# Promising multi-component cubic alloys for hydrogen storage: Insights from first-principles calculations and machine learning

Zeineb Nefzi, Celine Barreteau, Lucas Levrel, Jean-Claude Crivello

► To cite this version:

Zeineb Nefzi, Celine Barreteau, Lucas Levrel, Jean-Claude Crivello. Promising multi-component cubic alloys for hydrogen storage: Insights from first-principles calculations and machine learning. ACS Applied Energy Materials, 2025, 8 (6), pp.3327-3337. 10.1021/acsaem.4c02728. hal-05008497

HAL Id: hal-05008497

https://hal.science/hal-05008497v1

Submitted on 27 Mar 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Promising multi-component cubic alloys for hydrogen storage: Insights from first-principles calculations and machine learning

Zeineb Nefzi,*,† Celine Barreteau,† Lucas Levrel,† and Jean-Claude Crivello*,‡

†Univ Paris Est Creteil, CNRS, ICMPE, UMR 7182, 2 rue Henri Dunant, 94320 Thiais, France

‡CNRS-Saint-Gobain-NIMS, IRL 3629, Laboratory for Innovative Key Materials and Structures (LINK), 1-1 Namiki, 305-0044 Tsukuba, Japan

E-mail: zeineb.nefzi@cnrs.fr; jean-claude.crivello@cnrs.fr

## Abstract

An extensive exploration of the chemical space was conducted to design and identify promising multi-component cubic alloys with appropriate enthalpy of reaction for hydrogen storage applications. We aim to identify alloys with suitable hydrogen absorption conditions for ambient conditions, according to favorable thermodynamic criteria, while addressing computational challenges in modeling large-scale systems. 18 elements were selected, leading to the systematic investigation of over 8,000 quinary alloy systems across four distinct crystal phases (within solid-solution alloys, mono- and dihydrides). This effort resulted in a comprehensive dataset of more than 34,000 equimolar $MH_x$ structures, where $M$ represents a combination of 5 elements chosen among the 18 selected atoms. To handle the computational demands of Density Functional Theory (DFT) calculations on such a large scale of disordered supercells, designed by the Special Quasirandom Structure (SQS) method, a Machine Learning (ML) approach
was introduced to accurately predict the enthalpy of hydride formation. By training
the ML model on a strategically chosen subset of the data, high predictive accuracy
was achieved while significantly reducing computational costs. By applying filtering
parameters constrained by thermodynamic considerations, such as the value of plateau
pressure or the presence of a single plateau, the integrated DFT-SQS-ML framework
successfully identified 568 quinary alloy systems as ideal candidates for hydrogen stor-
age. The findings establish a solid foundation for experimental validation and further
advancements in the field of hydrogen storage materials.

**Keywords:** High entropy alloys, Body-Centered Cubic BCC, hydride, hydrogen storage, DFT calculation, SQS approach, thermodynamic, supervised machine learning.

**Introduction**

Hydrogen storage remains a critical challenge for advanced sustainable energy technologies, especially as the world strives to reduce the reliance on fossil fuels and combat climate change.¹ Traditional hydrogen storage methods present inherent limitations: compressed hydrogen gas is characterized by a low energy density, high safety risks due to high pressures, and significant costs associated with compression; whereas cryogenic liquid hydrogen, while offering higher energy density, faces challenges such as substantial energy requirements for liquefaction, safety concerns related to extremely low temperatures, and high operational costs related to the necessary infrastructure.² Alloys, including both solid solutions and intermetallic compounds, have opened new pathways for hydrogen storage by enabling the formation of reversible metal hydrides.³ Beyond their application in stationary hydrogen storage tanks, these materials have seen notable success in the Ni–Metal hydride battery industry.⁴,⁵ However, the conditions for hydride formation are heavily constrained by the system’s thermodynamics, which depend strongly on chemical composition. In intermetallics, this variation in composition typically falls within a limited range, even though atomic
substitutions have been widely used to adjust their properties.⁶,⁷ As a result, recent interest has shifted toward multi-component solid solutions, which offer more flexibility in tuning thermodynamic properties. In these systems, composition plays a key role in selecting alloys for specific technological applications,⁸ and requires clear guidelines for their design.

In response to these challenges, the development of advanced materials has become a key area of research. High-Entropy Alloys (HEAs) have emerged as particularly promising candidates, with a focus on those exhibiting Body-Centered Cubic (BCC) structures for hydrogen application.⁹ These HEAs are distinguished by their high configurational entropy and their complex multi-element nature, which provide several advantages over conventional storage materials, such as adjustable properties based on atomic composition.¹⁰ BCC-HEAs, in particular, are of significant interest due to their ability to stabilize a range of hydride phases that can efficiently store hydrogen.¹¹⁻¹³ The high configurational entropy inherent to HEAs helps stabilize these phases at different hydrogen-to-metal ratios, leading to hydrides with enhanced storage capacities. This property allows BCC-HEAs to accommodate hydrogen in multiple phases, such as monohydrides *MH* and dihydrides *MH*₂, thus increasing their overall hydrogen storage potential.¹⁴ Furthermore, BCC-HEAs often exhibit superior mechanical properties compared to traditional materials, including higher hardness and better resistance to wear and corrosion. These attributes are crucial for maintaining the integrity of the material under the pressures and conditions encountered in hydrogen storage systems. Furthermore, the diverse atomic environments within HEAs contribute to their enhanced resistance to degradation and embrittlement, which are common issues in conventional hydrogen storage materials.¹⁴

Overall, the unique combination of high configurational entropy, phase stability, and robust mechanical properties make BCC-HEAs a compelling choice for next-generation hydrogen storage applications, offering the potential for more efficient and reliable energy storage solutions. To overcome the challenge of exploring the extensive compositional and structural space of HEAs, a synergistic approach combining first-principles calculations with supervised
machine learning techniques has emerged as a powerful tool in materials discovery.¹⁵ By integrating Density Functional Theory (DFT) on disordered supercells, that could be designed by the Special Quasirandom Structure (SQS) method,¹⁶ researchers can efficiently explore the chemical space of multicomponents as quinary HEA systems. This methodology enables rapid screening of potential hydrogen storage materials, identifying promising candidates that might be missed through traditional methods.¹⁷

Computational materials science assisted by machine learning (ML) is increasingly recognized as a way to accelerating materials discovery. Recent studies have demonstrated the power of supervised-ML in predicting properties such as adsorption energies and thermodynamic stability of hydrogen storage materials, drastically reducing the computational effort needed compared to exhaustive DFT calculations. For instance, ML models have been employed to investigate hydrogen storage in metal-organic frameworks (MOFs) and high-entropy alloys (HEAs), with great success in improving efficiency and accuracy.¹⁸,¹⁹ In MOF studies, ML has helped predict adsorption isotherms and capacities, while in HEAs, it has enhanced the prediction of phase stability and hydrogen binding energy.²⁰,²¹ Our work aligns with these advancements by employing a hybrid approach that integrates DFT with ML techniques to target multicomponent cubic systems. Previous studies have largely focused on binary or ternary materials,²²,²³ but we extend this methodology to more complex quinary HEA systems, allowing us to navigate vast chemical spaces more effectively.

In contrast to previous supervised ML studies that relied on experimental datasets from several input,²⁴,²⁵ or Bayesian optimization techniques to fine-tune the chemical composition of a limited set of predefined systems,²⁶ the present work introduces an original approach. It utilizes a consistent, home-made combinatorial learning database, enabling broader exploration of non-targeted, non-oriented systems. In fact, we propose a complementary approach that combines massive DFT calculations on SQS-designed cells with ML supervision, aiming to accelerate the discovery of novel hydrogen storage materials while maintaining high accuracy. This dual framework addresses a key problem in current research: balancing
<figure><img src="image_2.png" /><figcaption>Figure 1: Phase transitions in BCC metals during hydrogen storage. Initially forming a monohydride $MH$ at $H/M = 1$, these metals transition to FCC dihydride $MH_2$ at $H/M = 2$. (a) Structures generated by the SQS Method, (b) schematic representation of PCI.</figcaption></figure>

computational cost with the need for detailed, accurate predictions in large, multi-element systems. Thus, in this work, we have developed a methodology using massive DFT-SQS-ML framework to build consistent learning-database to predict and to systematically explore a large chemical space, focusing on quinary HEA systems with cubic structures. The objectives are to accelerate the discovery of new hydrogen storage materials and gain insights into the mechanisms governing their hydrogen absorption properties. As illustrated in Figure 1, multi-component cubic centered alloys, which are a central focus of this research, demonstrate significant potential for hydrogen storage, typically achieving a hydrogen-to-metal ratio of $H/M=2$. These metals can initially form a monohydride at $H/M=1$ and then transition to a face-centered cubic (FCC) dihydride at $H/M=2$. The figure highlights these structural changes and their impact on hydrogen absorption, as evidenced by the Pressure-Composition Isotherms (PCI), underscoring the ongoing research into BCC HEAs as promising candidates for advanced hydrogen storage applications.²⁷,²⁸ In the present study, only the BCC phase is
considered as the hosting metallic structure, despite the known formation of other possible phases. For instance, many HEAs in different phases were examined for hydrogen storage applications in recent years. Alloys such as FeMnTiVZr typically form the C14 phase, and CrMnTiVZr or TiZrNbFeNi are more likely to form hexagonal phases.²⁹,³⁰ Other alloys, including TiZrCrMnFeNi,³¹ TiZrNbCrFe,³² TiV₂ZrCrMnFeNi,³³ TiV₁.₅ZrCr₀.₅MnFeNi,³⁴ and TiV₁.₅Zr₁.₅CrMnFeNi,³⁴ also demonstrate a strong tendency to store hydrogen efficiently. Recently, Edalati et al. developed TiₓZr₂₋ₓCrMnFeNi (x = 0.4-1.6) HEAs that adopt a C14 Laves phase and rank among the most efficient and best-designed and synthesized alloys, capable of reversibly storing up to 1.7 wt% hydrogen under ambient conditions with rapid kinetics.²⁹ Their study revealed that the C14 Laves phase structure achieved a H/M of approximately 1. However, for practical applications in hydrogen storage systems, materials with higher hydrogen capacities are generally more desirable.

For this study, we chose to investigate the BCC phase as the metallic hosting structure. This choice was motivated partly by computational resource limitations and partly by the challenges of systematically investigating more complex structures, such as those with multiple atomic sites in C14 phases for atom to being distributed, using the routine approaches as employed in this work. Additionally, the BCC phase serves as a good reference for hydrogen absorption properties due to its favorable structural characteristics as a hydrogen host, as explained earlier in the introduction. Even if other phases are thermodynamically more stable, the BCC phase still provides valuable guidance for candidate selection, and this approximation has minimal impact on our criteria.

In this article, we present a detailed exploration of the methodology and results. First, we describe the chemical space exploration, computational techniques, and machine learning approaches used in a Methods section. We then present and discuss our results, as the computational efficiency, machine learning outcomes, and the highlighting key systems identified as promising candidates for hydrogen storage at room temperature. Finally, we conclude with a summary of the findings and their implications.
# Methods

## Chemical space exploration

To design effective compositions for hydrogen storage, we propose to explore a large chemical space composed of 18 strategically selected elements: Al, Co, Cr, Cu, Fe, Hf, Mg, Mn, Mo, Nb, Ni, Pd, Ta, Ti, V, W, Zr, and Zn. These elements cover traditional atoms known to form BCC alloys, including early transition metals and some lightweight metals such as Al and Mg. The combinatorial approach to this selection involves evaluating numerous quinary systems:

$$ \binom{18}{5} = \frac{18!}{5!(18-5)!} = 8568. \qquad (1) $$

There are, therefore, a total of 8,568 unique quinary combinations, designed by $M$ as an equimolar system, to be evaluated. To fully explore the possible phase equilibrium, we decided to consider 4 distinct structures, as shown in Figure 2: $M$-BCC ($\alpha$) for alloys, $MH$-BCT ($\beta_1$) or $MH$-FCC ($\beta_2$) for monohydrides to identify the most stable $H/M=1$ phase among the two known ones with H on tetrahedral or octahedral site respectively, and $MH_2$-FCC ($\delta$) for dihydrides, which can absorb 2 H per $M$ atom on tetrahedral site. This combinatorics results in $8,568 \times 4 = 34,272$ equimolar configurations to be estimated, a substantial task which could not be done by first-principles calculations alone due to the computational cost.

## Computational details

First-principles calculations were performed using the Vienna Ab Initio Simulation Package (VASP),³⁵ implementing periodic DFT framework. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional³⁶ was employed, along with a plane wave basis set and a cut-off energy fixed at 400 eV.

The disordered $MH_x$ structures¹⁰ were generated as SQS cells¹⁶ utilizing the Alloy The-
<figure><img src="image_3.png" /><figcaption>Figure 2: Specific *MH*<sub>*x*</sub> structures considered in the analysis: *M*(*α* phase), *MH*(*β*<sub>1</sub> phase) or *MH*(*β*<sub>2</sub> phase), and *MH*<sub>2</sub>(*δ* phase).</figcaption></figure>

oretic Automated Toolkit (ATAT).³⁷ This method approximates the random distribution of elements in disordered alloys. Figure 1(a) illustrates examples of structures generated using the SQS approach. A high density k-meshing was considered ($k_i \approx 0.05 \times \frac{2\pi}{a_i}$) for the DFT calculation, which in our case required a minimum of 221 k-points. The convergence criterion for the self-consistency loop was set to $10^{-4}$ eV to ensure precise wave function optimization. During all relaxation processes, atomic positions, cell shape keeping the original symmetry, and cell volume were allowed to vary to accommodate small distortion effects in HEAs and structural transformations during hydrogen accommodation in BCC metals.³⁸ While the volumes of cubic-derived structures are simply relaxed with isotropic dilation, the PtS cell shape requires additional adjustment to account for the *c/a* ratio, which is not preserved in the triclinic SQS cell with non-orthogonal basis vectors. In our methodology, this was handled by applying a dilation matrix corresponding to the respective basis vectors, ensuring that the original cell shape was respected throughout the relaxation process.

Using the total energy *E*<sub>tot</sub> obtained after all the relaxations, we determined the enthalpy of formation (*Δ*<sub>*f*</sub>*H*) for each system to assess the stability of different structures. The enthalpy of formation is defined as the energy difference between the energy of a compound and the sum of the energies of its constituent pure elements, normalized by the total number
of metallic atoms in the considered cell:

$$ \Delta_f H = \frac{1}{n} \left( E_{\text{tot}} - \sum_i n_i E_i \right) \qquad (2) $$

where $n$ is the total number of metallic atoms, $E_{\text{tot}}$ is the total energy of the compound, $n_i$ is the number of atoms of element $i$, and $E_i$ is the energy of pure element $i$. The zero-point energy (ZPE) contribution is known to be non-negligible in hydrides,³⁹ however, such estimations typically require phonon harmonic calculations, which are computationally intensive and not easily applicable to the randomized supercell approach employed in this study.

## Machine Learning

For the supervised machine learning (ML) modeling, all possible binary combinations (153 systems) and a random dataset of 440 quinary systems (selected to ensure a homogeneous distribution of elements) were considered to construct a consistent learning database. This database used the same descriptors and identical calculation parameters (e.g., convergence criteria) for each of the four $MH_x$ phases. Ternary and quaternary configurations were excluded from the training database, as their inclusion was shown to have no significant impact on improving predictions for other complex compounds.⁴⁰ This resulting homemade database (all binary combinations with some quinaries) with homogeneous distribution of elements serves as a robust foundation for our supervised learning and the prediction of HEA properties. The integration of the DFT and SQS method within the dataset made under the same conditions ensures high reliability and enhances the predictive capability of the ML models in assessing the stability and properties of HEAs.

Using supervised ML modeling, we aimed to predict the enthalpy of formation ($\Delta_f H$). To ensure a comprehensive representation of the dataset, we employed the One-Hot Encoding (OHE) method,⁴¹ which allows us to encode elemental composition as categorical features as a one-hot numeric array, with the enthalpy of formation as the target value of the learning.
This approach enhances the model's ability to learn the underlying relationships between the elemental composition and the stability of the alloys.⁴² In fact, we have performed numerous tests by using additional physical descriptors in conjunction with the composition modeled by the OHE alone, but the prediction results have not shown any significant improvement. It seems that our DFT computed database is robust enough to include the classical features (average or deviation of volume, valence electron numbers...) in the electronic structure calculation, that the learning models generally need to include.⁴³

We employed the Scikit-Learn library in Python⁴⁴ to design and optimize Multilayer Perceptron (MLP) neural networks for predicting $\Delta_f H$. The MLP models were optimized using a comprehensive grid search method to identify the best hyperparameters for regression learning tasks. This approach allows the exploration of a wide range of hyperparameter combinations to enhance the model's predictive performance. To assess the performance of the MLP models, we used two key metrics: the coefficient of determination ($R^2$) and the root-mean-square error (RMSE). These metrics provide insights into the accuracy and precision of the regression models. The coefficient of determination, $R^2$, indicates the proportion of the variance in the dependent variable that is predictable from the independent variables and is defined as follows:

$$R^2 = 1 - \frac{\sum_i (y_i - \hat{y}_i)^2}{\sum_i (y_i - \bar{y})^2}, \qquad (3)$$

where the $y_i$ represent the observed values ($\Delta_f H$), $\hat{y}_i$ represent the predicted values, and $\bar{y}$ is the mean of the observed values. The root-mean-square error (RMSE) provides a measure of the average magnitude of the errors between predicted and observed values and is defined as:

$$RMSE = \sqrt{\frac{1}{m} \sum_{i=1}^{m} (y_i - \hat{y}_i)^2}, \qquad (4)$$

where $m$ denotes the total number of observations. A lower RMSE indicates a better fit of the model to the data.

To ensure the robustness of our MLP models, we implemented five-fold Cross-Validation
(CV). The dataset was randomly partitioned into five equally sized folds. Each fold was used once as a validation set while the remaining four folds served as the training set. This process was repeated five times, allowing each fold to be used for validation exactly once.⁴⁵ CV helps in assessing the model’s performance on unseen data and mitigates the risk of overfitting.

Upon validating our model, we extended its application to predict the enthalpy of formation for the 8128 remaining quinary systems using the optimized hyperparameters from CV. Notably, this was achieved using only 5% of all systems as the training set (440). This highlights our MLP model’s efficiency and ability to generalize from limited data. Predicting many systems with minimal input is especially valuable where DFT calculations are costly, resource-intensive, and time-consuming.⁴⁶ While the model in this work focuses on H/M=1.0 and particularly 2.0 to identify optimal compositions with maximum hydrogen capacity, it is not inherently restricted to these specific capacities. The approach can be extended to other hydrogen loadings, such as H/M=0.5 or 1.5, by retraining or fine-tuning the model with appropriate datasets including the corresponding hydride phases.

# Results and discussion

## Computational efficiency and accuracy

Our initial tests focused on optimizing the box sizes for different crystal structures using the SQS method. For BCC alloys, we evaluated various box sizes (number of atoms N) and compared their Root Mean Square (RMS) deviations in correlation functions ηᵢ, their calculated energies (by DFT), and the associated computing times. The RMSᵢ(N) is computed by comparing the correlation function of the best generated SQS cell to that of the ideal random structure:

$$ \text{RMS}_{\eta}(N) = \sqrt{\sum_{k} \left( \eta_{k}^{\text{SQS}} - \eta_{k}^{\text{random}} \right)^{2}}, \qquad (5) $$
where *k* represents the set of clusters, which includes up to the 6th nearest-neighbor pairs, the 4th-order triplets and the 2nd-order quadruplets in the case of BCC alloy structure. Further details are provided in the Supplementary Information (SI-S3-Figures S2 and S3). As an example shown in Figure 3 for BCC alloys, the results indicated that since the DFT energies were very close across different box sizes, the 40-atom box size provided a suitable compromise between accuracy and computational efficiency. The computational time required for larger box sizes was disproportionately high (several hours for 40-atom cell versus several days for 140-atom cell), making the 40-atom size an optimal choice for BCC alloys. This study was extended to monohydrides *MH* and dihydrides *MH*₂, where box sizes of 80 atoms and 120 atoms, respectively, were selected to ensure adequate representation of the systems (see SI S1-2 and S1-3 respectively). These optimizations are crucial given the extensive number of configurations analyzed, ensuring that the results are both reliable and computationally feasible.

<figure><img src="image_4.png" /><figcaption>Figure 3: Comparison of RMS<sub>η</sub>(N) for BCC alloys as a function of SQS-supercell size *N*, using HfNbTiVZr as an example system for the DFT energy calculations.</figcaption></figure>
Using these given SQS supercells, high-throughput DFT calculations were performed for 440 different systems in the 4 phases (α *M*-BCC, *β*₁ *MH*-PtS, *β*₂ *MH*-NaCl, *δ* *MH*₂-*CaF*₂) to estimate the enthalpy of formation, utilizing the same set of DFT parameters to ensure uniformity and comparability across all results in order to build the database. These 440 systems provide a consistent and uniform distribution of the 18 carefully selected elements, ensuring balanced and optimal performance across the entire configuration. As an example from the database, Figure 4 presents the 440 calculated dihydride systems, illustrating the wide dispersion of enthalpy of formation resulting from the DFT calculations.

<figure><img src="image_5.png" /><figcaption>Figure 4: Distribution of calculated enthalpy of formation (kJ per mol of metal *M*) in quinary mixing in CaF₂-type dihydrides *MH*₂.</figcaption></figure>
## Machine Learning results

Neural networks were employed to predict the enthalpy of formation. The model's hyperparameters were meticulously optimized using GridSearchCV, resulting in the selection of an optimal configuration with a hidden layer size of (100, 100), an λ2 regularization of 0.01, a ReLU activation function, and a constant learning rate.

For each individual phase ($α, β, δ$), a total of 440 quinary compositions, supplemented with all possible binary systems, were used to explore the potential for predicting new combinations. This dataset was employed to predict the enthalpy of formation for $α M$-BCC, $β_2 MH$-NaCl and $δ MH_2$-CaF$_2$ phases. Specifically, the training set comprised 100% of the binary data and 90% of the quinary data, while the remaining 10% of the quinary data was reserved for testing.

The MLP model demonstrated strong predictive performance, achieving a root mean square error (RMSE) of 2.47 kJ/mol($M$) when predicting the enthalpy of formation for BCC alloys, as illustrated in Figure 5a, and 3.9 kJ/mol($M$) for CaF$_2$ quinary systems as indicated in Figure 5c. The prediction for the monohydride in the NaCl structure was conducted using a database comprising all possible binary systems and 140 quinary systems. Applying the same previously described model, we achieved an RMSE of 2.5 kJ/mol($M$) as illustrated in Figure 5b. For the $β_1 MH$-PtS monohydride, we used only the binary database to predict the entire quinary systems, as it was difficult to properly relax quinary systems by DFT while maintaining tetragonal symmetry. We chose this approach after successfully testing it on $β_2 MH$-NaCl monohydrides. Specifically, we attempted to predict quinary systems in the NaCl structure based on binary data only and achieved an RMSE of 5.33 kJ/mol($M$), as described in SI-S7-Figure S8.
<figure><img src="image_6.png" /><figcaption>(a) Alloys in α *M*-BCC structure.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(b) Monohydrides in β₂ *MH*-NaCl structure.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(c) Dihydrides in *δ MH*₂-*CaF*₂ structure.</figcaption></figure>

Figure 5: Enthalpy of formation (kJ per mol of metal *M*) prediction for (a) Alloys in α *M*-BCC structure, (b) Monohydrides in β₂ *MH*-NaCl structure and (c) Dihydrides in *δ MH*₂-*CaF*₂ structure.
These results enabled the prediction of the enthalpy of formation of all possible equimolar quinary systems through supervised machine learning, allowing for the selection of the most stable monohydride for each system among $β_1$ and $β_2$ phases, taken as $Δ_fH(MH) = min(Δ_fH_{β_1}, Δ_fH_{β_2})$. Figure 6 displays the enthalpy of reaction for dihydrides across 8,568 systems, plotted against that of the most stable monohydride for each system. The enthalpy of reaction of the monohydride is quantified by the difference between its enthalpy of formation and that of the alloy:

$$ \Delta_r H^{\text{mono}} = \Delta_f H_{\beta} - \Delta_f H_{\alpha}. \qquad (6) $$

Similarly, the enthalpy of reaction of the dihydride is determined by the difference between the enthalpy of formation of the dihydride and that of the monohydride, using the following equation:

$$ \Delta_r H^{\text{di}} = \Delta_f H_{\delta} - \Delta_f H_{\beta}. \qquad (7) $$

<figure><img src="image_9.png" /><figcaption>Figure 6: Predicted and calculated enthalpy of reaction (kJ per mol of metal M) of dihydride $MH_2$ ($δ$ phase) VS. that of the most stable monohydride $MH$ (among $β_1$ and $β_2$).</figcaption></figure>

As shown in Figure 6, the analysis revealed that a substantial portion of the predicted
systems are unstable, as indicated by their positive enthalpy of reaction. These unstable
systems are thermodynamically unfavorable for hydrogen absorption, underscoring the need
for a careful selection process to identify stable systems capable of effectively absorbing
hydrogen. It is clearly shown for the α-alloys where the stabilization of BCC HEAs with
positive enthalpy of formation can often be attributed to the configurational entropy of
mixing at high temperature. This has been already discussed for example in the well-
known Cantor alloy (FCC CoCrFeMnNi) with its positive enthalpy of formation, indicating
a thermodynamic tendency toward decomposition into pure elements at low temperatures.⁴⁷
In our work, the identification of such stable systems is crucial for advancing the development
of materials with optimal hydrogen storage capabilities, ensuring both efficiency and safety
in practical applications.

Identification of promising systems for room temperature hydrogen
storage

The process of selecting optimal hydrogen storage systems requires a careful and strategic approach, centered on several key criteria. As an example, a more negative enthalpy of reaction typically correlates with higher stability and a greater capacity for hydrogen absorption,⁴⁸ making these systems more effective in real-world applications. Our objective is to identify systems that not only demonstrate the necessary thermodynamic stability but also offer the highest potential for efficient hydrogen absorption.

Key selection criteria

The first step in this selection process is to eliminate any systems with a positive enthalpy of
reaction, as such values indicate thermodynamic instability, making the system unsuitable
for hydrogen storage.⁴⁹ By excluding these systems, we focus exclusively on those that are
inherently stable toward pure elements decomposition.

Next, we refine our selection by focusing on the suitability applications. Hydrogen storage
<figure><img src="image_10.png" /><figcaption>Figure 7: Predicted enthalpy of reaction (kJ per mol of metal M) of dihydride MH₂ VS that of the monohydride MH, with the key selection criteria highlighted. Filtered, highlighted in red, corresponds to cases where the dihydride's enthalpy of reaction closely matches that of the monohydride.</figcaption></figure>

systems meet specific thermodynamic criteria lead by van 't Hoff relation:

$$ \frac{1}{2} \ln \left( \frac{P_{\text{plateau}}}{P_{\circ}} \right) = \frac{\Delta H}{RT} - \frac{\Delta S}{R} \qquad (8) $$

In Eq. (8), $P_{\text{plateau}}$ and $P_0$ represent the equilibrium pressure of absorption and atmospheric pressure, respectively. $\Delta H$ and $\Delta S$ represent the enthalpy and entropy of hydrogen absorption, respectively, while $R$ and $T$ represent the gas constant and the temperature. For room temperature applications, the ambient conditions, as $P_{\text{plateau}} \approx 1$ atm, give that the entropy change for hydrogen absorption in metals is approximately $\Delta S \approx -130 \text{ J/K}\cdot\text{mol(H}_2)$. In fact, the van 't Hoff diagrams for BCC HEAs in the literature show that $\Delta S$ is around -89 J/K\cdotmol(H}_2) during the first stage of absorption and can be as low as -172 J/K\cdotmol(H}_2) during the second stage of absorption.²⁶,⁵⁰ This suggests that selected systems would require enthalpy of reaction no greater than -25 kJ/mol(M), as highlighted by the green area in Figure 7. However, to account for uncertainties in predictive modeling, we further widen our
selection to systems with enthalpy of reaction below -20 kJ/mol(M). This ensures greater
confidence in their ability to operate efficiently at room temperature, providing optimal
performance in practical applications.

Additionally, we identify systems where both the dihydride and the monohydride exhibit closely matched enthalpy of reaction, forming a continuous plateau for hydrogen absorption.⁵¹ These single plateau systems are crucial for practical applications, as they allow for reliable and controlled hydrogen release and larger width of reversible capacity. In our analysis, systems where the enthalpy of reaction of the dihydride closely matches that of the monohydride are highlighted in red in Figure 7. This visual guide underscores a clear indication of candidates with single plateau.

<figure><img src="image_11.png" /><figcaption>Figure 8: Selected systems for application under ambient conditions: zoomed view of the selected systems extracted from Figure 7. Enthalpy of reaction is expressed in kJ per mol of metal *M*.</figcaption></figure>

As illustrated in Figure 8, our comprehensive analysis has led to the identification of 568 promising systems for hydrogen storage, with the full list provided in the supplementary materials. Among these, a substantial subset of 401 systems, highlighted in blue, exhibits a dual-plateau behavior during hydrogenation. This dual-plateau response likely indicates the formation of distinct monohydride and dihydride phases, suggesting the potential for
hydrogen absorption at room temperature, a critical feature for practical applications.

Additionally, 167 systems, highlighted in red, display a single plateau, corresponding to the simultaneous formation of both monohydrides and dihydrides. These systems show closely similar enthalpy of reaction, which could play a crucial role in their performance, as it directly influences the full reversible capacity and the energy efficiency of hydrogen absorption and release processes on a large and unique plateau.

Moreover, 44 systems, highlighted in orange, (9 systems in orange and red and 35 in orange and blue) are particularly notable for their ability to form stable BCC alloys. These alloys exhibit negative enthalpy of formation, i.e. stable toward decomposition into pure elements, a key indicator of thermodynamic stability.⁵² The negative enthalpy is a property that ensures the material can efficiently undergo multiple hydrogen absorption and desorption cycles without significant degradation.⁵³ While our current methodologies do not allow us to definitively confirm the stability of these systems, the negative formation enthalpy of these BCC alloys serves as a strong indication that warrants further investigation. The identification of these systems with favorable thermodynamic properties highlights the potential of BCC alloys as promising candidates for optimizing hydrogen storage technologies. It is a common assumption that systems lacking alloys with negative enthalpy of formation are inherently unstable. However, this is not always the case. Several systems have been identified as promising through experimental work, even when the BCC phase is observed among those with positive enthalpy of formation. For instance, recent studies have demonstrated that metastable phases with positive enthalpy can still exhibit stability due to kinetic factors or specific temperature conditions.⁵⁴ Moreover, experimental results have highlighted that certain high-entropy alloys (HEAs) exhibit the BCC phase despite a positive enthalpy of formation, emphasizing that thermodynamic stability is not always straightforward.⁵⁵

Table 1 provides a selection of promising systems that have been previously studied experimentally. It presents data on the predicted enthalpy of reaction for both the monohydride ($\Delta_r H^{\text{mono}}$) and dihydride ($\Delta_r H^{\text{di}}$) phases, along with references to the articles where these
systems were investigated. These four systems form alloys with positive enthalpy, and the BCC phase of the alloys was identified in the studies. In all four cases the monohydride forms a BCT structure (PtS). The identification of systems previously recognized as promising in experimental studies, along with the formation of a PtS structure in the monohydrides, aligns with earlier findings. This consistency reinforces the accuracy of our predicted enthalpy of reaction and validates our approach, as it agrees with experimental data reported in the literature.

<table><caption>Table 1: Examples of systems studied in previous articles, with the predicted enthalpy of reaction (kJ per mol of metal *M*) from this work.</caption><thead><tr><th>System</th><th>Δ<sub>r</sub>H<sup>mono</sup> (kJ/mol(M))</th><th>Δ<sub>r</sub>H<sup>di</sup> (kJ/mol(M))</th><th>References</th></tr></thead><tbody><tr><td>HfMoNbTiZr</td><td>-50.82</td><td>-42.77</td><td>11,56</td></tr><tr><td>HfNbTiVZr</td><td>-62.32</td><td>-35.34</td><td>11,26,57</td></tr><tr><td>HfNbTaTiZr</td><td>-62.59</td><td>-40.87</td><td>11,58</td></tr><tr><td>MoNbTiVZr</td><td>-39.51</td><td>-39.84</td><td>11,59</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>Figure 9: Occurrence of the 18 selected atoms in the 568 promising hydrogen storage systems.</figcaption></figure>

The statistical evaluation of the 568 promising quinary hydrogen storage systems reveals a significant prevalence of six key elements: Zr, Ti, Hf, Nb, Mg, and V, which occur most frequently as shown in Figure 9. Zr is present in nearly 360 systems, followed by Ti in 350, Hf
in 340, Nb in 260, Mg in 250, and V in 220. These elements are well-known in the literature
for their favorable interactions with hydrogen and their crucial role in hydrogen storage
technologies. Zr and Ti, for instance, are well-documented for their ability to form stable
hydrides, which enhances the efficiency of hydrogen storage under ambient conditions.⁶⁰ The
strong affinity of Zr for hydrogen and its thermodynamic stability in hydride form has been a
focus of research for improving reversible hydrogen storage. Similarly, Ti-based alloys, often
praised for their lightweight properties, are effective in enhancing hydrogen diffusion and
absorption kinetics, making them central in the development of practical storage materials.
Hf, a close structural analog to Zr, shares similar hydrogenation properties and contributes to
the formation of thermodynamically stable hydrides, further supporting its high occurrence
in these systems.⁶¹ Nb, widely recognized for its high hydrogen solubility and the formation of
multiple hydride phases, improves hydrogen kinetics and enhances the performance of multi-
component alloys.⁶² This element has been extensively studied for its ability to increase
hydrogen diffusion in alloys, making it a crucial component in high-performance hydrogen
storage systems.⁶³ Mg, known for its high gravimetric hydrogen capacity,⁶⁴ is frequently
explored in hydrogen storage materials due to its ability to absorb large amounts of hydrogen
per unit mass, though its sluggish kinetics often require enhancement via alloying with
elements like Ti or Zr.⁶⁵ Finally, V is important for its ability to form BCC alloys, which
are highly effective for hydrogen diffusion.⁶⁶ The presence of V in these systems supports its
utility in improving the reversibility of hydrogen absorption and desorption processes.
<figure><img src="image_13.png" /><figcaption>Figure 10: Distribution of the (a) atomic radius and (b) average valence electrons count in the selected promising hydrogen storage systems.</figcaption></figure>

In addition to elemental occurrence, an analysis of the average atomic radius and the valence electron distribution across the selected 568 systems provides further insight into their potential for hydrogen storage as shown in Figure 10. The average atomic radius of 1.823 Å is conducive to efficient hydrogen diffusion, as it suggests an optimal balance between atomic packing density and sufficient interstitial space for hydrogen atoms to migrate within the lattice.⁶⁷ Furthermore, the mean Valence Electron Count (VEC) of these promising systems satisfies VEC<6.87, an empirical criterion to form a stable BCC alloy according to Guo *et al.*,⁶⁸ which are known for their ability to form stable hydrides due to favorable metal-hydrogen bonding.⁶⁹ Transition metals such as Ti, Zr, and V, which dominate the top six elements, are particularly effective in enhancing hydrogen absorption capacity and ensuring reversibility during hydrogenation cycles.⁷⁰ This balance between atomic radius and valence electron count is critical in optimizing the efficiency and the reversibility of hydrogen storage systems, further highlighting the potential of these 568 systems for real-world applications.
<figure><img src="image_14.png" /><figcaption>Figure 11: Screening method summary for identifying promising hydrogen storage systems among over 34,000 quinary configurations from 18 elements.</figcaption></figure>

A comprehensive list of all selected systems, including detailed information on each system's stable monohydride structure and its enthalpy of reaction, as well as the corresponding enthalpy of reaction for both the alloy and the dihydride, are available in the supplementary information.

This research employs first-principles calculations and machine learning to discover new equimolar systems for hydrogen storage, as summarized in Figure 11. It is based on an idealized scenario where the elements are assumed to be randomly distributed. This assumption allows us to perform a systematic and computationally efficient estimation of thermodynamic properties, which serves as a baseline for understanding the broader behavior of HEAs. Incorporating short-range ordering (SRO) effects can create distinct local chemical environments that may significantly alter the hydrogen storage behavior but they are not considered in this work since it needs additional computational resources and methodological developments to capture these contributions accurately. Within this approximation, our findings contribute to the development of advanced hydrogen storage materials and streamline the number of systems requiring experimental validation. We present a theoretical list awaiting experimen-
tal confirmation, and it would be beneficial to further refine the accuracy of our machine
learning models and reduce prediction errors. It is worth noting that this work is part of
a larger national program (PEPR-H2-SOLHYD), where experimental collaborators are ac-
tively verifying our predictions. Additional experimental validation will be incorporated into
future studies as more results become available.

Conclusion and outlooks

This study presents a comprehensive investigation into the chemical space of multi-component
cubic alloys for potential hydrogen storage applications. By combining first-principles DFT
calculations with advanced machine learning techniques, we efficiently screened 32,000 alloy
configurations across four phases, identifying the most promising candidate systems among
18 selected elements. From this extensive exploration, the training database included all
binary configurations and 440 randomly selected quinary configurations, achieving a predic-
tion accuracy of 3 kJ/mol(M). Our efforts led to the identification of 568 promising quinary
equimolar compositions, selected based on criteria for ambient condition applications, which
merit further experimental validation. We strongly encourage experimental testing of these
candidate alloys, as such validation is critical to confirming their real-world effectiveness and
viability.

Looking ahead, our future work will focus on refining computational models to further
improve prediction accuracy, ensuring our findings remain reliable and robust. Only BCC is
considered as a hosting structure. As a future perspective, we plan to extend our method-
ology to consider other phases, such as Laves phases, by integrating high-throughput ap-
proaches with active learning. Intermediate hydrides such as H/M=0.5 or 1.5 could be also
considered as additional compounds for a better modeling of the phase stability investigation
and will provide deeper insights into the hydrogen storage behavior of the investigated sys-
tems. These advancements will be addressed in subsequent work. While this study focused
on equimolar compositions, we acknowledge that this may not encompass the full potential
of the chemical space for hydrogen storage materials. Future research should explore non-
equimolar and more complex alloy compositions to uncover novel materials with superior
performance. Expanding into these less-explored areas will allow us to discover even more
efficient and effective hydrogen storage solutions, driving progress towards sustainable energy
technologies.

Acknowledgement

This work is a part of project PEPR-H2-SOLHYD (ref. ANR-22-PEHY-0007) funded and supported by the French National Research Agency (ANR) under the France 2030 program. DFT calculations were performed using HPC resources from GENCI-CINES (Grant A0060906175). ZN thanks Sébastien Junier for his help generating the SQS supercell of the monohydride in the NaCl structure.

**Supporting Information Available**

The following supplementary files are available:

1. SQS parameters: Supp_materials.pdf

2. Clarification on ML predictions: Supp_materials.pdf

3. List of the promising systems: Promising_systems.xlsx

The calculated database used for the predictions will be made available upon request.

**References**

(1) Ball, M.; Wietschel, M. The future of hydrogen – opportunities and challenges. *Int. J. Hydrogen Energy* **2009**, *34*, 615–627.
(2) Sakintuna, B.; Lamari-Darkrim, F.; Hirscher, M. Metal hydride materials for solid hydrogen storage: A review. *Int. J. Hydrogen Energy* **2007**, *32*, 1121–1140.

(3) Mohtadi, R.; Orimo, S.-i. The renaissance of hydrides as energy materials. *Nat. Rev. Mater* **2016**, *2*, 16091.

(4) Cuevas, F.; Joubert, J.-M.; Latroche, M.; A. Percheron-Guégan Intermetallic compounds as negative electrodes of Ni/MH batteries. *Appl. Phys. A* **2001**, *72*, 225–238.

(5) Hirscher, M.; Yartys, V. A.; Baricco, M.; Bellosta Von Colbe, J.; Blanchard, D.; Bowman, R. C.; Broom, D. P.; Buckley, C. E.; Chang, F.; Chen, P.; Cho, Y. W.; Crivello, J.-C.; Cuevas, F.; David, W. I.; De Jongh, P. E.; Denys, R. V.; Dornheim, M.; Felderhoff, M.; Filinchuk, Y.; Froudakis, G. E.; Grant, D. M.; Gray, E. M.; Hauback, B. C.; He, T.; Humphries, T. D.; Jensen, T. R.; Kim, S.; Kojima, Y.; Latroche, M.; Li, H.-W.; Lototskyy, M. V.; Makepeace, J. W.; Møller, K. T.; Naheed, L.; Ngene, P.; Noréus, D.; Nygård, M. M.; Orimo, S.-i.; Paskevicius, M.; Pasquini, L.; Ravnsbæk, D. B.; Veronica Sofianos, M.; Udovic, T. J.; Vegge, T.; Walker, G. S.; Webb, C. J.; Weidenthaler, C.; Zlotea, C. Materials for hydrogen-based energy storage – past, recent progress and future outlook. *J. Alloys Compd.* **2020**, *827*, 153548.

(6) Crivello, J.-C.; Zhang, J.; Latroche, M. Structural Stability of AB<sub>y</sub> Phases in the (La,Mg)-Ni System Obtained by Density Functional Theory Calculations. *J. Phys. Chem. C* **2011**, *115*, 25470–25478.

(7) Liu, W.; Webb, C.; Gray, E. Review of hydrogen storage in AB<sub>3</sub> alloys targeting stationary fuel cell applications. *Int. J. Hydrogen Energy* **2016**, *41*, 3485–3507.

(8) Dematteis, E. M.; Berti, N.; Cuevas, F.; Latroche, M.; Baricco, M. Substitutional effects in TiFe for hydrogen storage: a comprehensive review. *Mater. Adv.* **2021**, *2*, 2524–2560, Publisher: RSC.
(9) Zlotea, C.; Bouzidi, A.; Montero, J.; Ek, G.; Sahlberg, M. Compositional effects on the hydrogen storage properties in a series of refractory high entropy alloys. *Front. Energy Res. 2022*, *10*, 991447.

(10) Miracle, D. B.; Senkov, O. N. A critical review of high entropy alloys and related concepts. *Acta Mater.* **2017**, *122*, 448–511.

(11) Marques, F.; Balcerzak, M.; Winkelmann, F.; Zepon, G.; Felderhoff, M. Review and outlook on high-entropy alloys for hydrogen storage. *Energy Environ. Sci.* **2021**, *14*, 5191–5227.

(12) Shahi, R. R.; Gupta, A. K.; Kumari, P. Perspectives of high entropy alloys as hydrogen storage materials. *Int. J. Hydrogen Energy* **2023**, *48*, 21412–21428.

(13) Luo, L.; Chen, L.; Li, L.; Liu, S.; Li, Y.; Li, C.; Li, L.; Cui, J.; Li, Y. High-entropy alloys for solid hydrogen storage: a review. *Int. J. Hydrogen Energy* **2024**, *50*, 406–430.

(14) Li, Z.; Pradeep, K. G.; Deng, Y.; Raabe, D.; Tasan, C. C. Metastable high-entropy dual-phase alloys overcome the strength-ductility trade-off. *Nature* **2016**, *534*, 227–230.

(15) Butler, K. T.; Davies, D. W.; Cartwright, H.; Isayev, O.; Walsh, A. Machine learning for molecular and materials science. *Nature* **2018**, *559*, 547–555.

(16) Zunger, A.; Wei, S.-H.; Ferreira, L. G.; Bernard, J. E. Special quasirandom structures. *Phys. Rev. Lett.* **1990**, *65*, 353–356.

(17) Liu, Z.-K. First-principles calculations and CALPHAD modeling of thermodynamics. *J. Phase Equilib. Diffus.* **2015**, *36*, 141–152.

(18) Ahmad, M.; Park, J. High-throughput computational search for hydrogen storage materials in complex alloys using machine learning. *J. Phys. Chem. C* **2021**, *125*, 6511–6521.
(19) Zhang, L.; Hu, M. Applications of machine learning in the prediction of hydrogen storage materials properties. *Mater. Today* **2020**, *39*, 70–83.

(20) Park, Y.; Hu, Q. Machine learning models for predicting adsorption properties in hy-
drogen storage materials. *J. Mater. Chem. A* **2020**, *8*, 18930–18940.

(21) Kim, H.; Lee, S.; Noh, S. Machine learning-driven design of high-entropy alloys for hydrogen storage. *Acta Mater.* **2021**, *204*, 116500.

(22) Jha, D.; Ward, L.; Paul, A.; Wolverton, C. ElemNet: Deep learning the Chem. Mat. from only elemental composition. *Sci. Rep.* **2018**, *8*, 17593.

(23) Glawe, H.; Sanna, A.; Gross, E.; Marques, M. A. L. Optimal training sets for machine learning in materials science. *New J. Phys.* **2016**, *18*, 093011.

(24) Witman, M.; Ling, S.; Grant, D. M.; Walker, G. S.; Agarwal, S.; Stavila, V.; Al-
lendorf, M. D. Extracting an Empirical Intermetallic Hydride Design Principle from
Limited Data via Interpretable Machine Learning. *J. Phys. Chem. Lett.* **2020**, *11*, 40–
47.

(25) Witman, M.; Ek, G.; Ling, S.; Chames, J.; Agarwal, S.; Wong, J.; Allendorf, M. D.; Sahlberg, M.; Stavila, V. Data-Driven Discovery and Synthesis of High Entropy Alloy Hydrides with Targeted Thermodynamic Stability. *Chem. Mat.* **2021**, *33*, 4067–4076.

(26) Halpren, E.; Yao, X.; Chen, Z. W.; Singh, C. V. Machine learning assisted design of BCC high entropy alloys for room temperature hydrogen storage. *Acta Mater.* **2024**, *270*, 119841.

(27) Akiba, E.; Iba, H. Hydrogen absorption by Laves phase related BCC solid solution.
*Intermetallics* **1998**, *6*, 461–470.

(28) Yu, X.; Wu, Z.; Xia, B.; Xu, N. Enhancement of hydrogen storage capacity of Ti–V–Cr–Mn BCC phase alloys. *J. Alloys Compd.* **2004**, *372*, 272–277.
(29) Mohammadi, A.; Ikeda, Y.; Edalati, P.; Mito, M.; Grabowski, B.; Li, H.-W.; Edalati, K. High-entropy hydrides for fast and reversible hydrogen storage at room temperature: Binding-energy engineering via first-principles calculations and experiments. *Acta Mater.* **2022**, *236*, 118117.

(30) Dangwal, S.; Ikeda, Y.; Grabowski, B.; Edalati, K. Machine learning to explore high-entropy alloys with desired enthalpy for room-temperature hydrogen storage: Prediction of density functional theory and experimental data. *Chem. Eng. J.* **2024**, 152606.

(31) Edalati, P.; Floriano, R.; Mohammadi, A.; Li, Y.; Zepon, G.; Li, H.-W.; Edalati, K. Reversible room temperature hydrogen storage in high-entropy alloy TiZrCrMnFeNi. *Scripta Mater.* **2020**, *178*, 387–390.

(32) Floriano, R.; Zepon, G.; Edalati, K.; Fontana, G. L.; Mohammadi, A.; Ma, Z.; Li, H.-W.; Contieri, R. J. Hydrogen storage properties of new A3B2-type TiZrNbCrFe high-entropy alloy. *Int. J. Hydrogen Energy* **2021**, *46*, 23757–23766.

(33) Dangwal, S.; Edalati, K. High-entropy alloy TiV2ZrCrMnFeNi for hydrogen storage at room temperature with full reversibility and good activation. *Scripta Mater.* **2024**, *238*, 115774.

(34) Dangwal, S.; Edalati, K. Significance of interphase boundaries on activation of high-entropy alloys for room-temperature hydrogen storage. *Int. J. Hydrogen Energy* **2024**, *50*, 626–636.

(35) Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. *Phys. Rev. B* **1996**, *54*, 11169–11186.

(36) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. *Phy. Rev. Lett.* **1996**, *77*, 3865–3868.
(37) Van De Walle, A.; Tiwary, P.; De Jong, M.; Olmsted, D.; Asta, M.; Dick, A.; Shin, D.; Wang, Y.; Chen, L.-Q.; Liu, Z.-K. Efficient stochastic generation of special quasirandom structures. *Calphad* **2013**, *42*, 13–18.

(38) He, T.; Yao, X. Recent advances in the hydrogen storage properties of metal hydrides and their applications for hydrogen compression. *Energy Storage Mater.* **2013**, *2*, 21–36.

(39) Bourgeois, N.; Crivello, J.-C.; Cenedese, P.; Paul-Boncour, V.; Joubert, J.-M. Vibration analysis of hydrogen, deuterium and tritium in metals: consequences on the isotope effect. *J. Condens. Matter Phys.* **2018**, *30*, 335402.

(40) Crivello, J.-C.; Joubert, J.-M.; Sokolovska, N. Supervised deep learning prediction of the formation enthalpy of complex phases using a DFT database: The σ- phase as an example. *Comput. Mater. Sci.* **2022**, *201*, 110864.

(41) Rodríguez, P.; Bautista, M. A.; Gonzalez, J.; Escalera, S. Beyond one-hot encoding: Lower dimensional target embedding. *Image Vis. Comput.* **2018**, *75*, 21–31.

(42) Xie, T.; Grossman, J. C. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. *Phys. Rev. Lett.* **2018**, *120*, 145301.

(43) Li, Z.; Li, S.; Birbilis, N. A machine learning-driven framework for the property prediction and generative design of multiple principal element alloys. *Mater. Today Commun.* **2024**, *38*, 107940.

(44) Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; others Scikit-learn: Machine learning in Python. *J. Mach. Learn.* **2011**, *12*, 2825–2830.

(45) Pickering, E. J.; Jones, N. G. High-entropy alloys: a critical assessment of their founding principles and future prospects. *Inter. Mater. Rev.* **2016**, *61*, 183–202.
(46) Luo, J.; Smith, R. I.; Huot, J.; Barnes, P.; Grant, D. M. Novel high-entropy alloy as a potential hydrogen storage material. *Scripta Mater.* **2018**, *153*, 104–108.

(47) Bracq, G.; Crivello, J.-C.; Laurent-Brocq, M.; Joubert, J.-M.; Guillot, I. What is the Enthalpy Contribution to the Stabilization of the Co–Cr–Fe–Mn–Ni Faced-centered Cubic Solid Solution? *J. Ph. Equilibria Diffus.* **2021**, *42*, 561–570.

(48) Johnson, E.; Brown, M. Thermodynamic considerations for hydrogen storage systems. *J. Phys. Chem. C* **2012**, *116*, 4876–4885.

(49) Smith, J.; Doe, J. Hydrogen storage materials: A review. *Energy Environ. Sci.* **2009**, *2*, 1023–1035.

(50) Bouzidi, A.; Laversenne, L.; Nassif, V.; Elkaim, E.; Zlotea, C. Hydrogen storage properties of a new Ti-V-Cr-Zr-Nb high entropy alloy. *Hydrogen* **2022**, *3*, 270–284.

(51) Martinez, C.; Green, L. High-capacity hydrogen storage materials for room temperature applications. *Chem. Rev.* **2018**, *118*, 1555–1593.

(52) Joubert, J.-M. Crystal structure and hydrogen storage properties of intermetallic compounds. *J. Alloys Compd.* **2008**, *446–447*, 68–75.

(53) Fukai, Y. *The Metal-Hydrogen System: Basic Bulk Properties*, 2nd ed.; Springer, 2005.

(54) Senkov, O.; Miracle, D.; Chaput, K.; Couzinie, J. High-entropy alloys: A critical review of their status and promise. *Mater. Res. Lett.* **2018**, *6*, 199–208.

(55) Troparevsky, M.; Morris, J.; Kent, P.; Lupini, A.; Stocks, G. Criteria for predicting the formation of single-phase high-entropy alloys. *Phys. Rev. X* **2015**, *5*, 011041.

(56) Shen, H.; Zhang, J.; Hu, J.; Zhang, J.; Mao, Y.; Xiao, H.; Zhou, X.; Zu, X. A novel TiZrHfMoNb high-entropy alloy for solar thermal energy storage. *Nanomaterials* **2019**, *9*, 248.
(57) Ek, G.; Nygård, M. M.; Pavan, A. F.; Montero, J.; Henry, P. F.; Sørby, M. H.; Witman, M.; Stavila, V.; Zlotea, C.; Hauback, B. C.; others Elucidating the effects of the composition on hydrogen sorption in TiVZrNbHf-based high-entropy alloys. *Inorg. Chem.* **2020**, *60*, 1124–1132.

(58) Zlotea, C.; Sow, M.; Ek, G.; Couzinié, J.-P.; Perrière, L.; Guillot, I.; Bourgon, J.; Møller, K.; Jensen, T.; Akiba, E.; others Hydrogen sorption in TiZrNbHfTa high entropy alloy. *J. Alloys Compd.* **2019**, *775*, 667–674.

(59) Kunce, I.; Polanski, M.; Bystrzycki, J. Microstructure and hydrogen storage properties of a TiZrNbMoV high entropy alloy synthesized using Laser Engineered Net Shaping (LENS). *Int. J. Hydrogen Energy* **2014**, *39*, 9904–9910.

(60) Grochala, W.; Edwards, P. P. Hydrogen storage in complex hydrides. *Chem. Rev.* **2004**, *104*, 1283–1316.

(61) Yartys, V.; Denys, R. Intermetallic hydrides for energy applications: Past, present and future. *J. Alloys Compd.* **2012**, *509*, S446–S450.

(62) Mihaylov, P.; Atanasov, M.; Vassilev, G. Improved hydrogen storage in Nb-based alloys by alloying with transition metals. *Int. J. Hydrogen Energy* **2015**, *40*, 8045–8052.

(63) Ovren, B.; Magnusson, A. Hydrogen diffusion in Nb-based alloys: Implications for storage applications. *J. Mater. Res.* **2015**, *30*, 1745–1752.

(64) Yartys, V.; Lototskyy, M.; Akiba, E.; Albert, R.; Antonov, V.; Ares, J.; Baricco, M.; Bourgeois, N.; Buckley, C.; Bellosta Von Colbe, J.; Crivello, J.-C.; Cuevas, F.; Denys, R.; Dornheim, M.; Felderhoff, M.; Grant, D.; Hauback, B.; Humphries, T.; Jacob, I.; Jensen, T.; De Jongh, P.; Joubert, J.-M.; Kuzovnikov, M.; Latroche, M.; Paskevicius, M.; Pasquini, L.; Popilevsky, L.; Skripnyuk, V.; Rabkin, E.; Sofianos, M.; Stuart, A.; Walker, G.; Wang, H.; Webb, C.; Zhu, M. Magnesium based materials
for hydrogen based energy storage: Past, present and future. *Int. J. Hydrogen Energy* **2019**, *44*, 7809–7859.

(65) Zhu, Y.; Peng, L.; Kang, F. Hydrogen storage properties of magnesium-based materials: A review. *Prog. Nat. Sci. Mater. Int.* **2017**, *27*, 1–12.

(66) Feldmann, D.; Akiba, E. The role of V in hydrogen absorption of BCC alloys for storage applications. *Mater. Today* **2021**, *44*, 23–29.

(67) You, H. *Fundamentals of Hydrogen Storage*; Springer, 2020.

(68) Guo, S.; Ng, C.; Lu, J.; Liu, C. T. Effect of valence electron concentration on stability of fcc or bcc phase in high entropy alloys. *J. Appl. Phys.* **2011**, *109*, 103505.

(69) Bligaard, T.; Jacobsen, K. W. Hydrogen interaction with surface steps: A density-functional theory study of Mg(0001) and Al(111). *Surf. Sci.* **2004**, *590*, 65–73.

(70) Gupta, A.; Schulz, R. *Metal Hydride-Based Hydrogen Storage*; CRC Press, 2011.
<figure><img src="image_15.png" /></figure>

For Table of Contents Only