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Article

Integrating Density Functional Theory Calculations and Machine Learning to Identify Conduction Band Minimum as a Descriptor for High-Efficiency Hydrogen Evolution Reaction Catalysts in Transition Metal Dichalcogenides

by
Xiaolin Jiang
1,
Guanqi Liu
1,
Lifu Zhang
2,* and
Zhenpeng Hu
1,*
1
School of Physics, Nankai University, Tianjin 300071, China
2
School of Biomedical Engineering and Technology, Tianjin Medical University, 22 Qixiangtai Road, Tianjin 300070, China
*
Authors to whom correspondence should be addressed.
Catalysts 2025, 15(4), 309; https://doi.org/10.3390/catal15040309
Submission received: 27 February 2025 / Revised: 17 March 2025 / Accepted: 21 March 2025 / Published: 25 March 2025
(This article belongs to the Special Issue Two-Dimensional (2D) Materials in Catalysis)
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Abstract

:
Identifying efficient and physically meaningful descriptors is crucial for the rational design of hydrogen evolution reaction (HER) catalysts. In this study, we systematically investigate the HER activity of transition metal dichalcogenide (TMD) monolayers by combining density functional theory (DFT) calculations and machine learning techniques. By exploring the relationship between key electronic properties, including the conduction band minimum (CBM), pz band center, and hydrogen adsorption free energy (ΔG*H), we establish a strong linear correlation between the CBM and ΔG*H, identifying the CBM as a reliable and physically meaningful descriptor for HER activity. Furthermore, this correlation is validated in vacancy-defected TMD systems, demonstrating that the CBM remains an effective descriptor even in the presence of structural defects. To enable the rapid and accurate prediction of the CBM, we develop an interpretable three-dimensional model using the Sure Independence Screening and Sparsifying Operator (SISSO) algorithm. The SISSO model achieves a high predictive accuracy, with correlation coefficients (r) and coefficients of determination (R2) reaching 0.98 and 0.97 in the training and 0.99 and 0.99 in the validation tests, respectively. This study provides an efficient computational framework that combines first-principles calculations and machine learning to accelerate the screening and design of high-performance TMD-based HER catalysts.

1. Introduction

The hydrogen evolution reaction (HER), as a key half-reaction in water electrolysis for hydrogen production, plays a crucial role in clean energy conversion [1]. With the growing global demand for sustainable resources, the need for low-cost, stable, and efficient HER catalysts is also increasing. The efficiency of the HER is closely tied to the electrocatalytic activity of the catalyst, making the development of efficient new catalysts a critical factor for enabling large-scale hydrogen energy applications [2,3,4,5,6,7]. In recent years, transition metal dichalcogenides (TMDs) have emerged as a prominent research focus for HER. TMD materials possess unique layered structures, good electrical conductivity, excellent electronic properties, and a tunable catalytic performance, all while being cost-effective [8,9,10,11]. The catalytic performance of TMD materials is significantly influenced by their crystal structure (e.g., 1T, 2H, and 1T’ phases), electronic structure, and surface adsorption properties [12,13,14,15,16,17]. In terms of the electronic structure, the 1T phase of TMDs typically exhibits higher conductivity and catalytic activity than the 2H phase, leading to higher hydrogen evolution rates [18]. Surface adsorption properties, along with lattice strain, surface defects, and electronic structure, are important factors that influence the HER activity of TMD materials and should be considered in the design and optimization of efficient catalysts.
Traditional Density Functional Theory (DFT) calculation methods have successfully revealed the microscopic mechanisms of catalytic processes, providing a theoretical foundation for understanding the HER catalytic activity of TMDs [17]. Through DFT calculations, researchers can investigate key factors of catalytic materials, such as reaction energy barriers, electronic structures, hydrogen adsorption energy, offering valuable insights into the nature of catalytic reactions [19]. However, DFT calculations are computationally expensive, and when screening large-scale materials, they can consume significant resources and time. Therefore, using simple and effective descriptors to characterize material properties provides a more concise and efficient approach. Descriptors are quantifiable parameters that map the complex characteristics of materials into simplified numerical forms. In HER research, commonly used descriptors [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] include the d band center, bond electronegativity, band gap width, and surface structure. Thus, selecting appropriate descriptors can effectively streamline the material screening and optimization process.
Algorithms such as Sure Independence Screening and Sparsifying Operator (SISSO) [35], which are based on machine learning, can efficiently identify key descriptors and establish mathematically rigorous models with physical significance. Unlike traditional physical models, data-driven approaches analyze large amounts of first-principles and experimental data to identify potential interrelated patterns and features, thereby accelerating material design and performance prediction [36,37,38,39,40,41,42,43]. Therefore, combining DFT calculations with data-driven machine learning methods allows for the rapid and effective extraction of key descriptors related to HER activity from diverse data sources, enhancing both the accuracy and efficiency of HER catalytic performance predictions.
In this study, we focus on the HER activity of TMDs by systematically analyzing the role of electronic properties, such as the conduction band minimum (CBM) and pz band center, in determining catalytic performance. Based on first-principles (DFT) calculations of 55 TMD materials with varying transition metals, chalcogens, and crystal structures (1T, 2H, and 1T’), we establish a strong linear relationship between the CBM and hydrogen adsorption free energy (ΔG*H), demonstrating that the CBM serves as a reliable and physically meaningful descriptor for HER activity. To examine the universality of this descriptor, we further validate the CBM–ΔG*H correlation in vacancy-defected TMD systems, confirming that the CBM remains effective for characterizing catalytic performance even in the presence of structural defects. We further leverage the Sure Independence Screening and Sparsifying Operator (SISSO) algorithm to develop a machine learning model for predicting the CBM, enhancing the accuracy and efficiency of the catalyst screening process. This work combines computational chemistry with machine learning to streamline the discovery of new materials for HER, offering a promising framework for future catalytic design.

2. Results and Discussion

2.1. Crystal Structure

There are some polytypic structures within TMDs, and 1T, 2H, and 1T’ are the common phases investigated. The crystal structures of 1T, 2H, and 1T’ are shown in Figure 1. Each cation of the 1T-TMD monolayer binds to six neighboring anions, and one anion binds to three cations. 1T-TMDs possess the space group of P 3 ¯ m 1 , and the cation is in the octahedral coordination (Figure 1a,d,e). Similar with 1T-TMDs, each cation of the 2H-TMD monolayer binds to six neighboring anions, and one anion binds to three cations. 2H-TMDs possess the space group of P 6 ¯ m 2 , and the cation is in trigonal prism coordination (Figure 1b,f,g). 1T’-TMDs can be obtained when the cations of 1T-TMDs distort along the y direction slightly (Figure 1c,h,i). In the T, H, and T’ phases, the anion is the adsorption site.
A thermodynamic descriptor-based approach using density functional theory calculations was used to investigate the activity and nature of 55 different transition metal dichalcogenide catalysts for the hydrogen evolution reaction (HER). We considered variations in the transition metal (Si, Ti, V, Cr, Co, Ni, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Sn, Hf, Ta, W, Re, Os, Ir, and Pt), the chalcogen (O, S, Se, and Te), and the crystal structure (1T, 2H, and 1T’).

2.2. HER Activity on the Nanostructured TMDs

The HER is an electrochemical process that occurs on the surface of the electrode in contact with the electrolyte. It is widely accepted that there are two reaction mechanisms including the Volmer–Tafel and Volmer–Heyrovsky mechanism [11,12]. The Volmer reaction indicates the electrochemical hydrogen adsorption ( ∗ + H + + e − → H ∗ ); then, the Heyrovsky or Tafel reaction would happen for the whole HER process. In the Volmer–Heyrovsky reaction, the adsorbed hydrogen atom reacts with a proton from solution to produce a H2 molecule ( H ∗ + H + + e − →   ∗ + H 2 ). In the Volmer–Tafel reaction, the adsorbed hydrogen atom reacts with another contiguous adsorbed hydrogen atom to form a H2 molecule. Usually, the HER catalytic activity is determined by the free energy of adsorption for the hydrogen atom. According to the Sabatier principle, an ideal HER catalyst should exhibit moderate interaction with hydrogen atoms. This is quantitatively described as the free energy of hydrogen atoms on the catalyst surface being neither too strong to hinder desorption nor too weak to present effective adsorption. A ΔG*H value near zero ensures a good balance between the hydrogen adsorption and desorption processes, thereby enabling efficient HER performance [44].
The optimal structures of hydrogen atoms adsorbed on TMDs were investigated, where hydrogen was located on the top site of the anions. The ΔG*H values are listed in Table S2 and shown in Figure 2. The adsorption energy of hydrogen is related to the transition metal (Si, Ti, V, Cr, Co, Ni, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Sn, Hf, Ta, W, Re, Os, Ir, or Pt), the chalcogen (O, S, Se, or Te), and the crystal structure (1T, 2H, or 1T’).

2.3. CB, VB, and pz Band Center of TMDs

As shown in Figure 3, the 55 different transition metal dichalcogenide materials are divided into two groups according to the band gap. One group has an intrinsic band gap, including Ti2O4-T’, TiS2-H, TiS2-T, ZrS2-T, ZrSe2-T, HfS2-T, HfSe2-T, Nb2O4-T’, MoS2-H, MoSe2-H, MoTe2-H, WS2-H, WSe2-H, WTe2-H, TcS2-T, Re2O4-T’, ReS2-T, ReSe2-T, ReTe2-T, Ru2O4-T’, Os2O4-T’, PtS2-T, PtSe2-T, PdTe2-T, and PtTe2-T. Another group has no band gap, including TiSe2-T, TiTe2-T, ZrTe2-T, HfTe2-T, VS2-T, VSe2-T, VTe2-T, NbS2-T, NbS2-H, NbSe2-T, NbSe2-H, NbTe2-T, Ta2O4-T’, TaS2-T, TaS2-H, TaSe2-H, Cr2O4-T’, CrS2-T, CrSe2-T, Mo2O4-T’, MoS2-T, MoTe2-T’, W2O4-T’, WTe2-T’, ReSe2-H, CoTe2-T, RhTe2-T, Ir2O4-T’, IrTe2-T, and NiTe2-T. Most of the materials without a band gap are composed of T-phase and T’-phase crystal structure. They also include a small amount of H-phase crystal structure, such as NbS2-H, NbSe2-H, TaS2-H, TaSe2-H, or ReSe2-H. However, for materials with a band gap, they have different phase crystal structures. For the same cation, the pz band center is strongly related to the anion. The pz band center increases with the change in the anion from O, S, or Se to Te. This is consistent with the fact that the occupied state is mainly contributed by anions.

2.4. Selection and Analysis of HER Descriptors

To investigate the mechanism of hydrogen adsorption on TMDs, the Fermi level, electrostatic potential, projected density of states (PDOS), and the crystal orbital Hamilton population (COHP) (all energies with respect to vacuum level) of pristine and hydrogen absorbed TMDs were calculated. The CBM is adapted to the electron energies in vacuum by calculating the electrostatic potential.
It is known that weakly binding TMDs benefit from a higher CBM, whereas strongly binding TMDs favor a lower CBM to achieve good HER catalytic activity [45]. For semiconducting TMDs with a finite band gap, the CBM represents the first available state for hydrogen electron filling during adsorption. If the CBM is located too high, electron transfer from the hydrogen atom to the TMD surface becomes thermodynamically unfavorable, leading to weak orbital hybridization between the hydrogen and the chalcogen atoms, and, consequently, a high ΔG*H. In contrast, when the CBM lies lower, electron transfer is facilitated, enhancing the H–chalcogen interaction and stabilizing hydrogen adsorption. For metallic TMDs or those with negligible band gaps, partially filled orbitals at or near the Fermi level allow for more accessible electron transfer, resulting in stronger hydrogen adsorption. However, even in these systems, the relative position of the CBM can modulate the degree of orbital hybridization and thus affect the ΔG*H. As shown in Figure 4a, a strong linear correlation is observed between the ΔG*H and CBM, with the fitted Pearson correlation coefficient (r) and coefficient of determination (R2) being 0.90 and 0.80, respectively. Therefore, the CBM can be used as a descriptor for the HER performance of catalysts. Compared to the direct calculation of the ΔG*H, which requires explicit surface models and adsorption site analysis and is computationally demanding, the CBM is a bulk electronic property that can be computed with significantly lower computational cost and higher consistency across different TMD systems. This makes the CBM a highly efficient screening-level descriptor to pre-select promising HER candidates before performing costly ΔG*H evaluations. Meanwhile, we also explored the relationship between the CBM and electronic properties such as the CBM after hydrogen adsorption (Post_CBM), antibonding center, bonding center, difference between the antibonding center and the bonding center, difference between the antibonding center and Post_CBM, and the occupancy of the antibonding surface-adsorbate states ( N a b _ o ), as well as the correlation between the ΔG*H and these electronic properties. This additional analysis was performed along with that of the other electronic properties. As shown in Figures S1 and S2, the results indicate that the correlation of the CBM and ΔG*H with other electronic properties tends to be consistent.
We also investigated the relationship between the ΔG*H and pz band center. When the H atom is adsorbed on the surface of TMDs, the chalcogen atoms’ pz orbitals will interact with the H-s orbitals; then, the bonding and antibonding orbitals will be formed on the basis of molecular orbital theory. The bonding orbitals (σ) at lower energy levels are completely filled, while the antibonding orbitals (σ*) with higher energy levels are partially filled. Since the H-s orbitals are changeless, the different chalcogen atoms’ pz orbitals have a decisive influence on the bonding orbitals and antibonding orbitals. In other words, the bond strength between the H and chalcogen has a connection with the center of the chalcogen atoms’ pz orbitals. In this work, only the states below the Fermi energy level are considered during the calculation of the pz band centers. Also, the pz band centers are adapted to the electron energies in vacuum. The relationships between the hydrogen adsorption energy and pz band centers of the TMDs are shown in Figure 4b. Although there is a close correlation between the ΔG*H and pz band center, the relationship is not linear. Therefore, using the pz band center alone as a descriptor for the HER is not advisable.
To investigate the applicability of the CBM as a descriptor for HER activity, we analyzed vacancy-defected TMD structures (Figure S3) and examined the relationship between the CBM and the ΔG*H. As shown in Figure 5, a clear linear correlation is observed between the CBM and ΔG*H for these defective systems, with a correlation coefficient (r) of 0.95 and a coefficient of determination (R2) of 0.91. This result suggests that, even in the presence of vacancy defects, the CBM remains an effective descriptor to assess hydrogen adsorption properties.
As shown in Figure 6a,b, the monolayers of the 1T and 2H phases of TiS2 are both direct semiconductors, with band gaps of 0.0716 eV and 0.6990 eV, respectively. The CBM of the substrate before hydrogen adsorption positions are −5.8700 eV and −6.0089 eV. As shown in Figure 6c, when the CBM is positioned lower, the electron from the hydrogen atom is more easily excited to the filled state. As can be seen from the reaction step diagram in Figure S4, in the 2H phase TiS2, it is easier to carry out the hydrogen evolution reaction.

2.5. Construction of a 3D Descriptor for Predicting CBM Through Machine Learning

Traditional physicochemical methods frequently rely on complex models and significant computational resources. However, with the advancement of computational power and the rapid development of machine learning algorithms, machine learning-based predictive methods have become a key tool for efficiently exploring material properties.
Simple and easily accessible features are crucial factors in machine learning. In this study, we selected the structural and electronic properties of cations and anions in TMD systems as training features, with the CBM as the target, as shown in Table 1. The SISSO algorithm was applied to the machine learning training process, constructing a simplified physical model to predict the CBM.
To further evaluate the model’s performance and the importance of the features, we randomly split the data in an 8:2 ratio, with 80% of the data used for SISSO training to obtain a predictive model, and 20% used for prediction to assess the model’s generalization. As shown in Figure 7, the feature evaluation of the machine learning model is depicted in two parts. Figure 7a presents the correlation matrix of the selected features: the positive correlations are represented by shades of red, while the negative correlations are depicted in purple, with the intensity of the color indicating the magnitude of the correlation. Cde and Cve, Ade, and Ara display a strong positive correlation, suggesting they are highly related, whereas other pairs, such as Ara and AI, and Arc and Aχ show weaker correlations. Figure 7b shows the contribution of each feature to the training prediction of the CBM. A higher feature importance indicates a larger contribution of that feature to the prediction results. Cχ and Arc emerging as the most significant, contributing 31.51% and 14.50%, respectively, to the predictive power of the model. From a solid-state electronic structure perspective, the CBM of the TMDs arises from the hybridization of transition metal d-orbitals and chalcogen p-orbitals. The cation’s Pauling electronegativity affects the energy of d-orbitals: a higher electronegativity lowers the d-orbital energy and thus the CBM, while a lower electronegativity raises them. The anion’s covalent radius reflects the spatial extent of p-orbitals and determines their overlap with metal d-orbitals. Larger covalent radii reduce the orbital overlap, leading to narrower bands and a higher CBM, whereas smaller radii enhance coupling and lower the CBM. Similar effects have been reported in anion substitution studies on 2D materials, where changing the anion shifts the CBM and modulates the catalytic activity.
It should be noted that the Pearson correlation coefficients presented in Figure 7a only quantify the linear dependence between feature pairs and do not directly reflect their predictive relevance to the CBM. In contrast, the feature importance shown in Figure 7b measures the actual contribution of each feature to the model’s predictive accuracy, which accounts for linear and nonlinear interactions. Therefore, features with low pairwise correlation may still play critical roles in the model due to their unique influence on reducing the prediction error.
Figure 8a presents the SISSO training results: comparing the CBM values predicted by the SISSO model with the DFT-calculated values, we can observe that the orange points are distributed near the fitted regression line. The Pearson correlation coefficient (r) is 0.98, and the coefficient of determination (R2) is 0.97. The training results are very close to the DFT calculations, indicating that the SISSO model effectively captures the relationship between CBM and features, allowing for accurate predictions of CBM. Through the training of the SISSO algorithm, we obtained a 3D descriptor model for predicting CBM (Equation (1)). This descriptor model includes several key features, such as the cation’s first ionization energy, the anion’s first ionization energy, the cation’s valence electron number and Pauling electronegativity, and the numbers of anions. By incorporating these features, the model captures the complex relationships between the material’s electronic structure and the CBM.
C B M = − 0.783 × d 1 − 0.606 × d 2 − 0.337 × d 3 − 5.151 = − 0.783 × [ cos ⁡ A I C χ − cos ⁡ A I − sin ⁡ C I ] − 0.606 × | sin ⁡ e C V e − sin ⁡ ( C V e − A n a ) | − 0.337 × cos ⁡ A I × sin ⁡ A I cos ⁡ C I + cos ⁡ C V e − 5.151
We then used this descriptor model to predict the remaining data, as shown in Figure 8b. The results indicate that the r is 0.99, and the R2 is 0.99. These performance metrics suggest that the model provides highly accurate predictions, further confirming the effectiveness of the SISSO algorithm in predicting the CBM and its potential for use in larger-scale materials screening and design tasks.
To evaluate the generalization ability of the SISSO model, we used the model trained exclusively on pristine TMDs to predict the CBM of vacancy-defected systems. As shown in Figure 9, the SISSO-predicted CBM exhibits a good linear correlation with the DFT-calculated CBM, demonstrating the model’s ability to capture the overall CBM trend even for systems not included in the training data. However, a systematic deviation is observed, which we attribute to the absence of defect-related features in the initial training set. This highlights a current limitation of the model, as it lacks explicit descriptors to account for structural defects. Future work will focus on incorporating defect-specific information, possibly through advanced algorithms such as deep learning, to enable the accurate recognition and modeling of complex defect structures.
In this study, we chose to predict the CBM rather than directly predict the hydrogen adsorption free energy ΔG*H using machine learning. While ΔG*H prediction is possible, it requires detailed, system-specific descriptors to account for adsorption sites and local configurations, which would significantly increase the complexity and limit the generalizability. In contrast, the CBM is an intrinsic electronic property that can be efficiently and consistently calculated across different TMDs. Moreover, the strong linear correlation observed between the CBM and ΔG*H in both pristine and defective systems supports the use of the CBM as a practical and physically meaningful descriptor for HER activity screening. Nonetheless, the verification by experiments is the final aim, so obtaining a good trend would be better here for a certain value of the ΔG*H.

3. Computational Method

3.1. DFT Calculation and Parameters

All spin-polarized DFT calculations were performed using the projector augmented plane-wave (PAW) [46] method, as implemented in the Vienna Ab initio Simulation Package (VASP) (Version 5.4.4, Company VASP Software GmbH, Vienna, Austria) [47,48,49,50]. The calculations are based on the strongly constrained and appropriately normed (SCAN) meta-GGA [51,52]. SCAN has been reported to improve the predictive power of DFT largely for geometries and energies of diversely bonded materials and molecules (including covalent, metallic, ionic, hydrogen, and van der Waals bonds) [53]. It should be noted that while the SCAN functional is employed in this study for its improved accuracy over standard GGAs in predicting adsorption energies and electronic properties, it may still underestimate or overestimate the absolute band edge positions such as CBM. However, since our focus is on establishing trends and correlations between the CBM and ΔG*H across TMDs, SCAN provides a reasonable balance between computational efficiency and accuracy. The identified CBM–ΔG*H relationship thus reflects relative trends rather than absolute energy levels [54,55,56,57].
An energy cutoff of 500 eV was employed for the plane-wave expansion. All of the atoms were fully relaxed. The convergence criterion of maximum force during the optimization of each atom was less than 0.02 eV/Å, and the convergence criterion for the total energy was 10−5 eV. For the 2D TMDs monolayer slab model, the vacuum space was set at about 20 Å to avoid the interaction between two periodic images. The dipole correction [58,59] was applied to remove the artificial dipole interaction caused by using the slab supercell method for surface calculations. The mechanism of HER and the analytical principle of COHP are described in the Supporting Information.

3.2. Machine Learning Model and Performance Evaluation

To predict the hydrogen adsorption energy, we applied the Sure Independence Screening and Sparsifying Operator (SISSO) algorithm. The model training parameters are detailed in the Table S1.
We randomly divided the data into a training set and a testing set according to a 8:2 ratio. Two machine learning performance indicators, the Pearson correlation coefficient (r), and coefficient of determination (R2) are adopted (see Supporting Information for detailed formulas). These metrics were applied to both the training and test sets to ensure that the model generalized well and maintained a high level of accuracy when applied to unseen data.

4. Conclusions

In summary, this work establishes the CBM as an effective and computationally efficient descriptor for evaluating the HER activity of TMDs. Based on first-principles calculations of 55 TMD materials, a strong linear relationship between the CBM and ΔG*H is demonstrated. This correlation is further validated in vacancy-defected TMDs, confirming the robustness of CBM as a descriptor even in defective systems. To enable the rapid prediction of the CBM, we developed an interpretable SISSO-based machine learning model with high accuracy. By using the CBM instead of ΔG*H, the computational cost of screening can be significantly reduced. Overall, this study provides both a mechanistic understanding of HER activity and a practical framework for the rational design of TMD-based HER catalysts.

Supplementary Materials

The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/catal15040309/s1: Figure S1: Pre_CBM as a function of the (a) CBM after hydrogen adsorption (Post_CBM), antibonding center, bonding center, antibonding center-Post_CBM, antibonding center-bonding center, and Nab_o.; Figure S2: ΔG*H as a function of the (a) CBM after hydrogen adsorption (Post_CBM), antibonding center, bonding center, antibonding center-Post_CBM, antibonding center-bonding center, and Nab_o.; Figure S3: Anion vacancy structure: (a) Mo2O4-T’, (b) Mo2S4-T, (c) Te2Mo-H, and (d) Te2Mo-T. Figure S4: HER step diagram of TMDS with different anions: (a) S anion, (b) Se anion, (c) Te anion, and (d) O anion.; Table S1: Parameter setting of SISSO.; Table S2: ΔG*H of TMD structures. References [44,60,61,62,63,64,65,66,67,68,69] are cited in the Supplementary Materials.

Author Contributions

Conceptualization, X.J., L.Z. and Z.H.; methodology, X.J., L.Z. and Z.H.; software, Z.H.; validation, X.J., G.L. and L.Z.; formal analysis, X.J. and L.Z.; investigation, X.J. and L.Z.; resources, Z.H.; writing—original draft preparation, X.J.; writing—review and editing, X.J., L.Z., G.L. and Z.H.; visualization, X.J.; resources and funding acquisition, Z.H. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (No. 21933006, 21773124), the Tianjin Municipal Education Commission Scientific Research Project (No. 2024KJ229), the Fundamental Research Funds for the Central Universities Nankai University (No. 63243091, 63213042) and the Supercomputing Center of Nankai University (NKSC).

Data Availability Statement

The original contributions presented in this study are included in the article and Supplementary Materials; further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. (a–c): top views of the structure of 1T, 2H, and 1T’ TMD monolayer. (d,e), (f,g), and (h,i): Side views of the structure of 1T, 2H and 1T’ TMDs monolayer. The supercells are used in the calculations (dotted purple lines). Blue and yellow balls represent cations and anions, respectively.
Figure 1. (a–c): top views of the structure of 1T, 2H, and 1T’ TMD monolayer. (d,e), (f,g), and (h,i): Side views of the structure of 1T, 2H and 1T’ TMDs monolayer. The supercells are used in the calculations (dotted purple lines). Blue and yellow balls represent cations and anions, respectively.
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Figure 2. The ΔG*H of hydrogen atom on the TMDs. Blue, purple, and orange regions represent 1T, 2H, and 1T’ phase.
Figure 2. The ΔG*H of hydrogen atom on the TMDs. Blue, purple, and orange regions represent 1T, 2H, and 1T’ phase.
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Figure 3. Band centers of CB, VB, and pz of TMDs: (a) intrinsic band gap TMDs, (b) band gap-free TMDs.
Figure 3. Band centers of CB, VB, and pz of TMDs: (a) intrinsic band gap TMDs, (b) band gap-free TMDs.
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Figure 4. ΔG*H as a function of the (a) CBM before hydrogen adsorption (Pre_CBM) and (b) pz band center before hydrogen adsorption.
Figure 4. ΔG*H as a function of the (a) CBM before hydrogen adsorption (Pre_CBM) and (b) pz band center before hydrogen adsorption.
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Figure 5. ΔG*H as a function of the Pre_CBM in defective systems.
Figure 5. ΔG*H as a function of the Pre_CBM in defective systems.
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Figure 6. PDOS with respect to the vacuum level reference of (a) the 1T-phase and (b) the 2H-phase monolayers of TiS2. (c) Schematic of the accommodation of the H electron for CBM. (The red dashed line represents “standard hydrogen electrode potential relative to vacuum level −4.44eV”).
Figure 6. PDOS with respect to the vacuum level reference of (a) the 1T-phase and (b) the 2H-phase monolayers of TiS2. (c) Schematic of the accommodation of the H electron for CBM. (The red dashed line represents “standard hydrogen electrode potential relative to vacuum level −4.44eV”).
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Figure 7. Feature evaluation for machine learning: (a) feature correlation and (b) feature importance.
Figure 7. Feature evaluation for machine learning: (a) feature correlation and (b) feature importance.
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Figure 8. Evaluation of machine learning results: (a) training of the model, and (b) testing of the model’s universality.
Figure 8. Evaluation of machine learning results: (a) training of the model, and (b) testing of the model’s universality.
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Figure 9. Evaluation of SISSO model generalization ability in vacancy defect system.
Figure 9. Evaluation of SISSO model generalization ability in vacancy defect system.
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Table 1. Feature selection for machine learning training of CBM.
Table 1. Feature selection for machine learning training of CBM.
CategoryFeatureAbbreviation
Cationd-electron numberCde
Valence electron numberCVe
Pauling electronegativityCχ
1st ionization energyCI
Covalent radiusCrc
Atomic radiusCra
Aniond-electron numberAde
Pauling electronegativityAχ
1st ionization energyAI
Covalent radiusArc
Atomic radiusAra
Numbers of anionsAna
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Jiang, X.; Liu, G.; Zhang, L.; Hu, Z. Integrating Density Functional Theory Calculations and Machine Learning to Identify Conduction Band Minimum as a Descriptor for High-Efficiency Hydrogen Evolution Reaction Catalysts in Transition Metal Dichalcogenides. Catalysts 2025, 15, 309. https://doi.org/10.3390/catal15040309

AMA Style

Jiang X, Liu G, Zhang L, Hu Z. Integrating Density Functional Theory Calculations and Machine Learning to Identify Conduction Band Minimum as a Descriptor for High-Efficiency Hydrogen Evolution Reaction Catalysts in Transition Metal Dichalcogenides. Catalysts. 2025; 15(4):309. https://doi.org/10.3390/catal15040309

Chicago/Turabian Style

Jiang, Xiaolin, Guanqi Liu, Lifu Zhang, and Zhenpeng Hu. 2025. "Integrating Density Functional Theory Calculations and Machine Learning to Identify Conduction Band Minimum as a Descriptor for High-Efficiency Hydrogen Evolution Reaction Catalysts in Transition Metal Dichalcogenides" Catalysts 15, no. 4: 309. https://doi.org/10.3390/catal15040309

APA Style

Jiang, X., Liu, G., Zhang, L., & Hu, Z. (2025). Integrating Density Functional Theory Calculations and Machine Learning to Identify Conduction Band Minimum as a Descriptor for High-Efficiency Hydrogen Evolution Reaction Catalysts in Transition Metal Dichalcogenides. Catalysts, 15(4), 309. https://doi.org/10.3390/catal15040309

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