Hydrogen adsorption energy trends in Mo/WXY (X, Y = S, Se, Te) regular and Janus TMD monolayers: a first-principles and machine learning study

Abstract

The hydrogen adsorption behavior on two-dimensional (2D) transition metal dichalcogenides (TMDs) plays a key role in their photocatalytic properties. Using density functional theory (DFT), we investigate H adsorption energy (E_ads) trends in regular MX₂ and Janus MXY TMDs (M = Mo/W, X, Y = S/Se/Te). Our analysis identifies energetically favorable adsorption sites, revealing non-uniform site preferences influenced by covalency differences and atomic radii. Notably, Janus MXY TMDs exhibit significantly lower E_ads due to their built-in dipole moment, with stronger adsorption on surface atoms of higher electronegativity (EN). We develop machine learning (ML) models in order to predict E_ads based on the DFT dataset. Four energetic descriptors are used. Linear regression (LR) performs poorly with an R² score of 0.31, but partitioning the data by adsorption surface improves its accuracy (R² > 0.83), indicating a piecewise linear trend. Non-linear models, support vector regression (SVR) and multilayer perceptron regression (MLPR), capture this behavior more effectively, with one-hidden-layer MLPR achieving the highest accuracy (R² = 0.98). These results underscore the surface-dependent nature of H adsorption and highlight the effectiveness of our ML approach in predicting E_ads, offering a reliable alternative to complex theoretical simulations for the hydrogen evolution reaction.

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1 Introduction

Currently, the world is facing a huge energy crisis due to the growing population and their energy needs, the continuous burning of fossil fuels whose availability is limited, and geopolitical wars that affect energy production and transportation.¹ The usage of conventional energy sources will cause environmental pollution due to the emission of greenhouse gases. Thus, focused research has been going on to find alternative energy sources with zero carbon emissions. One such source is hydrogen, which is abundant on the earth in the form of water.^2,3 The water-splitting reaction is an uphill reaction that requires an energy of 1.23 eV to split an H₂O molecule into H₂ and O₂ molecules.^4,5 There exist several methods such as photocatalysis^6–8 and photoelectrocatalysis⁹ to produce H₂ with zero carbon emissions via solar-assisted water splitting.¹⁰ Among these methods, photocatalysis, which resembles natural photosynthesis, is a method requiring a semiconductor with an active band gap in the visible region (380–750 nm). The entire process involves two reactions, namely the oxygen evolution reaction (OER) and hydrogen evolution reaction (HER), which can be written as^11–13

2H₂O + 4 h⁺ → 4H⁺ + O₂(OER) (1)

2H⁺ + 2e⁻ → H₂(HER) (2)

In the OER process (see eqn (1)), photogenerated holes (h⁺) will react with water molecules and produce H⁺ ions and oxygen molecules. The H⁺ ions released in the OER process participate in the HER (eqn (2)) by combining with the photogenerated electrons, and H₂ molecules get released. The Gibbs free energy change (ΔG) of the redox reactions mentioned above ideally screens and characterizes the material as a candidate material for photocatalytic activity. The key process in water splitting is the HER, which proceeds through two main steps: the adsorption of hydrogen atoms on the catalyst surface (Volmer reaction) and their subsequent recombination to form a H₂ molecule (Heyrovsky or Tafel reaction).^13–15 As a critical step in green hydrogen production, the HER plays a vital role in sustainable energy systems. The first reaction, which involves atomic H adsorption, is the rate-determining step of the HER.¹⁶ Thus, it is important to probe and understand the adsorption of H on various catalytic surfaces.

The isolation of single-layer graphene from bulk graphite sparked interest in 2D materials.¹⁷ Following graphene, many atomically thin monolayers such as borophene, silicene, germanene, and stanene were theoretically predicted and experimentally synthesized subsequently.^18–20 Similarly, transition metal dichalcogenides (TMDs), layered materials with the stoichiometry MX₂,^21,22 offer excellent choices for the synthesis of respective 2D layers. Bulk TMDs feature strong covalent intralayer bonds and weak van der Waals interlayer interactions, which are utilized in the exfoliation method for 2D layer synthesis. These TMD single layers generally adopt either the trigonal prismatic (2H) or octahedral (1T) phase, with significantly varying properties by phase. For example, 2H-MoS₂ is a semiconductor, while 1T-MoS₂ exhibits metallic behavior.²³ Owing to their diverse characteristics, TMDs are widely employed across various fields, including photocatalysis,²⁴ energy storage,²⁵ photoelectric devices,²⁶ thermoelectric materials,²⁷ and optoelectronics.²⁸ Apart from regular MX₂ type TMDs, 2D Janus TMDs with MXY stoichiometry have gained much interest in recent years, due to their broken out-of-plane symmetry.^29–31 Janus structures are experimentally realized via replacing one of the sublayers X fully with another chalcogen, with a different electronegativity (EN), in MX₂.^32,33 Many Janus materials based on elements from group-III,³⁴ group-IV,³⁵ and transition-metals³⁶ are proposed by theoretical investigations. The MoSSe layer was the first Janus layer synthesized in 2017 by substituting H in the place of the top S layer of MoS₂ followed by selenization via vaporizing the Se powders.³² Following that, the WSSe layer was synthesized using the pulsed laser deposition technique.³⁷ The broken out-of-plane symmetry in the Janus structures leads to interesting phenomena such as the Rashba effect, vertical piezoelectricity, etc.^38,39 The built-in electric field in Janus TMDs is expected to drive the photoexcited electrons and holes in opposite directions, leading to an increased lifetime of the carriers due to the reduction in recombination.

In addition to extensive theoretical and experimental investigations of 2D materials for catalytic applications, recent advances in machine learning (ML) approaches have garnered significant attention for predicting the catalytic activity of materials which are yet to be studied. For instance, a kernel ridge regression-based ML model utilizing structural descriptors as training features has been applied to 2D MoS₂ and copper–gold clusters to predict adsorption energies E_ads.¹⁶ Similarly, Shanping et al. developed ML potential leveraging neural networks (NNs) to accurately probe the adsorption and diffusion properties of H₂ molecules on metal–organic frameworks.⁴⁰ Usuga et al. introduced an ML model that incorporates geometric and electronic descriptors to predict catalytic activity in bimetallic alloys. Their approach successfully estimated adsorption energies for several species, including C, N, S, O, and H atoms.⁴¹ Furthermore, supervised ML models by Bakır et al., trained using experimental H adsorption energies, have been applied to predict hydrogen adsorption in a range of materials such as LaFeO₃, LaRuO₃, and BiFeO₃.⁴²

Other than H adsorption, the adsorption strength of metals has also been explored. Dou et al. investigated the adsorption energies of alkali metal atoms (Li, Na, K, and Rb) on 2D TMDs using an LR-based ML model.⁴³ Detailed studies on Li adsorption in both regular and Janus TMDs were conducted by Chaney et al. to develop a universal ML model using just three descriptors for predicting Li adsorption.⁴⁴ Beyond adsorption energies, ML has been applied to predict other properties related to photocatalytic activity. Jiang et al. employed a crystal graph convolutional NN to predict photocatalytic degradation rate constants of metal oxide photocatalysts across a wide range of organic contaminants.⁴⁵ Similarly, Javed et al. utilized 13 distinct ML techniques (including extreme gradient boosting (XGB), decision trees, and artificial neural networks (ANNs) to estimate photocatalytic degradation rates of TiO₂ for air contaminants.⁴⁶

To date, the theoretical modeling combined with ML techniques for the H adsorption energy prediction remains limited for the family of 2D regular and Janus TMDs, hindered by the varying nature of chemical interactions of H with these TMDs. Notably, the chemical bonding may be of covalent or ionic type or a mix of these two, making it challenging to devise an accurate ML model for E_ads. For the present investigation, we focus on the adsorption properties of atomic hydrogen on 2H polymorphs of regular MX₂ and MXY Janus structures, confining to M = Mo, W, and X, Y = S, Se, and Te. For this purpose, we carried out detailed DFT-based electronic structure calculations at the GGA level. We also probed the effect of coverage on E_ads by choosing different supercells. After confirming the non-linearity of E_ads with respect to the adsorption site and choice of 2D materials, we employ two non-linear methods to develop the ML model. The article is structured as follows: the next section gives the computational details. After this, we explore the structural properties and energetics of the TMD single layers (SLs). Next, we identify the energetically favorable adsorption site for atomic hydrogen. We then delve into the trends in charge transfer and adsorption energy. Using these insights, we train the ML models for E_ads prediction. The predictions are initially tested using the linear regression (LR) model, followed by two non-linear regression models: support vector regression (SVR) and multilayer perceptron regression (MLPR).

2 Computational details

The periodic electronic structure calculations are performed using density functional theory (DFT) implemented in the Vienna Ab initio Simulation Package (VASP).⁴⁷ The electron–ion interactions are treated with the projected augmented wave (PAW) method.⁴⁸ The Perdew, Burke, and Ernzerhof (PBE) functional of the generalized gradient approximation is used to consider the exchange and correlation effects.⁴⁹ The chosen valence electron configurations for the Mo, W, S, Se, and Te elements are 4p⁶ 5s² 4d⁴, 6s² 4f¹⁴ 5d⁴, 3s² 3p⁴, 4s² 4p⁴, and 5s² 5p⁴, respectively. A plane wave basis set with a kinetic energy cutoff of 650 eV is used in the simulations along with Γ-centered k-mesh for Brillouin zone sampling. The energy convergence criterion is set as 10⁻⁶ eV. The forces on each atom are converged below 0.001 eV Å⁻¹. The energy levels are Gaussian smeared by 0.05 eV. To understand the electron transfer in the materials, we perform Bader charge analysis^50,51 and to attribute the coverage dependence, we use four different sizes of supercells 1 × 1 × 1, 2 × 2 × 1, 3 × 3 × 1, and 4 × 4 × 1. The corresponding k-mesh used for these different supercells is 16 × 16 × 1, 8 × 8 × 1, 4 × 4 × 1, and 1 × 1 × 1, respectively. A vacuum of 15 Å is provided along the direction perpendicular to the layer direction to prevent the interaction between neighboring layers. We have performed spin-polarized calculations for a few selected structures as the H adsorption leads to an open-shell configuration. From the comparison of the total energies with the non-spin polarized energies, we found a negligible difference. Thus, we have carried out non-spin polarized calculations for all structures. The VASPKIT interface is used to post-process the VASP-generated output files.⁵²

For the prediction of H adsorption energy from the ML model, we apply the LR model and two non-linear models, namely SVR with the radial basis function (RBF) kernel and MLPR with the rectifying linear unit (ReLU) activation function. A brief overview of SVR and MLPR models is given in the ESI.†

3 Results and discussion

3.1 Structural information

Initially, we optimize the regular MX₂ TMDs where M = Mo/W and X = S/Se/Te. The results for the optimized lattice constants are given in Table 1. From the optimized regular SLs, the Janus SLs are designed by replacing one of the X layers with a distinct chalcogen atom Y. For example, the Janus MoSSe SL is formed as a result of replacing an S (Se) layer of the MoS₂ (MoSe₂) SL with a Se (S) layer. The lattice constant of MXY SLs is found to be the arithmetic mean of the lattice constants of their respective parent layers, i.e., the analogous MX₂ TMDs. For instance, the lattice constant of MoSSe is the average of the lattice constants of MoS₂ and MoSe₂. Due to the different chemical environments on both sides of the Janus transition metal dichalcogenides, the charge transfer is not uniform along all bondings. To understand the amount of charge transfer quantitatively, the Bader analysis^50,51 is carried out and the results are summarized in Table 1. The Bader charge on each atom shows that the electron transfers from the Mo/W atom to the chalcogen atoms in all MX₂. However, as expected, the chalcogen atom with a higher EN gains more negative charge from the metal atoms in MXY SLs. For example, in the MoSeTe SL, electron transfer to Se is higher (−0.48e) than that to Te (−0.22e). The maximum charge transfer is observed between W and S atoms (−0.75e) in the WSTe SL, and the minimal charge transfer is between Mo and Te atoms (−0.21e) in the MoSTe SL. In all layers, electron transfer to the S atom is stronger compared to Se and Te atoms.

Table 1 The calculated lattice constant (a), Bader charge of metal (ΔQ_M) and chalcogen (ΔQ_X and ΔQ_Y) atoms, cohesive energy (E_coh), the electrostatic potential difference (Δϕ) between X and Y sublayers in MXY SLs, and the GGA band gap (E_g) are summarized. The calculated E_coh, Δϕ, and E_g values for various SLs are compared with the reported values from ref. 44 and 53–60, provided in parentheses

System	a (Å)	ΔQM (e)	ΔQX (e)	ΔQY (e)	E coh (eV)	Δϕ (eV)	E g (eV)
Regular
MoS2	3.16	1.18	−0.59		−5.34 (−5.34)	—	1.75 (1.68)
MoSe2	3.30	0.92	−0.46		−4.80 (−4.86)	—	1.51 (1.44)
MoTe2	3.52	0.50	−0.24		−4.25 (−4.57)	—	1.14 (1.18)
WS2	3.17	1.41	−0.70		−5.96 (−6.09)	—	1.78 (1.82)
WSe2	3.30	1.09	−0.64		−5.36 (−5.56)	—	1.61 (1.55)
WTe2	3.53	0.59	−0.28		−4.76 (−5.02)	—	1.14 (1.04)
Janus
MoSSe	3.23	1.04	−0.61	−0.43	−5.06 (−5.10)	0.66 (0.77)	1.64 (1.55)
MoSeTe	3.41	0.70	−0.48	−0.22	−4.50 (−4.59)	0.61 (0.72)	1.34 (1.34)
MoSTe	3.34	0.85	−0.64	−0.21	−4.73 (−4.79)	1.11 (1.49)	1.14 (0.94)
WSSe	3.23	1.23	−0.71	−0.52	−5.65 (−5.82)	0.62 (0.71)	1.76 (1.85)
WSeTe	3.41	0.84	−0.57	−0.28	−5.02 (−5.26)	0.56 (0.71)	1.42 (1.54)
WSTe	3.34	1.02	−0.75	−0.27	−5.28 (−5.47)	1.21 (1.42)	1.32 (1.31)

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Furthermore, we calculate the cohesive energy per atom, which is defined as follows:

where E_MX2 and E_MXY represent the ground state energy of the MX₂ and MXY SLs respectively. The terms E_M and E_X/Y represent the ground state energy of reference transition metal (Mo/W) and chalcogen (S/Se/Te) atoms, respectively. Negative values of E_coh indicate that all 2D materials studied in this work form bound systems, and the larger magnitude of E_coh indicates the stronger binding of the atoms in the material. In the present scenario, we observe larger cohesive energies for the systems involving the S atoms, see Table 1 and SFig. 1.† Furthermore, we calculate the electrostatic potential difference between two opposite surfaces for all the Janus structures, which are given in SFig. 2.† All these results discussed in the section so far are summarized within Table 1.^44,53–59

3.2 Hydrogen adsorption

3.2.1 Energetics and sites of H adsorption. The strength of atomic H adsorption (E_ads), which is important for the HER, is analyzed on each SL, focusing on its dependency on the H coverage ratio and site preference. The H coverage ratio is simulated using supercells of sizes 1 × 1 × 1, 2 × 2 × 1, 3 × 3 × 1, and 4 × 4 × 1, corresponding to coverage ratios of 100%, 25%, 11.11%, and 6.25%, respectively. To explore site preference, the H atom is adsorbed on five distinct sites, applicable to both MX₂ and MXY layers, (a) C-site: on top of the chalcogen atom, (b) T-site: on top of the transition metal atom, (c) M-site: at the midpoint between two chalcogen atoms, (d) B-site: at the center of the M-X/Y bond (bridge site) and (e) H-site: at the hollow site between chalcogen atoms, located at the center of the hexagonal ring, cf.Fig. 1 for schematics of these adsorption sites. The calculated ground state energies indicate that for H adsorption on S in TMDs, the C-site is preferred, whereas for H adsorption on Se and Te, the M-site is energetically favored. This preference stems from the significant EN difference between S (2.58) and H (2.20) atoms, as well as the smaller atomic radius of S (0.87 Å) compared to Se (1.03 Å) and Te (1.23 Å). The smaller atomic radius and higher EN of S concentrate electron density near the S atom, which makes the C-site more favorable for H adsorption. In contrast, the larger atomic radii and lower EN of Se and Te result in longer Se–H and Te–H bond distances. Additionally, the increasing covalent character from S to Te alters charge distribution across the Se and Te atomic planes. As a result, H preferentially adsorbs at the M-site when interacting with Se/Te surfaces, forming bonds with two neighboring Se/Te atoms. Adsorption at the T-, B-, and H-sites is more endothermic compared to that at the C- and M-sites. Thus, unlike Li adsorption, H atom adsorption on TMDs does not follow a uniform trend. This may be attributed to the fact that the Li atom tends to give away electrons to the TMD surface, but the H atom can either give or take, depending upon the EN of the surface atom.⁴⁴ Further studies focus on the most energetically favorable adsorption sites: the C-site for S and the M-site for Se and Te.

Fig. 1 The side and top views of the TMD when the H atom is adsorbed on (a) C-site, (b) T-site, (c) M-site, (d) H-site, and (e) B-site, respectively. The H atom in the light pink color is highlighted with a red-colored elliptical shape.

3.2.2 Adsorption energetics of H. The adsorption energy is defined usingwhere E_TMD+H and E_TMD are the total energies of the TMD with and without the adsorbed H atom, respectively. The term E_H2 represents the ground state energy of the H₂ molecule. The calculated E_ads for all TMDs is shown in Fig. 2 and also listed in STable 2.† The positive (negative) value of E_ads indicates that the H adsorption is endothermic (exothermic), which requires (releases) an additional energy of E_ads for the adsorption to take place. In general, the 3 × 3 × 1 and 4 × 4 × 1 supercells result in almost the same E_ads in all TMDs. Fig. 2 shows that the adsorption reaction is endothermic in all considered cases. However, no universal variation of E_ads is observed as the cell size increases. Overall, the average E_ads calculated from MoX₂ and MoXY is lower than that of W-based TMD layers. Hence, the presence of Mo in the W atomic sites improves the H adsorption.

Fig. 2 The calculated H adsorption energies on various TMDs with respect to various supercell sizes. The energetically favorable sites are chosen for the study, i.e., the C-site for S surfaces and M-site for Se and Te surfaces. The olive and purple colored dotted lines indicate the average adsorption energies of the Mo- and W-based SLs.

First, we discuss the H adsorption on the S side of the TMDs. Initially, the adsorption energy E_ads decreases when the size of the supercell increases from 1 × 1 × 1 to 2 × 2 × 1 and then it increases for the 3 × 3 × 1 supercell. The adsorption energies for the 3 × 3 × 1 and 4 × 4 × 1 supercells are almost equal. For the Se side H adsorption, the E_ads shows an increasing trend with the decrease in coverage, except in the case of Janus TMDs when the other surface is of Te. For this particular case, E_ads in the full coverage limit is higher than the 25% coverage. In the case of the Te side H adsorption, E_ads increases as the H concentration decreases. Among the MX₂ TMDs, the adsorption energies are found to be larger for MSe₂ SLs and are found to be lower for the MTe₂ SLs. In the case of MXY Janus TMDs, if the H atom is adsorbed on the X side and if the Y atom has less EN than X, the E_ads is decreased compared to the respective regular MX₂ TMDs. If the EN of X is lower than that of Y, not much deviation is observed in the E_ads of the MXY SL compared to the MX₂ SL. For example, in the MoSTe Janus SL, when H is adsorbed on the S (X) side, the other surface is Te (Y), which has less EN than S. Hence, we observe a large decrease in the E_ads compared to MoS₂. When the H atom is adsorbed on the Te (X) side and the other surface is S (Y) which has a large EN over Te, the adsorption energy is not much deviated from that of the MoTe₂ SL. This may depend on the direction of the built-in electric field arising from the EN difference between the top and bottom surfaces of the Janus TMDs, which is directed from the less EN surface to the higher EN surface. In the case of MoSTe, the direction of the field will be from Te to S. Thus, when the H atom is adsorbed on the S surface, the field also exists towards S and reduces the E_ads. When the H atom is adsorbed on Te, i.e., in a direction opposite to that of the built-in electric field, the E_ads is not changed much from that of the regular TMD MoTe₂. The larger the EN difference between the top and bottom surfaces of the SLs, the larger the built-in electric field and hence the reduction in E_ads. However, a large amount of electron transfer is required for H₂ reduction. Of all the considered Janus SLs, MoSTe with H adsorption on the S side is found to have the smallest E_ads. However, the electron transfer is negligible here. Overall, the Te surfaces have good charge transfer and low adsorption energies compared to the other S and Se surfaces. The E_ads is found to be larger for Se surfaces due to the very small electron transfer between the surface and the H atom. In addition to the intrinsic local charge distribution quantified by the Bader analysis, the required negative charge for the HER may be supplied by photoinduced electron–hole pairs. These pairs get charge separated due to the gradient of the electrostatic potential, which is enhanced by the electric polarization in the Janus SLs.

From Fig. 2 we can observe that the E_ads shows a similar trend in the Mo and W-based TMDs. Here, in Fig. 2, we have drawn dotted lines for the Mo and W-based TMDs, which indicate the average value of E_ads in these systems. The average value of E_ads of the W-based TMDs lies at higher energies than those of the Mo-based TMDs. This may be due to the larger EN of W (2.36) than Mo (2.16). The reason for this can be explained by calculating the highest occupied state (E_HOS) for all the TMDs, from/to where the electron transfer takes place. When we analyze the position of the (E_HOS) of the TMDs, which is given in SFig. 3,† the average value of W-based TMDs lies at higher energies than those of the Mo-based TMDs. Thus, the position of the E_HOS directly influences the E_ads.

3.2.3 Trend in charge distribution. The effect of cell size (therefore coverage ratio) on the interaction of H with TMDs is enlightened by charge transfer occurring between H and SLs in the hydrogenated layer. The calculated charge on H is depicted in Fig. 3 and is listed in STable 2.† The positive (negative) charge on H indicates the electron depletion from (accumulation on) the H atom.The interaction of H on S-surfaces exhibits an intriguing behavior characterized by the direction switching of charge (electron) transfer. At high hydrogen coverage (i.e., 100% in the 1 × 1 × 1 cell), charge transfer occurs from the layer to the H atoms. This direction of charge transfer is reversed at lower H coverage ratios. This switching behavior can be attributed to the dominance of the EN of S over H, coupled with the decreasing H–H interaction as the coverage ratio is reduced. The TMD layers (regular and Janus) containing S atoms have a lower lattice constant compared to the ones with Se and Te atoms due to the smaller atomic radius of S compared to that of Se and Te atoms. Thus, it is expected that the electrostatic repulsion between the protons (hydrogen) to be slightly larger for TMD layers with S atoms. To reduce the total energy of the system in these layers, a small amount of charge is being transferred from the layer to the H atom in a 1 × 1 × 1 cell of TMD layers containing an S atom, and this will partially shield the repulsive interaction between the H atoms. To verify this aspect, we carried out the analysis of the partial density of states and checked the contribution of s-orbitals of H atoms. The results for MoS₂ as a representative case are given in SFig. 4,† which clearly shows that the contribution of the s-orbital of H atoms has significantly increased for the states just below the Fermi level. This leads to a reversal of charge transfer between the layer and H atoms when we reduce the cell size to the 1 × 1 × 1 cell despite the S atom having a larger electronegativity than that of the H atom.Unlike the S–H interaction in MoS₂, the variation of the H coverage on MoSe₂ and MoTe₂ does not alter the direction of electron transfer between TMDs and H atoms. These observations reveal that the non-uniform charge transfer behavior arises from the EN differences between the chalcogens (S: 2.58, Se: 2.55, and Te: 2.10) and H (H: 2.20). Specifically, the higher electronegativities of S and Se compared to H result in a relatively small amount of electron transfer to H on S or Se surfaces with varying H coverage. Recently, similar trends in the charge transfer difference for S-surfaces and Se-/Te-surfaces when the H atom adsorption takes place on the MX₂ TMDs were reported by Ke-Xin Hou et al.⁶¹ The behavior observed in MX₂ SLs also extends to MXY SLs.

Fig. 3 The charge on the H atom after getting adsorbed on various TMDs with different coverage ratios. The energetically favorable sites are chosen for the analysis, i.e., the C-site for S surfaces and the M-site for Se and Te surfaces.

To get insights into the spatial distribution of the charge transfers, we calculated the differential charge density (DCD) of the TMDs when the H atom is adsorbed. The DCD Δρ is defined as

Δρ = ρ_MXY+H − ρ_MXY − ρ_H, (5)

where ρ_MXY+H and ρ_MXY are the electron densities of the MXY TMDs with and without H, respectively, and ρ_H is the charge density of H. The DCD for the H adsorbed on S and Se surfaces of MoSSe, as well as for the Te surface of MoSTe, is shown in Fig. 4(a)–(c). The DCD for the rest of the SLs is presented in SFig. 5.† From Fig. 4 it is clear that for S surfaces, the charge is depleted from the H and accumulated on the S surface. For the Se and Te surfaces, we observe charge accumulation on the H atom and depletion at the Se/Te surface. Note that the shape of the isosurface for S and Se/Te side adsorption is different because of the difference in the adsorption site (C vs. M). This DCD analysis aligns with the results of the Bader analysis.

Fig. 4 The charge density difference when the H is adsorbed on (a) S surface of MoSSe (C-site), (b) Se surface of MoSSe (M-site), and (c) Te surface of MoSTe (M-site), respectively, of the 4 × 4 × 1 supercell. The yellow and cyan-colored isosurfaces around the atoms represent the regions of charge accumulation and depletion, respectively. The isosurface value is chosen to be 0.002 e ⁻³.

3.2.4 Trend in chalcogen–H bond distance. Along with charge distribution analysis, we measure the optimized H-chalcogen bond distances for all structures at various coverage ratios, as shown in Fig. 5. These bond distances form three distinct clusters, based on the adsorption site and surface. As expected, due to the smaller atomic radius of the S atom, the average S–H bond distances are the shortest, measuring 1.38 Å and 1.40 Å for Mo- and W-based regular and Janus TMDs, respectively. Compared to their Mo counterparts, W-based TMDs exhibit slightly longer bond distances due to the increased atomic radius of the W atom. At varying H concentrations, the bond distances do not vary much. The average Se–H bond distances for Mo- and W-based TMDs are 1.86 Å and 1.91 Å, respectively. Due to the larger atomic radius of Te, the H–Te bond distances are correspondingly larger for MTe₂ and MXTe structures.

Fig. 5 The bond distances (X–H) between the adsorbed H atom and the surface of adsorption in all the considered regular and Janus TMDs. The energetically favorable sites are chosen for the study, i.e., the C-site for S surfaces and M-site for Se and Te surfaces.

In addition to the bond distances, we have also measured the vertical H–X/Y distances (hereafter represented as l_X–H) along the direction perpendicular to the sheet of the layer, which are depicted in SFig. 6 and are listed in STable 2.† A general trend of l_X–H distances is derived, which mainly depends on the specific adsorption site (C and M) and atomic size relations of H–S, H–Se, and H–Te. The l_S–H of H and S is the same when H is adsorbed on the S sublayer in TMDs consisting of S (i.e., in MoS₂, MoSSe, MoSTe, WS₂, WSSe, and WSTe). The l_S–H is the same as the S–H bond distance since H favors the C-site. Similarly, the l_Se–H of H and Se in TMDs consisting of Se (i.e., in MoSe₂, MoSSe, MoSeTe, WSe₂, WSSe, and WSeTe) is almost the same. Likewise, the l_Te–H of H and Te in TMDs consisting of Te (i.e., in MoTe₂, MoSTe, MoSeTe, WTe₂, WSTe, and WSeTe) is almost the same. The average values of l_S–H, l_Se–H and l_Te–H are estimated to be 1.40 Å, 0.95 Å, and 1.06 Å respectively, showing the trend of l_S–H > l_Te–H > l_Se–H. The vertical distances l_Se–H and l_Te–H are smaller compared to l_S–H because the H atom prefers the M-site for these cases.

3.3 Descriptor design

The DFT-based simulation of H adsorption at low-level concentrations is computationally expensive. In order to avoid DFT calculations of E_ads for each particular case, we develop an ML model for its prediction. For training the ML model, we select energetic descriptors that capture key properties influencing adsorption. To ensure a universal representation, we incorporate descriptors that account for the polarity of Janus materials, as well as site and coverage dependencies, following the approaches proposed by Chaney et al.⁴⁴ and Dou et al.⁴³ Below, we discuss the specific descriptors used in training the model.

(1) E_coh: for the identification of the TMD on which the H adsorption is taking place, we use the cohesive energy of the material, see eqn (3).

(2) E_HOS: in the HER process, the electron transfer occurs from the highest occupied state (HOS) of the TMD to the H atom. Thus, we consider this as a descriptor whose values are corrected with respect to vacuum energy.

(3) E_TMD-H: when the H atom gets adsorbed on the TMD surface, there exists an interaction between the atoms of the TMD and H. This is calculated as an electrostatic interaction energy between each sub-layer of the TMD with the H. In the process of the HER, the electron transfer occurs from the TMD to the H atom. Thus, we assume that the H atom has a negative charge of −1e. The electrostatic interaction of the 1e⁻ on H with photogenerated holes is neglected. The charge on each sublayer is obtained from the Bader analysis of the relaxed structure. The perpendicular distance between the H atom and the TMD surface is calculated as the sum of the atomic radius of the H and the chalcogen atoms on which it is adsorbed in the case of the S surface. In the case of Se and Te surfaces where the H atom is at the bridge site, the perpendicular distance is calculated by applying the Pythagorean theorem. This term, E_TMD-H, is calculated with the code used by Chaney et al.⁴⁴

(4) E_H–H: in the HER, H₂ will be formed after the formation of many such H* complexes. Thus, for multiple H atoms to get adsorbed on the material, the H atoms have to overcome the repulsive interaction of H–H. This is calculated as the DFT total energy of the H atom alone in the relaxed cell, excluding the other atoms corresponding to the TMD. Here, we assume that the H atom has a charge of −1e.

3.4 Machine learning approaches

In this work, we apply ML approaches to predict the H adsorption energies (output variable) on the TMD surfaces from the four energy descriptors, E_coh, E_HOS, E_TMD-H and E_H–H (input variables). The values of the descriptors E_TMD-H and E_H–H for the considered TMD SLs at different coverage ratios are listed in STable 3.† We use standard supervised learning methods such as linear regression,⁶² support vector regression,⁶³ and multilayer perceptron regression.⁶⁴

3.4.1 Linear regression (LR). Here, we assume a linear relationship between the input and output variables and apply linear regression, as

E_ads = aE_coh + bE_HOS + cE_TMD-H + dE_H–H + e (6)

In this approach, the coefficients a, b, c, d, and e of the linear model are determined or learned from the data points. We have collected 72 data points from 12 different structures having 18 surfaces in 4 different coverages of atomic hydrogen. Among these points, 70% of the data is used for training the linear model. The training and testing data division is done randomly. The coefficients are determined so that the sum of squared residual error (difference between the actual and predicted values based on the model coefficients) over all the above training points is minimized. The model performance has been evaluated using the root mean squared error (RMSE) and the coefficient of determination (R² score).⁶² The LR method is applied to different subsets of the above-mentioned training data, and the coefficients and related performance metrics are measured.

When the model is implemented by considering 70% of the whole data points as training data, the optimized feature coefficients and performance metrics are given in the first line of Table 2. The corresponding DFT calculated versus ML predicted E_ads scatter plots are shown in Fig. 6(a). These results indicate the poor prediction of the LR model with RMSE 0.2183 eV and R² score 0.31 for the testing data. In the dense H concentration levels, i.e., in the 1 × 1 × 1, the repulsive H–H interaction is largest for the 1 × 1 × 1 cell and the values are found to be approximately 3.5 times compared to the next cell, which is 2 × 2 × 1. Also, in the case of the S-surface, there is an electron transfer to the H atom at dense concentrations, and at lower H concentrations the electron transfer is reversed, see Fig. 3. Thus, we consider the LR model that excludes the 1 × 1 × 1 cell data. The corresponding results are given in Fig. 6(b) and in the second line of Table 2. The performance of the model is slightly improved with the exclusion of 1 × 1 × 1 cell data. We have also implemented the ML model, including only Se- and Te-surface data and excluding 1 × 1 × 1 cell data and the S-surface data, see Fig. 6(c) for the scatter plot and the third line of Table 2 for regression statistics. The performance of the model is improved significantly when we exclude the S-surface data. The RMSE and R² scores for this case are found to be 0.052 eV and 0.741, respectively. This is due to the significant change in the E_ads values of S-surfaces of regular TMDs from those of Janus TMDs, which are not captured by the linear model when 70% of the whole data is used for training. From Fig. 3, we can observe that the accumulation of the charge on the H atom shows large differences for Se and Te surfaces. Thus, we consider only test and training data for individual S, Se, and Te surfaces and apply the linear regression model for each group of data. The corresponding scatter plots between actual and predicted energy values are shown in Fig. 6(d)–(e). After segregation into groups, the performance of the model is improved, which can be observed from the error metrics given in Table 2. From Fig. 6 we observe that the adsorption energy shows a piecewise linear trend (matching actual and predicted values) with respect to the surface of adsorption. The surface-wise segregation results presented here do not include the 1 × 1 × 1 cell data. The same experimentation is done by including the 1 × 1 × 1 cell data for each of the S-, Se-, and Te-surfaces, and the performance metrics are given in STable 4.† From these results, it is observed that the exclusion of the 1 × 1 × 1 cell data gives a better prediction for E_ads over the inclusion.

Table 2 The regression coefficients and the performance metrics RMSE and the R² score for the 30% testing (70% training) data of the linear regression model when implemented on various subsets of training data. Each line from top to bottom represents the results when the LR model is applied to the whole data, excluding the 1 × 1 × 1 cell data, excluding the S surface data, and after segregating into three groups depending on the surfaces of adsorption of S, Se, and Te

Data	a	b	c	d	e	RMSE (eV)	R ^2 score
Including 1 × 1 × 1	−0.1077	−0.0123	−0.0597	0.0602	1.7295	0.2183 (0.1638)	0.307 (0.362)
Excluding 1 × 1 × 1	−0.1139	−0.0039	−0.0799	0.0549	1.7607	0.1755 (0.1812)	0.323 (0.393)
Excluding S-surfaces	−0.1287	−0.0079	0.0680	0.0485	1.8259	0.0525 (0.0545)	0.741 (0.874)
S-surfaces	−0.2073	−0.0118	−0.1216	0.0863	1.6303	0.0386 (0.0312)	0.988 (0.986)
Se-surfaces	−0.1992	−0.0445	0.0643	0.0794	1.9228	0.0423 (0.0321)	0.832 (0.944)
Te-surfaces	−0.0559	0.0108	0.0204	0.0327	1.7290	0.0335 (0.0301)	0.419 (0.825)

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Fig. 6 The comparison between DFT-simulated and ML-predicted E_ads values when the LR model is implemented on various subsets of data. The results are shown when the LR model is implemented on (a) the whole data, (b) excluding the 1 × 1 × 1 cell data, (c) excluding the S-surface data, and after making the data into three parts based on the surface of adsorption, i.e., (d) S-surfaces, (e) Se-surfaces, and (f) Te-surfaces, respectively.

With all these trials, the results of regression coefficients (Table 2) indicate that a, which is the coefficient of E_coh, is the most significant descriptor in predicting the E_ads. The material dependency is taken care of by the E_coh as it is calculated with respect to the single atom energies. The value of a is the largest for the adsorption on S-surfaces where there are large differences in the E_ads of regular and Janus TMDs. The value of this coefficient remains almost the same for Se surfaces as the E_ads is affected by the presence of the other surface in the Janus TMDs. For the data of the Te-surfaces, the coefficient a is decreased by a factor of 3.5 compared to the S-/Se-surfaces. This is because the value of E_ads is not affected much for the H adsorption on Te surfaces. On the other hand, we observe that the coefficient c of E_TMD-H for the S-surface is negative, and the exclusion of this surface leads to a positive value. This is attributed to the direction of charge transfer between the TMD and H, which is found opposite for S surfaces compared to those of the Se/Te surfaces, see Fig. 3. The coefficient d indicates the role of E_H–H in the prediction of E_ads. Among the different experiments, it is found to be the least important for Te surfaces. Overall, among all the regression coefficients b of E_HOS is found least significant in the E_ads prediction, whereas it is found to be the most significant descriptor in predicting the Li adsorption energy in the TMDs.^43,44 In the case of H adsorption energy prediction, E_HOS is the least significant because of the very small charge transfer from the VBM of the TMD to H. Finally, the parameter e corresponds to the E_ads on various TMDs whose descriptors have the mean values of the data set.⁴⁴ For partitioned S-surface data, the mean values of E_coh, E_HOS, E_TMD-H, and E_H–H are −5.33, −5.43, 2.39, and −2.74 eV, respectively. These mean values of the four descriptors closely match the H adsorption on the S surface of MoSSe at a concentration of 11.11%. A similar observation can be extended to the Se and Te surface data separately. From the above discussion, we can conclude that the proposed LR model is not suitable to match all the known E_ads data.

3.4.2 Support vector regression (SVR). A brief overview of the SVR model is provided in the section ML approaches of ESI.† At first, we applied SVR with a non-linear RBF kernel function, and the corresponding scatter plot is shown in Fig. 7(a). Compared to the LR approach, there is a significant improvement in the model performance. On the test data (30% of data points), the observed R² score is 0.93, see Table 3. Subsequently, when we excluded the 1 × 1 × 1 cell data (see Fig. 7(b)), the R² score decreased to 0.89. When we exclude the S-surface data further, the R² scores of training and testing data is further dropped, which are given in Table 3. The corresponding scatter plot is given in Fig. 7(c). Also, the model has been implemented on the surface-wise segregated data, and the corresponding plots are shown in Fig. 7(d)–(f). The performance metrics for each case are tabulated in Table 3 for the training and testing data. It can be observed that the performance of the SVR model is better on the complete data than on the other experiments with data division. This may be due to the smaller number of available data points after excluding certain data. We have also implemented the SVR model on the surface-wise segregated data by including the 1 × 1 × 1 cell data, and performance metrics are provided in STable 4.† From these metrics, we can understand that similar to the LR model the SVR model also performs well when we exclude the 1 × 1 × 1 cell data. From the above results with SVR, we can conclude that the hydrogen adsorption energy follows a non-linear trend with respect to the input descriptors.

Fig. 7 The comparison between DFT-simulated and ML-predicted E_ads values when the SVR model is implemented on various subsets of data. The results are shown when the SVR model is implemented on (a) the whole data, (b) excluding the 1 × 1 × 1 cell data, (c) excluding the S-surface data, and after making the data into three parts based on the surface of adsorption, i.e., (d) S-surfaces, (e) Se-surfaces, and (f) Te-surfaces, respectively.

Table 3 Performance of the models LR, SVR, and MLPR on the training and testing data under different partitions

Method	R ^2 score
	Training data			Testing data
	LR	SVR	MLPR	LR	SVR	MLPR
Including 1 × 1 × 1	0.36	0.87	0.97	0.31	0.93	0.99
Excluding 1 × 1 × 1	0.39	0.89	0.98	0.32	0.89	0.98
Excluding S-surfaces	0.89	0.78	0.98	0.74	0.48	0.94
S-surfaces	0.98	0.92	0.99	0.98	0.94	0.99
Se-surfaces	0.94	0.75	0.98	0.83	0.68	0.94
Te-surfaces	0.83	0.29	0.97	0.42	−1.07	0.74

---

3.4.3 Multilayer perceptron regression (MLPR). We have designed a feedforward neural network MLPR, with an input layer, one hidden layer, and an output layer. Input descriptors are fed at the input layer, and the output variable, H adsorption energy E_ads, is learned at the output layer. The activation of neurons at the hidden layer and output layer is controlled by the ReLU function ReLU(x) = max(0, x).^64,65 The schematic of the neural network with an input layer, one hidden layer, and an output layer is given in SFig. 8.†The MLPR model, with one input layer and one hidden layer, correctly predicts the E_ads, cf.Fig. 8(a). We have tested the performance of the MLPR model by changing the number of neurons in the hidden layer, see STable 5.† The optimal performance is achieved with 20 neurons in the hidden layer. Also, we have analyzed the permutation importance to understand the correlation between the four considered descriptors and the target property E_ads, which is given in SFig. 9.† From the permutation analysis it is found that the E_HOS is playing a major role in predicting the E_ads whereas the E_H–H is found least significant among the four descriptors. Fig. 8(b) presents the results obtained by excluding the data of the 1 × 1 × 1 cell. Similar performance can be observed after considering only the Se- and Te-surface data (see Fig. 8(c)) and partitioning them into three groups (see Fig. 8(d)–(f)) based on the surface of adsorption. For the surface-wise partitioned data, we have done the experimentation by including the 1 × 1 × 1 cell data as well. The performance metrics for this case are given in STable 4.† The results with the MLPR model indicate that the performance of the model is slightly improved when we exclude the 1 × 1 × 1 cell data. Among the three considered ML models, MLPR is giving the best prediction for E_ads.

Fig. 8 The comparison between DFT-simulated and ML-predicted E_ads values when the MLPR model is implemented on various subsets of data. The results are shown when the MLPR model is implemented on (a) the whole data, (b) excluding the 1 × 1 × 1 cell data, (c) excluding the S-surface data, and after making the data into three parts based on the surface of adsorption, i.e., (d) S-surfaces, (e) Se-surfaces, and (f) Te-surfaces, respectively.

4 Data augmentation

In the present model, we have a total of 72 data points for which we have selected the energetically favorable sites for the atomic H adsorption. Due to the relatively small size of the dataset, we have used data augmentation techniques to generate synthetic data points. To add the new data points to the existing data, we have used the Synthetic Minority Over-Sampling Technique for Regression (SMOTER). The SMOTER technique generates synthetic data points using the K-Nearest Neighbor (KNN) algorithm. We have done several trials by the addition of 36, 72, and 144 new synthetic data points to the existing dataset. The performance of the MLPR model is tested for three data sets, which contain a total of 108, 144, and 216 data points. As in the previous selection, we have used 70% data for training and 30% for testing. The results of the implemented MLPR algorithm for three different sets of data and the performance metrics are given in Fig. 9. The observed R² score for the testing is 0.90, 0.92, and 0.95 for the three sets of the considered data points. The performance of the MLPR model is slightly reduced compared to the case when it is implemented on the original data. The details of the RMSE and R² scores for the training data are given in STable 6.†

Fig. 9 The performance of the MLPR with the inclusion of the (a) 36, (b) 72, and (c) 144 new synthetic data points to the existing data.

5 Summary of the application of ML algorithms

We have compared and summarized the performance of the three models (R² values) without and with partitioning in Table 3, which shows the scores for the 70% training data and 30% testing data. From these results, we can observe and conclude the following points:

(1) The LR model has shown poor performance on the complete data, but has shown good performance in cases of different data partitioning.

(2) Non-linear SVR has shown good performance on the complete data, but slightly poor performance (particularly on the Se and Te sides) after partitioning.

(3) The MLPR model has shown good performance on the complete data as well as the partitioned data.

From the above points, we can conclude that the atomic H adsorption energy follows a non-linear trend with respect to the input descriptor.

In the linear regression model, we have noticed that when the model is implemented on the whole data, the E_ads prediction was poor. The prediction is improved if we exclude the dense coverage data (1 × 1 × 1) for all the structures. This is due to the larger relaxation energy (E_relax) per unit area. We define E_relax as the energy difference between the relaxed and unrelaxed TMD SL with H. We consider the 1 × 1 × 1 cell area as unity, and to get the E_relax per unit area for the 2 × 2 × 1, 3 × 3 × 1, and 4 × 4 × 1 cells, we divide E_relax by 4, 9, and 16, respectively. The calculated relaxation energy trends are given in Fig. 10. It is observed that at dense H coverage limit (in 1 × 1 × 1 cell), the E_relax per unit area is very large compared to the other coverage ratios. Thus, the performance of the LR model is improved when we exclude the 1 × 1 × 1 dataset. The second reason for the non-linear behavior is the difference in the adsorption site. This was seen in Fig. 6(c) and (d). When we partitioned data into S and Se/Te surfaces, the performance of the model was tremendously improved.

Fig. 10 The calculated relaxation energies per unit area in regular and Janus TMDs when the H atom is adsorbed. The energetically favorable sites are chosen for the study, i.e., the C-site for S surfaces and M-site for Se and Te surfaces.

Furthermore, we have analyzed the energy variation ΔE with respect to the electrostatic potential difference Δϕ in the Janus SLs, see Fig. 11. Here, ΔE is defined as

ΔE = E_RTMD − E_JTMD, (7)

where E_JTMD and E_RTMD are the H adsorption energies of the Janus SL and the corresponding regular TMD SL. We have performed this analysis for the three surfaces S, Se, and Te at distinct coverage ratios. In the case of S surfaces, we observe that the ΔE is large when the electrostatic potential difference between the two surfaces of the Janus SL is large, i.e., when the other surface consists of Te. This is due to the built-in electric field in the material, which is directed from the Te to the S surface. The variations in ΔE are smaller for Se and Te surfaces, irrespective of Δϕ. In the case of Se surfaces, the ΔE remains almost constant when the other surface is of S atoms. If the other surface is of Te atoms, the ΔE is found to be large for the structure MoSeTe. But, in the case of WSeTe, the change is not significant due to the smaller Δϕ (0.56) over the MoSeTe (0.61). This confirms the statement that we made in the earlier section, i.e., the E_ads of the Janus TMDs is reduced by a significant amount if the other surface has less EN than the surface on which the H atom is adsorbed. In the case of the Te surfaces, the ΔE remains constant irrespective of Δϕ and the presence of the other surface. These dependencies on the adsorption surface and the built-in electric field bring non-linearity into the picture, which is further confirmed by the implementation of ML algorithms.

Fig. 11 The change in adsorption energy of the Janus TMDs with respect to the regular TMDs as a function of the electrostatic potential difference Δϕ present in the Janus TMDs when we considered only the (a) S-surfaces, (b) Se-surfaces and (c) Te-surfaces respectively. Here, the X-labels are indicated by Janus TMD corresponding to particular Δϕ.

The involved complexity in the prediction of the H adsorption energy arises from the Janus TMDs. The presence of the built-in electric field in the Janus TMDs brings non-linearity into the picture. With the exclusion of the Janus TMD data points, the simplest LR model is able to predict the H adsorption energy accurately, cf. SFig. 10.† The calculated RMSE and R² score for the testing data are 0.0380 eV and 0.939, whereas these values for the model without the exclusion of Janus TMDs are 0.2183 eV and 0.307, respectively. The built-in electric field in the Janus TMDs has shown a significant effect on the H adsorption energy. The values of the H adsorption energy are reduced for the Janus TMDs when the H atom is adsorbed on the high electronegative surface, as discussed in Section 3.2.2. The universal approximation theorem says that a neural network, even with a single hidden layer and a sufficient number of neurons, can accurately represent any continuous function within a given range. Overall, the MLPR model performs well on both the complete and segregated datasets.

6 Conclusion

In summary, we performed a comprehensive study on atomic H adsorption energy in 2D regular MX₂ and Janus MXY TMDs (M = Mo, W; X, Y = S, Se, and Te), providing valuable insights into the HER activity of photocatalysts. Our analysis, based on DFT-simulated data, explored site and coverage dependency, adsorption energy, and charge transfer trends. The studied TMD surfaces consist of S, Se, and Te, and their interaction with H varies due to differences in atomic radii and EN. The smaller atomic radius of S and the larger EN difference between S and H make the C-site where H sits on top of the S atom energetically favorable. In contrast, the larger atomic radii of Se and Te favor adsorption at the M-site. Notably, the S–H bond exhibits a more ionic character, whereas Se–H and Te–H bonds are covalent, with a single H atom bonding to two nearby Se or Te atoms. This non-uniform site preference is reflected in the charge (electron) transfer behavior: on Se and Te surfaces, charge transfer consistently occurs from the TMD to H across all coverage ratios, while on S surfaces, it reverses at coverage ratios of 25%, 11.11%, and 6.25%. Compared to regular TMDs in Janus TMDs, a significant reduction in H adsorption energy is only observed when H is adsorbed on the X surface if the opposite Y surface has a lower EN. This suggests that the built-in electric field in Janus TMDs plays a crucial role in lowering the H adsorption energy. Among all the Janus SLs, MoSTe and WSTe are well-suited for the HER process. In the MoSTe SL, we found a large reduction in the E_ads when H is adsorbed on the S surface, but the electron transfer is not to the H but to the S surface. For the desorption process of the H atom, to form H₂ by reacting with the nearby H atom, the electron transfer should be to the H atom. On the surface of Te, we have appreciable electron transfer to the H from the Te surface, making the desorption process easier. Thus, the MoSTe and WSTe SLs with Te side adsorption are more favorable for H₂ production.

Based on the obtained DFT data, we developed an ML model to predict E_ads based on data from 18 surfaces at four different coverage ratios. Initially, a linear regression model trained on 70% of the data points using four input descriptors exhibited poor predictability. To improve performance, we grouped all S-surfaces and Se–Te surfaces separately, training the linear regression model on each group individually, leading to a noticeable improvement. Further refinement by dividing the data surface-wise into three groups—S, Se, and Te—resulted in even better E_ads predictions. These observations indicate that E_ads follows a piecewise linear trend depending on the adsorption surface. To capture the overall non-linear behavior, we tested two non-linear models, SVR and MLPR, on the complete dataset with and without partitioning. The SVR model performed well on the unpartitioned dataset, achieving an R² score of 0.89, but its performance declined when the data were divided. Among all models, the MLPR model demonstrated the best predictive accuracy for E_ads in both partitioned and unpartitioned cases with an R² score of 0.98. The inclusion of augmented data with varying sizes yields consistently almost similar performance scores, confirming the reliability of the proposed ML models for E_ads prediction. This also validates the adequacy of the original dataset. These findings highlight the surface-dependent adsorption behavior in regular and Janus TMDs and demonstrate how tailored ML models can effectively capture these trends. Our study not only provides a predictive framework for H adsorption energies with improved accuracy but also establishes our ML approach for TMDs as a reliable model for predicting E_ads, reducing the reliance on complex theoretical simulations for the HER.

Data availability

The trends in E_HOS, electrostatic potential of the considered Janus structures, the vertical distance between the H atom top layer, and DCD when the H atom is adsorbed on the TMD SLs are provided in the ESI.† Also, tables containing the information of the E_ads, charge on the H atom, the two descriptors used in model training E_TMD-H and E_H–H, and a brief overview of the SVR and MLPR models are given.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

G. T. and D. M. acknowledge funding from the DST-SERB within the project CRG/2022/006778. D. M. thanks the Department of Information Services and Computing at Helmholtz-Zentrum Dresden-Rossendorf for providing extensive computational facilities. For the computation of the electrostatic interaction of the TMD layer and the H atom E_TMD-H, we have used the code developed by Chaney et al.⁴⁴

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ta02028j

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