conditions. In other words, for example, the characteristics of adsorption and diffusion in a system taking into account the position of H on the surface, the surface area, the number of monolayers, the positions of H, and the chosen crystal directions were difficult or impossible to determine from the kinetics and thermodynamics of equilibrium adsorption of gases (adatoms) on a solid surface.

4. Conclusions

In this study, first-principles calculations of the surface hydrogen H atom adsorption and diffusion behavior of H in the cubic crystal lattice of Ti₃Sb (110) were performed (2 × 1 × 1, 2 × 2 × 2, and 5 × 5 × 5 supercells). We performed first-principles calculations of the Ti₃Sb–H electronic structure using the DFT SGGA-PBE method. The calculated relaxed lattice constant of Ti₃Sb–H (hydrogen coverage (Θ) of the monolayer (ML) of the surface Θ = 0.25 ML; a = 5.217 Å) turned out to be in agreement with the known experimental data (a = 5.223 Å) of Ti_3−xSbH_x.

On the Ti₃Sb (110) surface, three possible sites for hydrogen adsorption were considered: the H site on top (TS), site in the bridge bond between metal atoms (BS) of the surface, and the H site in the hollow site (HS) of the lattice (or in the HS hole position). For the first time, the adsorption energy of H $\left(E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}\right)$ on the surface of the crystal lattice of the Ti₃Sb (110) compound was calculated. A more stable configuration of adsorption of H atoms with their coverage of $\Theta$ = 0.25–1 ML was the hollow site HS on the surface of the Ti₃Sb lattice. The energetic stability of the hollow site for H adsorption was determined by the high coordination number of the hollow site containing H with the nearest neighboring atoms and the minimal repulsion between the electron orbitals of the metal and the hydrogen atoms. The adsorption energy of H at a coverage of $\Theta$ = 0.25–1 ML of the HS site on the Ti₃Sb (110) surface of the 2 × 1 × 1 Ti₃Sb–H_ads supercell was practically constant ($E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$= −0.55 eV) and increased with increasing supercell size.

From the DOS spectra of Ti₃Sb–H, it followed that the adsorption of H on the HS surface corresponded to chemical adsorption. With an increase in ML coverage from $\Theta$ = 0.25 to 1, the adsorption energy of H first increased, which corresponded to a very weak state of physical adsorption, and then decreased. The adsorption energy of H on the top site TS and bridge site BS of the surface, and on the hollow site HS, remained constant in the Ti₃Sb–H_ads supercells. When H adatoms were located in the Ti–Ti regions, the value of $E_{\mathrm{a}\mathrm{d}\mathrm{s}}^{\mathrm{H}}$ was smaller than in the Ti–Sb regions of the supercells. This pattern was violated when the supercell size was reduced to the size of the primitive Ti₃Sb–H_ads cell.

Increasing the H coverage of the Ti₃Sb (110) surface formed Ti₃Sb_1−xH_x and Ti_3−xSbH_x solid solutions, in which the density of states changed, which increased stability and reduced the conductive properties. This was due to the interactions of Ti 3d–3d, Ti 3d–Sb 5p, and Ti 3d (Sb 5p)–H 1s electrons. In the band structures of Ti₃Sb_1−xH_x and Ti_3−xSbH_x, a narrow conductive band was localized near the Fermi level. Hydrogen absorption led to the increased localization of 3d states in Ti₃Sb_1−xH_x and Ti_3−xSbH_x due to a decrease in the d–d overlap of Ti.

The calculated partial magnetic moment in Ti₃Sb_1−xH_x solid solutions was small, 0.23 ${\mu }_{\mathrm{B}}$ (Ti) at Ti–Ti sites and 0.28 ${\mu }_{\mathrm{B}}$ (Ti) at Ti–Sb sites of the Ti₃Sb_0.98H_0.02 lattice.

The calculated energy of the migration barrier for H along the tetrahedral site → tetrahedral site route in the Ti₃Sb lattice was 1.57 eV. This energy, estimated by us taking into account the transition state model, corresponded to a saddle point on the dependence of the migration energy on the migration path for interstitial jumps of H atoms between two adjacent tetrahedral sites Ti.

The binding energy of a pair of substituted–interstitial Ti–Ti H sites was $E_{\mathrm{i}\mathrm{n}\mathrm{t}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}$ = –0.11 eV in Ti₃Sb_1−xH_x. Increasing the H surface coverage from 0.25 to 1 ML reduced the Ti–Ti binding energy between these non-identical sites to $E_{\mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{b}}$ = –0.14 eV.

The diffusion coefficient of H atom $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ in the ideal Ti₃Sb (110) lattice, calculated by the MD method taking into account the transition state model, was $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ = 1.4 × 10^–8 exp(–1.57/$k_{\mathrm{B}}T$) m²/s. This value was consistent with the values of $D_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}^{\mathrm{H}}$ in cubic structures of metals and alloys.

Thus, the results of this work are important for understanding the mechanism of interaction of H atoms with intermetallic surfaces. It provides a theoretical basis for further studies of intermetallic and/or hydride alloys, in particular, based on Ti₃Sb for storing H₂. The obtained data can be used to control the adsorption, diffusion, accumulation, and storage of hydrogen in Ti-based intermetallic materials, taking into account the control and stabilization of the crystal structure.