A systematic theoretical study of CO₂ hydrogenation towards methanol on Cu-based bimetallic catalysts: role of the CHO&CH₃OH descriptor in thermodynamic analysis

Abstract

The application of density functional theory (DFT) has enriched our understanding of methanol synthesis through CO₂ hydrogenation on Cu-based catalysts. However, variations in catalytic performance under different metal doping conditions have hindered the development of universal catalytic principles. To address these challenges, we systematically investigated the scaling relationships of adsorption energy among different reaction intermediates on pure Cu, Au–Cu, Ni–Cu, Pt–Cu, Pd–Cu and Zn–Cu models. Additionally, by summing the respective adsorption energies of two separate species, we have developed a dual intermediate descriptor of CHO&CH₃OH, capable of achieving computational accuracy on par with DFT results using the multiple linear regression method, all the while enabling the rapid prediction of thermodynamic properties at various stages of methanol synthesis. This method facilitates a better understanding of the coupling mechanisms between energy and linear expressions on copper-based substrates, and the universal linear criterion can be applied to other catalytic systems, with the aim of pursuing potential catalysts having both high efficiency and low cost.

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1. Introduction

With the extensive utilization of fossil fuels, the escalating anthropogenic emission of CO₂ has engendered a series of ecological and environmental predicaments. Facilitated by the burgeoning hydrogen industry, the hydrogenation conversion of CO₂ into high value-added products has emerged as one of the most promising approaches,^1,2 paving novel avenues for the rational development within the realm of carbon cycling. Against this backdrop, methanol synthesized from mixed CO₂/CO/H₂ feedstocks (CO₂ + 3H₂ → CH₃OH + 3H₂O) not only serves as a critical industrial raw material but also acts as a fuel source, contributing to the realization of the “methanol economy”.³ Currently, commercial catalysts for methanol production are primarily based on low-cost and high-activity copper materials. Based on this, the copper-based bimetallic catalysts, such as Au–Cu,⁴ Ni–Cu,^5,6 Pd–Cu,^7,8 Pt–Cu⁹ and Zn–Cu,^10,11 can enhance methanol production with higher selectivity at lower temperatures, addressing the sintering and deactivation issues. However, in previous studies, there has been limited comprehensive exploration of multiple bimetallic alloy combinations under the same computational standards. This deficiency hinders the consistent assessment of the catalytic performance of different systems. Therefore, there is an urgent need for comprehensive investigations across various systems to bridge this knowledge gap.

In the studies of catalytic performance, energy-related information has consistently remained a crucial focus. As is widely recognized, the energy studies concerning Cu-based slab models encompass multiple parameters, such as adsorption energy,^12–14 activation energy,^15–17 cohesive energy,¹⁸ dissociation energy,¹⁹ and others. Among them, adsorption energy is a fundamental factor for characterizing catalytic performance. However, despite the significance of adsorption energy in surface reactions, a prominent challenge persists, that is, how to circumvent the energy discrepancies arising from different surface structures, adsorbate geometries and electron distributions,²⁰ while establishing a unified descriptive paradigm. Hensen et al.²¹ compared the adsorption energies of intermediates involved in methanol synthesis on the Cu(1 1 1) facet and concluded that the adsorption energies of CO and O species could succinctly encapsulate the energy properties of other intermediates. However, the derivation process of CO and O descriptors was not expounded, and the key variables used to summarize the reaction performance were not consistent. In addition, earlier studies have reported individual linear correlations between adsorption energies of individual atoms and their hydrogenated species,^22–24 as well as the relationships between adsorption energy and activation energy,^25–29 but lack further theoretical exploration and universality verification. In view of the above, current research on adsorption energy descriptors remains inadequate, and a systematic analysis regarding scaling relationships will aid in revealing a unified adsorption energy paradigm and elucidating the energy relationships behind surface reactions.

Furthermore, considering that the adsorption energy is a determining factor for the magnitude of thermodynamic energy, refining the expression of thermodynamic energy based on the scaling relationship of adsorption energy would deepen our understanding of hydrogenation mechanisms on Cu-based catalysts. In recent years, significant advancements have been made in the combination of DFT with machine learning (ML) for unveiling the concealed information behind materials' properties,³⁰ particularly from the energy aspect. For instance, Pathak et al.³¹ employed the atomic-size difference factor and the ratio of mixed Gibbs free energy to predict surface stability, using which they proposed an effective model for predicting the adsorption energy of reaction intermediates. Zhang and Zhou et al.³² developed an extreme gradient boosting regression algorithm to predict the adsorption energy variations of CO molecules and obtained reliable results. Wu et al.³³ investigated the adsorption energy of CO and H species at various Pd–Au ratios using an active learning method and provided insights into the ensemble effect of Pd content on the CO₂ reduction reaction. Despite these advancements, a notable scarcity remains in the studies that directly compare the scaling relationship involving adsorption energy with machine learning, even though having the potential for offering illuminating insights from such comparison. To mitigate this limitation, one strategy is to incorporate scaling relationships into the training set and supplement with Sure Independence Screening and Sparsifying Operator (SISSO) variables^34,35 to generate highly correlated descriptors.

In the present work, we firstly explore the thermodynamic mechanism of methanol synthesis on copper-based substrates. A thermodynamic database tailored for CO₂ hydrogenation is constructed, designed specifically for Cu-based bimetallic alloys (Au–Cu, Ni–Cu, Pt–Cu, Pd–Cu and Zn–Cu). The adsorption patterns of 19 intermediates are elaborated and the thermodynamic pathways encompassing three reaction mechanisms are also investigated. Then, a systematic screening study of both individual and dual intermediate descriptors is conducted, and the descriptor of CHO&CH₃OH is proposed. By comparing the reaction energies obtained by DFT calculations, the accuracy of CHO&CH₃OH in depicting the thermodynamic properties of hydrogenation reactions on copper substrates is also determined. To further enhance the predictive accuracy of energy-linear descriptors, MLR and SISSO methods are introduced, and the feasibility of incorporating the CHO&CH₃OH descriptor into the linear regression model is confirmed. Finally, a comparison is conducted between the proposed descriptors from the published literature. Our findings enhance the credibility of the dual intermediate descriptor and present a novel avenue for exploring the thermodynamics underlying CO₂ reduction reactions.

2. Computational details

Herein, the bimetallic alloy model was constructed by substituting the copper atoms with other transition metal (TM) atoms (Au, Ni, Pd, Pt and Zn). The primary rationale for selecting TM atoms was based on a literature review, which identified Au–Cu,^4,36,37 Ni–Cu,^6,38,39 Pd–Cu,^7,40–42 Pt–Cu^43,44 and Zn–Cu^45–47 as extensively documented copper-based catalytic models. Choosing these models may help ensure that the combination of transition metals represented by Au, Ni, Pd, Pt and Zn demonstrates reliable catalytic activity when paired with copper. Additionally, this selection facilitates comparison with catalytic systems detailed in previous studies, thereby highlighting the significant advantage of the scaling relationships proposed in our work within relevant catalytic systems. Aided by the statistic disorder module,¹² the most thermodynamically stable structures with a periodic 1 × 1 × 2 slab configuration were taken, in which the dopant atoms were positioned at the vertices and edges of the Cu lattice (see Fig. S1, ESI†). The (1 1 0) facet was chosen as the default surface, and the resulting unit cells after extension are illustrated in Fig. 1, where the dopant atoms were distributed in the first and third layers. Among the four layers of catalysts, the top two layers remained relaxed to simulate surface reactions, while the bottom two layers were fixed to simplify computations. Note that the selection of the (1 1 0) facet over (1 1 1) and (1 0 0) facets is based on three factors. Firstly, as reported in the earlier literature,^48–51 the low-coordinated (1 1 0) facet exhibits higher reactivity. Secondly, the flat surface of the (1 1 0) facet is more suitable for investigating adsorption behaviors and benefits the formation of stable alloy phases compared to the stepped surfaces of (1 1 1) and (1 0 0) facets.⁵² Thirdly, the (1 1 0) facet is proved to hinder the interference of the reverse water–gas shift reaction, thus facilitating methanol synthesis.⁵³ As for the intermediate adsorption position, four predefined sites were designed, including top, long bridge, short bridge and hollow sites (as depicted in Fig. 2a).

Fig. 1 Top and side views of optimized structures of the (a) Cu(1 1 0) surface, (b) Au–Cu(1 1 0) surface, (c) Ni–Cu(1 1 0) surface, (d) Pd–Cu(1 1 0) surface, (e) Pt–Cu(1 1 0) surface and (f) Zn–Cu(1 1 0) surface.

Fig. 2 Schematic diagram of CO₂ conversion to methanol on Cu₃X alloys (X = Cu, Au, Ni, Pd, Pt and Zn). The default adsorption sites for intermediates are shown for reference.

All density functional theory (DFT) calculations were carried out using projector augmented wave (PAW) potentials⁵⁴ and the generalized gradient approximation (GGA) functional,⁵⁵ implemented through the Vienna ab initio simulation (VASP) code. Validated by our previous convergence tests,¹² a cutoff energy value of 450 eV was chosen for ensuring an accurate representation of wave functions, and a gamma mesh of 3 × 3 × 1 was employed to define the discretization of the Brillouin zone for numerical integration. For electronic structure optimization, a convergence criterion of 1.0 × 10⁻⁵ eV was applied, and geometry optimization was deemed to be completed when the force on each atom reached 0.02 eV Å⁻¹. To describe a more realistic representation of intermolecular forces, the corrections for dipole moment and van der Waals (DFT-D3) were included. Moreover, the influence of free energy correction was found to be negligible and was therefore not considered in this work.

The adsorption energy (E_ad) was calculated using eqn (1):

E_ad + E_Sub+ads − E_Sub − E_ads (1)

where E_Sub+ads represents the total energy of the system containing both the substrate and adsorbate and E_Sub and E_ads are the energies of the substrate and adsorbate, respectively. The reaction energy (ΔE) was determined using eqn (2):

ΔE = E_FS − E_IS (2)

where E_IS and E_FS represent the energies of initial and final states. Work function (∅) was used as a descriptor of system properties, which was formulated as eqn (3):

∅ = E_vac − E_Fermi (3)

where E_vac and E_Fermi are the energy levels of vacuum and Fermi energy. To quantify the structural variations, the “dist.pl” script developed by Henkelman et al.⁵⁶ was employed. The VASPKIT code⁵⁷ was also used for adjusting atomic coordinates and calculating basic properties. A widely recognized machine learning algorithm from the Scikit-learn package⁵⁸ was employed, which was multiple linear regression (MLR). The features were extracted from the charge and structural properties of alloy models, while the label was defined as the absolute thermodynamic energy (more details can be found in Section 3.4.1). During the training process, all models were trained on 75% of the available data and evaluated on the remaining 25% for testing. The evaluation metrics of predictive performance, namely the Pearson correlation coefficient (PCC), determination coefficient (R²) and mean-average error (MAE), were recorded in the ESI.†

3. Results and discussion

3.1 Construction of the thermodynamic database

In order to accurately discern the energy characteristics of CO₂ hydrogenation on copper-based bimetallic alloys, it is imperative to build a standardized thermodynamic database. This may assist in correcting the data errors arising from inconsistencies of calculation details in the existing datasets, and the complete chart is documented in the ESI,† for reproducibility. The database encompasses six adsorption substrate models (Cu, Au–Cu, Ni–Cu, Pd–Cu, Pt–Cu and Zn–Cu), four predefined sites (top, long bridge, short bridge and hollow), as well as nineteen reaction intermediates (H, O, H₂, OH, H₂O, CO, CO₂, CHO, mono-HCOO, bi-HCOO, mono-H₂CO, bi-H₂CO, trans-COOH, cis-COOH, H₂COO, HCOOH, CH₃O, CH₂OH and CH₃OH). For mono-HCOO species, its optimized configuration eventually transforms into bi-HCOO with dual sites and is thus categorized as the bi-HCOO form. Additionally, the database provides adsorption energy and thermodynamic values for both the substrate and adsorbate, along with computational times under 16 threads. Based on the principle of selecting the highest adsorption energy, the optimal structures are picked out from four predefined sites and serve as default models for subsequent studies, as displayed in Figs. S2–S7 (ESI†). For the Ni–Cu alloy, all adsorbed intermediates preferentially occupy near the Ni sites, signifying the superior adsorption properties of Ni compared to Cu. Specifically, the optimal adsorption sites for H, O, CO₂, CHO, bi-HCOO, bi-H₂CO, trans-COOH, cis-COOH and CHOH species are located on the long bridge sites between Ni and Ni atoms. As for H₂, H₂O, CO, mono-H₂CO, CH₂OH and CH₃OH species, their optimal adsorption sites are situated on the top sites of the Ni atom. Additionally, the optimal adsorption sites for OH, HCOOH and CH₃O species are located on the short bridge sites between Ni and Cu atoms, while H₂COO is preferentially placed on the hollow site adjacent to the Ni atom. In other words, copper catalysts decorated with nickel can effectively enhance the adsorption capacity of intermediates, thereby ameliorating thermodynamic behaviors.

3.2 Methanol synthesis analysis on bimetallic alloys

Fig. 2b presents a schematic diagram depicting the hydrogenation of CO₂ to produce methanol, with a primary focus on investigating the adsorption and thermodynamic characteristics of CO₂ as the initial reactant and its hydrogenated products in this section.

3.2.1 Reaction pathways of methanol synthesis. A complete energy profile for methanol synthesis can be constructed by supplementing simultaneous adsorption calculations of all reactants on a substrate. Emphasizing that the activation energy within the hydrogenation process is not taken here, instead the primary focus of this work is on establishing a thermodynamic matrix and providing foundational data for subsequent descriptor screening efforts. Three well-recognized hydrogenation pathways are explored, including the formate (HCOO), formic acid (HCOOH) and reverse water–gas shift (RWGS) pathways. The relevant model structures and thermodynamic energy curves are diagramed in Fig. 3, and the reaction energies for each elementary reaction are given in Table 1. As shown in Fig. 3(a), the adsorbed CO₂ molecule undergoes a series of transformations, including production of bi-HCOO, H₂COO, bi-H₂CO and CH₃O/CH₂OH, eventually culminating in the production of CH₃OH. During the hydrogenation of bi-H₂CO, two parallel routes exist, one involving hydrogen attacking the carbon site to form CH₃O species and the other involving hydrogen attacking the oxygen site to form CH₂OH species. The energy information is illustrated in the magnified inset. Adhering to the thermodynamic energy perspective, a higher absolute energy signifies more energy required for the completion of chemical reactions, which is unfavorable thermodynamically. Following this, the CH₃O route is more advantageous on pure Cu, Au–Cu, Ni–Cu, Pd–Cu and Zn–Cu substrates, while the CH₂OH route predominates on Pt–Cu substrates. As shown in the HCOOH pathway in Fig. 3(b), the CO₂ molecule is firstly converted into HCOOH or HCO + OH species and then transformed into bi-H₂CO for further hydrogenation. In the blue magnified inset, the HCOOH route is favored on pure Cu, Au–Cu, Pd–Cu, Pt–Cu and Zn–Cu systems, while the HCO + OH route exhibits thermodynamic advantages on the Ni–Cu model. In the yellow magnified inset, the comparison between CH₃O and CH₂OH routes is in line with the HCOO pathway. For the RWGS pathway in Fig. 3(c), the CO₂ molecule is hydrogenated to trans-COOH, followed by atomic rotation to form cis-COOH and decomposition into co-adsorption species containing CO and H₂O. The CHO species is produced by the hydrogenation of CO species, followed by further hydrogenation to yield either bi-H₂CO or CHOH species. Both of these species can be further hydrogenated to CH₃O and CH₂OH, and the subsequent pathways are consistent with the HCOO pathway.

Fig. 3 Thermodynamic energy profiles of methanol synthesis via (a) formate, (b) formic acid and (c) RWGS pathways, in which CH₃O vs. CH₂OH routes and HCOOH vs. CHO + OH routes are compared to screen the preferred reaction routes in thermodynamics.

Table 1 Reaction energies ΔE (eV) of each elementary step involved in methanol synthesis on Cu-based bimetallic alloys

Pathway	Number	Reaction	Cu	Au–Cu	Ni–Cu	Pd–Cu	Pt–Cu	Zn–Cu
HCOO	R1	CO2(g) + 3H2(g) → CO2* + 3H2(g)	−0.23	−0.25	−1.14	−0.43	−0.66	−0.23
	R2	CO2* + 3H2(g) → (CO2 + H)* + 2.5H2(g)	−0.36	−0.14	−0.49	−0.41	−0.40	−0.14
	R3	(CO2 + H)* + 2.5H2(g) → bi-HCOO* + 2.5H2(g)	−0.81	−0.62	−0.32	−0.52	−0.11	−0.97
	R4	bi-HCOO* + 2.5H2(g) → (bi-HCOO + H)* + 2H2(g)	0.20	−0.10	−0.53	−0.38	−0.62	−0.14
	R5	(bi-HCOO + H)* + 2H2(g) → H2COO* + 2H2(g)	−0.20	1.02	0.72	0.98	1.03	0.18
	R6	H2COO* + 2H2(g) → (H2COO + H)* + 1.5H2(g)	−0.21	−0.11	−0.58	−0.41	−0.66	−0.03
	R7	(H2COO + H)* + 1.5H2(g) → (bi-H2CO + OH)* + 1.5H2(g)	0.43	−0.56	0.41	0.07	0.04	0.34
	R8	(bi-H2CO + OH)* + 1.5H2(g) → (bi-H2CO + OH + H)* + H2(g)	−0.12	−0.11	−0.30	−0.13	−0.53	0.10
	R9	(bi-H2CO + OH + H)* + H2(g) → (bi-H2CO + H2O)* + H2(g)	−0.15	0.04	0.06	−0.43	−0.05	−0.24
	R10	(bi-H2CO + H2O)* + H2(g) → bi-H2CO* + H2(g) + H2O(g)	0.56	0.41	0.56	0.62	0.71	0.53
	R11	bi-H2CO* + H2(g) + H2O(g) → (bi-H2CO + H)* + 0.5H2(g) + H2O(g)	−0.27	−0.23	−0.55	−0.42	0.10	−0.11
	R12	(bi-H2CO + H)* + 0.5H2(g) + H2O(g) → CH3O* + 0.5H2(g) + H2O(g)	−0.62	−0.57	0.10	−0.09	−0.41	−1.04
	R13	CH3O* + 0.5H2(g) + H2O(g) → (CH3O + H)* + H2O(g)	−0.17	−0.10	−0.59	−0.44	−0.65	−0.16
	R14	(CH3O + H)* + H2O(g) → CH3OH* + H2O(g)	0.02	−0.58	0.52	0.00	0.15	0.07
	R15	(bi-H2CO + H)* + 0.5H2(g) + H2O(g) → CH2OH* + 0.5H2(g) + H2O(g)	−0.35	−0.29	0.59	0.25	0.12	−0.64
	R16	CH2OH* + 0.5H2(g) + H2O(g) → (CH2OH + H)* + H2O(g)	−0.28	−0.12	−0.55	−0.51	−0.65	−0.09
	R17	(CH2OH + H)* + H2O(g) → CH3OH* + H2O(g)	−0.62	−0.96	−0.04	−0.27	0.35	−0.82
HCOOH	R18	bi-HCOO* + 2.5H2(g) → (bi-HCOO + H)* + 2H2(g)	−0.30	−0.10	−0.53	−0.38	−0.62	−0.07
	R19	(bi-HCOO + H)* + 2H2(g) → HCOOH* + 2H2(g)	1.06	0.49	1.37	0.89	0.83	0.90
	R20	HCOOH* + 2H2(g) → (HCOOH + H)* + 1.5H2(g)	−0.39	−0.13	−0.54	−0.36	−0.62	−0.15
	R21	(HCOOH + H)* + 1.5H2(g) → (bi-H2CO + OH)* + 1.5H2(g)	−0.16	−0.01	−0.29	0.10	0.20	−0.32
	R22	(CO2 + H)* + 2.5H2(g) → trans-COOH* + 2.5H2(g)	0.12	0.28	0.29	0.21	0.15	−0.02
	R23	trans-COOH* + 2.5H2(g) → (trans-COOH + H)* + 2H2(g)	−0.33	−0.14	−0.54	−0.37	−0.62	0.15
	R24	(trans-COOH + H)* + 2H2(g) → (CHO + OH)* + 2H2(g)	0.31	0.53	−0.85	0.31	0.97	0.18
	R25	(CHO + OH)* + 2H2(g) → (CHO + OH + H)* + 1.5H2(g)	−0.46	−0.28	−0.30	−0.17	−0.53	−0.77
	R26	(CHO + OH + H)* + 1.5H2(g) → (bi-H2CO + OH)* + 1.5H2(g)	−0.24	−0.77	1.08	−0.24	−0.03	−0.18
RWGS	R27	trans-COOH* + 2.5H2(g) → cis-COOH* + 2.5H2(g)	−0.09	0.04	0.02	0.16	0.37	−0.05
	R28	cis-COOH* + 2.5H2(g) → (cis-COOH + H)* + 2H2(g)	−0.34	−0.39	−0.53	−0.37	−0.58	−0.11
	R29	(cis-COOH + H)* + 2H2(g) → (CO + H2O)* + 2H2(g)	−0.27	−0.34	−0.52	−0.72	−0.64	−0.32
	R30	(CO + H2O)* + 2H2(g) → CO* + 2H2(g) + H2O(g)	0.67	0.52	0.86	0.54	0.56	0.50
	R31	CO* + 2H2(g) + H2O(g) → (CO + H)* + 1.5H2(g) + H2O(g)	−0.32	−0.14	−0.51	−0.36	−0.61	−0.10
	R32	(CO + H)* + 1.5H2(g) + H2O(g) → CHO* + 1.5H2(g) + H2O(g)	0.60	0.47	0.92	0.91	1.22	0.32
	R33	CHO* + 1.5H2(g) + H2O(g) → (CHO + H)* + H2(g) + H2O(g)	−0.34	−0.12	−1.34	−0.39	−0.65	−0.12
	R34	(CHO + H)* + H2(g) + H2O(g) → bi-H2CO* + H2(g) + H2O(g)	−0.33	−0.38	0.83	−0.16	0.27	−0.34

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When comparing the trends in thermodynamic properties, the consistency of substrate preference becomes evident, with the order remaining Ni–Cu > Pt–Cu > Pd–Cu > pure Cu > Zn–Cu > Au–Cu among different mechanisms. The catalytic superiority of Ni–Cu in thermodynamic performance is demonstrated once again. A fascinating finding is that the substrate adsorption order for CO, CHO, bi-H₂CO and trans-COOH species closely mirrors the thermodynamic order mentioned above, and these intermediates, to some extent, can reflect the energy characteristics of the overall reaction. Given this discovery, these intermediates can be regarded as potential descriptors of thermodynamic features, with validation of CO species as a valuable descriptor in a prior study.²¹ To further elucidate the reasons behind the adsorption variations of these intermediates in different environments, CHO is taken as an example to present the charge density difference and electronic localization function (ELF), as depicted in Fig. 4(b). The VESTA software is utilized to present the charge density difference and ELF diagrams, wherein the yellow and cyan isosurfaces in charge density difference diagrams represent the charge accumulation and depletion regions, with default values of 2.0 × 10⁻⁴ and −2.0 × 10⁻⁴ e Bohr⁻³, respectively. In the Cu model, CHO species predominately exhibits an electrophilic state, accompanied by a notable accumulation of electrons. In the Ni–Cu model, the electron accumulation and depletion regions intertwine intricately, signifying substantial charge migration facilitated by the introduction of Ni and therefore the improvement of intermediate adsorption. Conversely, the Au–Cu model is primarily characterized by electron depleted regions, with limited charge transfer and weaker adsorption compared to Ni–Cu. In terms of ELF distribution, the peaks are arranged in the descending order of Ni–Cu > Pt–Cu > Pd–Cu > pure Cu > Zn–Cu > Au–Cu, consistent with the summarized adsorption performance order.

Fig. 4 Comparison of adsorption strength of reaction intermediates on Cu-based substrates, and property illustration for the adsorption performance (taking CHO species as an example). In the charge density difference diagrams, the yellow and cyan isosurfaces refer to the charge accumulation and depletion regions.

3.2.2 Adsorption performance of reaction intermediates. On the basis of the optimal adsorption structures, the adsorption energies of all intermediates are summarized in Fig. 4(a) for comparison. It can be observed that the energy distribution of each intermediate is relatively concentrated without significant energy deviation, and as such, the average energy values are employed as a metric to evaluate the adsorption performance. According to adsorption strength, the reaction intermediates can be categorized into three groups. The first comprises strongly adsorbed species with average energies exceeding 3 eV, including H₂COO (−5.09 eV), O (−5.03 eV), bi-HCOO (−3.65 eV) and OH (−3.59 eV). The second encompasses moderately adsorbed species with average energies ranging from 1 to 3 eV, including CH₃O (−2.82 eV), H (−2.63 eV), cis-COOH (−2.58 eV), trans-COOH (−2.57 eV), CHO (−2.27 eV), CH₂OH (−2.00 eV), CO (−1.59 eV) and bi-H₂CO (−1.10 eV). The third consists of weakly adsorbed species with average energies below 1 eV, including CH₃OH (−0.78 eV), HCOOH (−0.76 eV), H₂O (−0.63 eV), mono-H₂CO (−0.60 eV), CO₂ (−0.49 eV) and H₂ (−0.49 eV). When focusing on the impact of different substrates on adsorption, the following conclusion can be drawn: except for H, H₂ and CH₂OH species, all other intermediates exhibit the strongest adsorption on the Ni–Cu surface, which reaffirms the viewpoint discussed in Section 3.1 that Ni doping improves adsorption strength. Regarding H, H₂ and CH₂OH species, their adsorption strength on Ni–Cu is slightly lower than that on Pt–Cu, which can be partially attributed to the adsorption structure. For example, H₂ is adsorbed in a molecular form on Ni–Cu and in an atomic form on Pt–Cu, and the energy difference between them may originate from the dissociation energy of the H₂ molecule.

3.3. Exploration of scaling relationships during methanol synthesis

3.3.1 Scaling relationships of adsorption energy represented by individual intermediates. The central goal of establishing an adsorption energy scaling relationship is to develop a universal descriptor capable of effectively summarizing the thermodynamic behaviors of all or most intermediates within the entire hydrogenation reaction. This serves the purpose of streamlining the thermodynamic analysis of CO₂ reduction. Starting from the individual intermediate descriptors, the determination coefficients between all pairs of intermediates are compiled in Table S1 (ESI†), of which 24 pairs of combination with R² values exceeding 0.9 are showcased in Fig. S8 (ESI†). In the two-dimensional mapping diagrams of most adsorbates, Cu-based substrates modified with Ni, Pt and Pd conspicuously exhibit strong adsorption, primarily concentrated in the lower-left regions. Conversely, Cu-based substrates modified with Au and Zn display weak adsorption and are concentrated in the upper-right regions. These two categories are demarcated by a pure Cu substrate, and the adsorption energy distribution is basically concordant with the sequence elaborated in Section 3.2. From the aforementioned pairs, the specific intermediate combinations completely satisfying the thermodynamic order and remarkably high correlations (R² > 0.95) are identified, including CHO∼CO (0.95), bi-H₂CO∼CHO (0.95) and trans-COOH∼CHO (0.99). The intermediates appearing in these pairs encompass CO, CHO, bi-H₂CO and trans-COOH, all of which belong to the potential descriptors proposed in Section 3.2.2. It can be therefore inferred that these intermediates collectively reflect the thermodynamic properties of the overall reactions, which precisely accounts for their high mutual correlations. Furthermore, among the three pairs, CHO species demonstrates a highly consistent adsorption performance alongside CO, bi-H₂CO and trans-COOH, indicating that CHO species not only aptly represents the thermodynamic characteristics but also maintains a significant linear energy correlation with key intermediates. In summary, the adsorption energy of CHO species is proved to be an effective descriptor in the individual intermediate study, marking a significant milestone in its debut in the CO₂ reduction reaction research. Then, CHO is further applied in the subsequent investigations involving dual intermediates.

3.3.2 Scaling relationships of adsorption energy represented by dual intermediates. Referring to the methodology for constructing energy descriptors in ref. 21, the dual intermediate descriptors are obtained by directly summing the respective adsorption energies of two separate species, with a total of 306 possible combinations. Regarding the expression of dual intermediate descriptors, we adopt the format of “intermediate 1” followed by the “&” symbol and then “intermediate 2”, which signifies the sum of the adsorption energies of intermediate 1 and intermediate 2 individually on the catalytic surface. The correlation evaluation results are presented in Table S2 (ESI†). Amid the diverse array of dual intermediates, CHO&cis-COOH and HCOOH&cis-COOH have the highest R² value of 0.76, with cis-COOH serving as the common element. Following closely behind, there are 21 pairs with an average R² value of 0.75, slightly lower than the above value, specifically cis-COOH&H, H₂&O, CO&O, bi-HCOO&H₂, CHO&H₂O, bi-H₂CO&H₂O, cis-COOH&H₂O, bi-HCOO&CO, cis-COOH&CO, CHO&CO₂, bi-H₂CO&CO₂, trans-COOH&CO₂, bi-H₂CO&CHO, CH₂OH&bi-HCOO, bi-H₂CO&mono-H₂CO, trans-COOH&bi-H₂CO, cis-COOH&bi-H₂CO, cis-COOH&trans-COOH, CHO&CH₃OH, bi-H₂CO&CH₃OH and cis-COOH&CH₃OH. Further categorization of these combinations based on their components is presented in Table S3 (ESI†), where bi-H₂CO and cis-COOH species show higher frequencies, occurring 7 and 6 times, respectively. Considering the high correlations manifested by cis-COOH and bi-H₂CO, it may be summarized that these two elements are of great significance in the hydrogenation process, even though they may not exhibit as prominent scaling relationships as CHO species in the individual intermediate analysis. Their real function is to harmonize with other intermediates and enhance the conjunct linear correlations of adsorption energy when paired with other species. Nevertheless, not all reaction species are necessary to be considered in the actual catalytic research, that is, adjustments can be made according to different reaction mechanisms, allowing the exclusion of irrelevant intermediates to improve the linear accuracy of dual intermediate descriptors. One more point, for a more profound understanding of energetic characterization, a thorough analysis of both thermodynamics and kinetics is indispensable. Limited to the focus of this work solely on thermodynamic properties without delving into kinetics, previous literature reports are therefore recommended to elucidate the kinetic advantages and disadvantages of different reaction mechanisms. Building on insights from similar systems,^41,59–61 the activation energy of the rate-determining step in the RWGS pathway tends to be lower than that in HCOO and HCOOH pathways, which suggests that the RWGS pathway holds a kinetic advantage and is positioned as the primary route for hydrogenation.

Table 2 enumerates the top ten combinations of dual intermediates in HCOO, HCOOH and RWGS pathways, along with the corresponding average determination coefficients. In view of the comparative study, the descriptors within the RWGS pathway display a robust linear fitting degree (all exceeding 0.86), which is superior to those of HCOO (ranging from 0.743 to 0.753) and HCOOH pathways (ranging from 0.708 to 0.713). Compared to the highest R² of 0.76 recorded in Table S2 (ESI†), the dual intermediate descriptors have a better linear relationship with intermediates in the RWGS pathway, with values increasing from 0.76 to 0.865. On the other hand, there is a slight decline in the fitting degree observed in HCOO and HCOOH pathways. Such discrepancy may be attributed to the variations of involved intermediate species. In the RWGS pathway, bi-HCOO and HCOOH species are not included, which provide poor fitting results for HCOO and HCOOH pathways (determination coefficients of 0.52 for bi-HCOO and 0.68 for HCOOH). Additionally, the highly correlated intermediates, such as trans-COOH (0.72) and cis-COOH (0.72), are included in the RWGS pathway but are coincidentally absent in the other two pathways. As a result, the RWGS pathway is applied for the following investigations as the most favorable route, which not only features the lowest activation energy of the rate-determining step but also provides an accurate presentation of intermediate adsorption behavior via dual intermediate descriptors. Returning to Table 2, the descriptor formed by cis-COOH and bi-H₂CO presents the highest fitting value of 0.713 in the HCOO pathway, followed by cis-COOH&CH₃OH (0.710) and CO&O (0.709). In the HCOOH pathway, HCOOH&cis-COOH accurately depicts the adsorption performance during the hydrogenation process, achieving an R² value of 0.753, basically consistent with CO&O. In the RWGS pathway, CHO&CH₃OH and CHO&H₂O descriptors perform remarkably with R² values as high as 0.865, reiterating the remarkable superiority of the CHO descriptor in terms of linear correlation. Although the CO&O descriptor shows substantial linear correlation in HCOO and HCOOH pathways, its reduced applicability in the RWGS pathway implies that the suitability varies depending on different scenarios and CO&O is not appropriate as a universal descriptor. In the subsequent study, the combination of CHO&CH₃OH, boasting the highest R², is selected as the core descriptor, and the linear fitting results with different intermediates in the RWGS pathway are illustrated in Fig. 5(a) and (b). Notably, all reaction intermediates present strong correlations with CHO&CH₃OH, such as 0.96 for CO, 0.88 for H, 0.88 for CO₂, 0.81 for H₂O, 0.99 for CHO, 0.98 for trans-COOH, 0.83 for cis-COOH, 0.96 for bi-H₂CO, 0.87 for CH₂OH and 0.94 for CH₃OH. Moreover, the slopes of fitted equations for all scaling relationships are positive, suggesting a parallel rather than a competitive relationship between CHO&CH₃OH and adsorption energies of other intermediates.

Table 2 Scaling relationships of adsorption energy between dual intermediate descriptors and the corresponding intermediates in different hydrogenation pathways, along with R² for comparison

HCOO pathway		HCOOH pathway		RWGS pathway
Dual intermediates	R ^2	Dual intermediates	R ^2	Dual intermediates	R ^2
cis-COOH&bi-H2CO	0.713	HCOOH&cis-COOH	0.753	CHO&CH3OH	0.865
cis-COOH&CH3OH	0.710	CO&O	0.753	CHO&H2O	0.865
CO&O	0.709	bi-H2CO&CO2	0.751	trans-COOH&HCOOH	0.864
trans-COOH&bi-HCOO	0.709	bi-H2CO&H2O	0.750	CO&cis-COOH	0.864
trans-COOH&O	0.709	bi-H2CO&CH3OH	0.748	trans-COOH&CH3OH	0.863
CHO&O	0.708	cis-COOH&bi-H2CO	0.747	trans-COOH&H2O	0.862
CH2OH&bi-HCOO	0.708	bi-H2CO&CHO	0.746	trans-COOH&bi-H2CO	0.862
CH2OH&O	0.708	bi-H2CO&HCOOH	0.745	bi-HCOO&H2	0.861
cis-COOH&H2O	0.708	CHO&cis-COOH	0.743	bi-H2CO&CHO	0.861
bi-HCOO&CH2OH	0.708	bi-H2CO&mono-H2CO	0.743	CHO&HCOOH	0.860

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Fig. 5 Linear scaling relationships between dual intermediate descriptors and the corresponding intermediates in the RWGS pathway, wherein the combination of CHO with CH₃OH is recorded in Fig. 5(a) and (b) and the combination of CO with O is recorded in Fig. 5(c) and (d) for comparison.

3.3.3 Scaling relationships of reaction energy represented by the CHO&CH₃OH descriptor. The above study focuses on the advantages of the CHO&CH₃OH descriptor for characterizing the adsorption energy, and then an evaluation of its feasibility and accuracy is carried out in this subsection from a reaction energy perspective. The reaction energies of each elementary reaction, initially determined using DFT methods, are successively expressed by CHO&CH₃OH, and the relevant substitution expressions and energy values are detailed in Table 3, with graphical representations of the linear relationships provided in Fig. 6. Therein, the correction factors are introduced to rectify the inaccuracies arising from the co-adsorption of multiple adsorbates. In the energy comparison, a majority of absolute errors remain within 0.3 eV, where all data points are closely distributed around the fitted line with only a few outliers. This finding suggests that the energies derived from the scaling relationships are basically in agreement with the actual values obtained by DFT calculations throughout the entire hydrogenation process. For Au–Cu, Pd–Cu and Pt–Cu models, the coefficients of determination all exceed 0.9, with the specific values of 0.92, 0.96 and 0.92. In the Au–Cu model, the maximum absolute error between DFT calculations and predictions based on the scaling relationship is 0.26 eV, corresponding to the hydrogenation of cis-COOH species (R28), while the absolute errors of the remaining reaction steps are lower than 0.2 eV. The Pd–Cu model demonstrates the highest linear fit, in which all absolute errors are constrained within 0.2 eV. In other words, the doping of Pd allows for a more precise estimation of reaction energies synergistically expressed by CHO and CH₃OH among various bimetallic alloys, offering novel theoretical support for previous studies^41,62,63 that CHO and CH₃OH species play pivotal roles in methanol synthesis. In the Pt–Cu model, except for the hydrogenation of CH₂OH in R17 (with an absolute error of 0.37 eV), the reaction energies derived from CHO&CH₃OH show not much difference with DFT calculations. Energy discrepancies range from 0.04 eV (R28) to 0.24 eV (R15), and the average error is 0.15 eV. In pure Cu and Ni–Cu models, R² values are 0.89 and 0.85, respectively, presenting a slight decrease in linear fit. Discrete points deviating from the fitting line, evident in Fig. 6(c), are linked to the co-adsorption of CHO with H (R33) and the generation of bi-H₂CO (R34). In the Zn–Cu model, the determination coefficient decreases to 0.75, which primarily results from the significant energy deviation between the predicted and actual values during the hydrogenation of bi-H₂CO to CH₂OH (R15), leaving an energy error as high as 0.5 eV. To summarize, the fitting equations incorporating the CHO&CH₃OH descriptor are proved to be relatively accurate in predicting the reaction energy, which holds valuable theoretical guidance for comprehending the thermodynamics of CO₂ hydrogenation and improving catalyst design. However, there are still cases where predictions for specific reactions deviate noticeably, in which case a cautious assessment of the suitability of scaling relationships becomes imperative.

Table 3 Comparison of reaction energies obtained by DFT calculation with prediction values represented by the CHO&CH₃OH descriptor in the RWGS pathway

Number	Reaction	Reaction energy formulas by CHO&CH3OH descriptor
R1	CO2(g) + 3H2(g) → CO2* + 3H2(g)	ΔE = 0.60[Ead(CHO) + Ead(CH3OH)] + 1.33
R2	CO2* + 3H2(g) → (CO2 + H)* + 2.5H2(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.76
R22	(CO2 + H)* + 2.5H2(g) → trans-COOH* + 2.5H2(g)	ΔE = −0.12[Ead(CHO) + Ead(CH3OH)] − 0.18
R27	trans-COOH* + 2.5H2(g) → cis-COOH* + 2.5H2(g)	ΔE = −0.14[Ead(CHO) + Ead(CH3OH)] − 0.36
R28	cis-COOH* + 2.5H2(g) → (cis-COOH + H)* + 2H2(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.77
R29	(cis-COOH + H)* + 2H2(g) → (CO + H2O)* + 2H2(g)	ΔE = 0.06[Ead(CHO) + Ead(CH3OH)] − 0.34
R30	(CO + H2O)* + 2H2(g) → CO* + 2H2(g) + H2O(g)	ΔE = −0.16[Ead(CHO) + Ead(CH3OH)] + 0.17
R31	CO* + 2H2(g) + H2O(g) → (CO + H)* + 1.5H2(g) + H2O(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.76
R32	(CO + H)* + 1.5H2(g) + H2O(g) → CHO* + 1.5H2(g) + H2O(g)	ΔE = −0.54[Ead(CHO) + Ead(CH3OH)] − 0.89
R33	CHO* + 1.5H2(g) + H2O(g) → (CHO + H)* + H2(g) + H2O(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.76
R34	(CHO + H)* + H2(g) + H2O(g) → bi-H2CO* + H2(g) + H2O(g)	ΔE = −0.42[Ead(CHO) + Ead(CH3OH)] − 1.42
R11	bi-H2CO* + H2(g) + H2O(g) → (bi-H2CO + H)* + 0.5H2(g) + H2O(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.78
R13	CH3O* + 0.5H2(g) + H2O(g) → (CH3O + H)* + H2O(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.77
R15	(bi-H2CO + H)* + 0.5H2(g) + H2O(g) → CH2OH* + 0.5H2(g) + H2O(g)	ΔE = −0.52[Ead(CHO) + Ead(CH3OH)] − 1.48
R16	CH2OH* + 0.5H2(g) + H2O(g) → (CH2OH + H)* + H2O(g)	ΔE = 0.37[Ead(CHO) + Ead(CH3OH)] + 0.76
R17	(CH2OH + H)* + H2O(g) → CH3OH* + H2O(g)	ΔE = −0.79[Ead(CHO) + Ead(CH3OH)] − 2.81

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Number	Cu		Au–Cu		Ni–Cu		Pd–Cu		Pt–Cu		Zn–Cu
	DFT/eV	Prediction/eV	DFT/eV	Prediction/eV	DFT/eV	Prediction/eV	DFT/eV	Prediction/eV	DFT/eV	Prediction/eV	DFT/eV	Prediction/eV
R1	−0.23	−0.33	−0.25	−0.12	−1.14	−0.99	−0.43	−0.53	−0.66	−0.79	−0.23	−0.23
R2	−0.36	−0.26	−0.14	−0.14	−0.49	−0.67	−0.41	−0.39	−0.40	−0.55	−0.14	−0.20
R22	0.12	0.15	0.28	0.11	0.29	0.28	0.21	0.19	0.15	0.24	−0.02	0.13
R27	−0.09	0.03	0.04	−0.02	0.02	0.18	0.16	0.07	0.37	0.14	−0.05	0.00
R28	−0.34	−0.25	−0.39	−0.13	−0.53	−0.66	−0.37	−0.38	−0.58	−0.54	−0.11	−0.19
R29	−0.27	−0.51	−0.34	−0.49	−0.52	−0.57	−0.72	−0.53	−0.64	−0.55	−0.32	−0.50
R30	0.67	0.61	0.52	0.56	0.86	0.79	0.54	0.67	0.56	0.74	0.50	0.59
R31	−0.32	−0.26	−0.14	−0.14	−0.51	−0.67	−0.36	−0.39	−0.61	−0.55	−0.10	−0.20
R32	0.60	0.61	0.47	0.42	0.92	1.20	0.91	0.79	1.22	1.02	0.32	0.51
R33	−0.34	−0.26	−0.12	−0.14	−1.34	−0.67	−0.39	−0.39	−0.65	−0.55	−0.12	−0.20
R34	−0.33	−0.26	−0.38	−0.40	0.83	0.20	−0.16	−0.12	0.27	0.07	−0.34	−0.33
R11	−0.27	−0.24	−0.11	−0.12	−0.55	−0.65	−0.41	−0.37	−0.64	−0.53	−0.11	−0.18
R13	−0.17	−0.25	−0.10	−0.13	−0.59	−0.66	−0.44	−0.38	−0.65	−0.54	−0.16	−0.19
R15	−0.35	−0.04	−0.29	−0.22	0.59	0.53	0.25	0.13	0.12	0.36	−0.64	−0.13
R16	−0.28	−0.26	−0.12	−0.14	−0.55	−0.67	−0.51	−0.39	−0.65	−0.55	−0.09	−0.20
R17	−0.62	−0.62	−0.96	−0.90	−0.04	0.24	−0.27	−0.36	0.35	−0.02	−0.82	−0.76

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Fig. 6 Mapping relationships between reaction energy calculated using a dual intermediate descriptor of CHO&CH₃OH and DFT results on (a) Cu, (b) Au–Cu, (c) Ni–Cu, (d) Pd–Cu, (e) Pt–Cu and (f) Zn–Cu substrates.

3.4. Multiple linear regression (MLR) analysis

The dual intermediate descriptor relates only to the energy characteristics of CHO and CH₃OH species, lacking other information about the reaction process. It is therefore expected that the limitations in linear fitting of reaction energy expressions can be compensated by supplementing additional variables to improve the simulation accuracy. Referring to our previous work,⁶⁴ the MLR method with the same linear characteristics as the above descriptor is able to enhance the predictive capacity aided by the supplemental eigenvalues, and this subsection focuses on the investigation of the MLR method for thermodynamic energy prediction. Note that the thermodynamic energies addressed in this section share the same fundamental attributes as the reaction energy investigated in Section 3.3, both of which represent the thermodynamic energy characteristic data.

3.4.1 Extraction of additional descriptors and establishment of the dataset. A set of 14 property parameters are consolidated as the fundamental data for the MLR study, drawing from the types of common features in prior scientific research. These parameters can be classified into three categories according to the properties, as elaborated in Table 4. The first category is related to the intrinsic properties of reaction intermediates, including the atomic number, molar mass, electronegativity, electron affinity, outermost electron count, first ionization energy, total charge quantity and work function. The second category is the charge and structural properties of TM–Cu alloy substrates, encompassing the overall charge quantity, work function and deformation degree of the substrate before and after the reaction. The third category takes the intermediate and substrate as an integrated whole, representing the properties of the entire reaction system. It contains the transferred charge quantity, work function difference and atomic spatial distance between the both. The correlation heatmap regarding the interrelationships among all data is presented in Fig. 7, where the color legend manifests the strength of linear correlation, and a lighter color signifies a stronger positive correlation. With thermodynamic energy calculated by DFT methods as a reference, two elements demonstrate a strong positive correlation, i.e., Chg_Sub (PCC of 0.95) and SD_Sub (PCC of 0.66). Conversely, the correlation between other elements and DFT results is relatively weak, with PCC values spanning a range from −0.29 to 0.20. Although the compelling PCC value of 0.95 for Chg_Sub signifies that the charge quantity carried by the substrate seemingly serves as a direct estimator of thermodynamic energy during CO₂ reduction, the utilization of Chg_Sub may lead to overfitting issues. To prevent the potential interference with the evaluation study, Chg_Sub is therefore excluded from further analysis. Instead of this, SD_Sub shows a moderate yet viable correlation with the target variable, rendering it an acceptable input in MLR investigations. Fig. 7 also unveils ten pairs of elements with correlation exceeding 0.9, which are AN_IM ∼ EN_IM, AN_IM ∼ EA_IM, AN_IM ∼ FIE_IM, MM_IM ∼ OE_IM, MM_IM ∼ Chg_IM, EN_IM ∼ EA_IM, EN_IM ∼ FIE_IM, EA_IM ∼ FIE_IM, OE_IM ∼ Chg_IM and WF_IM ∼ WF_IM-Sub. Interestingly, apart from WF_IM ∼ WF_IM-Sub, all other nine combinations reflect the inherent properties of the intermediates themselves, as highlighted in the enlarged region. The prevailing overfitting situation is primarily due to the substantial repetitive information intricately interwoven in the above features. To ensure the data independence, a discerning approach necessitates the removal of certain descriptors, despite their modest correlation with the target variable. An exhaustive vetting process culminates in the exclusion of AN_IM, MM_IM, EA_IM, OE_IM, FIE_IM and WF_IM-Sub as redundant data. The remaining seven features, namely EN_IM, Chg_IM, WF_IM, WF_Sub, SD_Sub, Chg_IM-Sub and BL_IM-Sub, are collected to constitute the optimized dataset as input for further research endeavors.

Table 4 Summary of elements in the MLR dataset, including the intermediate part, substrate part, intermediate-substrate part and SISSO part

Category	Name	Abbreviation
Intermediate	Atom number of intermediate	ANIM
	Molar mass of intermediate	MMIM
	Electronegativity of intermediate	ENIM
	Electron affinity of intermediate	EAIM
	Outer electron of intermediate	OEIM
	First ionization energy of intermediate	FIEIM
	Bader charge of intermediate	ChgIM
	Work function of intermediate	WFIM
Substrate	Bader charge of substrate	ChgSub
	Work function of substrate	WFSub
	Shape deformation of substrate	SDSub
Intermediate and substrate	Bader charge between intermediate and substrate	ChgIM-Sub
	Work function between intermediate and substrate	WFIM-Sub
	Bond length between intermediate and substrate	BLIM-Sub
SISSO	SDSub + (BLIM-Sub/WFSub)	SISSO1
	SDSub × (SDSub/WFSub)	SISSO2
	(ENIM × WFIM) − BLIM-Sub	SISSO3
	BLIM-Sub + (ENIM × WFIM)	SISSO4
	WFSub × (WFSub × BLIM-Sub)	SISSO5
	(WFSub × BLIM-Sub) − BLIM-Sub	SISSO6
	(WFSub × BLIM-Sub) − SDSub	SISSO7
	1.16 × 10^6 × SISSO1 + 1.01 × 10^3 × SISSO2 + 8.79 × 10^5 × SISSO3	SISSO8
	−8.79 × 10^5 × SISSO4 − 1.21 × 10^4 × SISSO5 − 9.96 × 10^5 × SISSO6
	+1.16 × 10^6 × SISSO7 − 2.73 × 10^2

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Fig. 7 Heat map of PCC between absolute energies obtained by DFT calculations and characteristic parameters, along with an enlarged view of overfitting data. The closer the absolute value reaches 1, the stronger the correlation between two features. On the contrary, the weaker the correlation between two features. AN_IM, MM_IM, EN_IM, EA_IM, OE_IM, FIE_IM, Chg_IM, WF_IM, Chg_Sub, WF_Sub, SD_Sub, Chg_IM-Sub, WF_IM-Sub and BL_IM-Sub represent the atom number of intermediates, molar mass of intermediates, electronegativity of intermediates, electron affinity of intermediates, outer electron of intermediates, first ionization energy of intermediates, charge of intermediates, work function of intermediates, charge of substrates, work function of substrates, shape deformation of substrates, charge difference between intermediates and substrates, work function difference between intermediates and substrates, and bond length between intermediates and substrates, respectively.

3.4.2 MLR evaluation after removing the over-related descriptors. The corrected heatmap after removing the over-related descriptors is presented in Fig. 8, in which five pairs are notably correlated (labeled in red). These pairs consist of DFT results ∼SD_Sub, EN_IM ∼ Chg_IM, Chg_IM-Sub ∼ EN_IM, Chg_IM-Sub ∼ Chg_IM and Chg_IM-Sub ∼ SD_Sub. As previously discussed, the linear correlation observed between SD_Sub and DFT results prompts us to reevaluate the potential impact of substrate properties on adsorption reactions. Following this insight, the thermodynamic performance can be finely tuned through adjusting the substrate morphology. The positive correlation, as indicated by a PCC value of 0.66, implies that as substrate structure variations become more pronounced, the absolute values of thermodynamic energy in the reaction system decrease, resulting in weaker adsorption capacity. From an experimental standpoint, this can be explained as follows: Cu-based catalysts are attacked by hydrogen during the hydrogenation process, and structures with lower stability undergo drastic morphological changes, which make it difficult to maintain the capture of adsorbates, ultimately leading to reduced adsorption capacity. The high correlation between EN_IM and Chg_IM can be attributed to both descriptors belonging to the category of electronic properties of intermediates, sharing a mutual property association. The remaining three pairs are centered around the charge difference between the substrate and intermediate (Chg_IM-Sub), manifesting as the coupling of internal electron transfer with other factors. Next, energy predictions are conducted on a total of 192 thermodynamic data points using the MLR method, and the data information is detailed in Table S4 (ESI†). Fig. 9(a) depicts the prediction capacity of the MLR method for DFT data, with its MAE values of 13.04 and linear R² of 0.76. These results suggest that the MLR model exhibits a relatively high prediction error for thermodynamic data without any modification, which cannot be employed to predict the thermodynamic properties of CO₂ hydrogenation on Cu-based substrates. As shown in the diagrams, only a few data points strictly coincide with the fitted line, and the majority are discretely distributed, even though some points exhibit significant deviations. Therefore, more attention should be paid to additional auxiliary methods for enhancing the assessment precision of the MLR model.

Fig. 8 Correlation between absolute thermodynamic energies obtained by DFT calculations and filtered parameters after removing over-related descriptors.

Fig. 9 The fitting results of the calculated energy and predicted energy for absolute thermodynamic values by (a) the MLR method and under the guidance of (b) the SISSO method and (c) the CHO&CH₃OH descriptor, in which the values of R² and MAE are computed to evaluate the prediction performance. The inset shows the data histogram distribution. (d) Comparison of energy profiles via the RWGS pathway on the Ni–Cu substrate by DFT calculations, the SISSO method and the CHO&CH₃OH descriptor, respectively.

3.4.3 Predictive comparison between SISSO and CHO&CH₃OH strategies. Given the findings from Section 3.4.2, which indicates that relying solely on the conventional physicochemical features as input information fails to achieve the expected predictive performance, two strategies are proposed to enhance the predictive accuracy. One introduces the dual intermediate descriptor of CHO&CH₃OH into the dataset. The other involves incorporating the linear summary descriptor obtained through the SISSO method based on the features in Table 4 into the input dataset. This method has been validated in prior research for delivering dependable predictive accuracy in related issues,^65–67 and the specific SISSO calculation details are recorded in the ESI,† for reference. Its inclusion in this study is designed for a comparative examination with the CHO&CH₃OH descriptor. Such arrangement allows us to assess whether the dual intermediate descriptor developed from thermodynamic scaling relationships demonstrates efficacy similar to the SISSO method. Concurrently, it serves to validate the applicability of the dual intermediate descriptor in the MLR study. The impact of the SISSO strategy on MLR is first explored. The research object is the 114 thermodynamic data points appearing in the RWGS pathway, on the basis of which seven SISSO sub-descriptors (SISSO₁ to SISSO₇) and one comprehensive descriptor (SISSO₈) covering the entire basic information are obtained, with specific information presented in Table 4 and the relevant data recorded in Table S5 (ESI†), wherein SISSO₈ denotes the linear combination with the highest predictive performance based on the efficient elements (SISSO₁ to SISSO₇). SISSO₈ is introduced as a new descriptor into the dataset, and the prediction results are shown in Fig. 9(b). R² for MLR rises from 0.76 to 0.94, which indicates that SISSO₈ significantly reinforces the overall fitting performance of MLR. Additionally, considering the MAE results, the intervention of SISSO results in reduced prediction errors, reaching 4.84 for the MLR model. Despite this advancement, it is important to acknowledge that the MLR model has not achieved the desired precision typical of regression models. Thus, the import of comprehensive variables refined by the SISSO method as descriptors into the dataset proves beneficial for enhancing predictive capabilities to some extent, but achieving precise predictions remains a challenge.

Then, the CHO&CH₃OH descriptor is incorporated into the training set to compare the performance with the SISSO approach, and the predictive result is shown in Fig. 9(c). The corresponding MAE values show a reduction to 0.14, while the R² values are improved to 0.99, representing a significant enhancement. This improvement suggests that, with the support of the CHO&CH₃OH descriptor, MLR can provide accurate predictions for thermodynamic parameters. Moreover, Fig. 9(d) presents a comparison between the actual DFT calculation values and multiple linear regression predictions guided by SISSO₈ and CHO&CH₃OH descriptors, as showcased by the absolute thermodynamic energy values along the RWGS pathway on the Ni–Cu substrate. Evidently, the blue energy curve related to CHO&CH₃OH closely tracks the black actual values, with an average relative error of merely 0.16 eV. The largest energy difference is observed in the co-adsorption of CHO and H, with a difference of 0.73 eV. In contrast, the red energy curve related to SISSO₈ shows remarkable deviations, particularly in the case of CHO adsorption, with an error as high as 12.04 eV. The average relative error concerning DFT values is 1.44 eV, which is considerably higher than that of CHO&CH₃OH. Hence, in contrast to the traditionally effective SISSO method, the integration of dual intermediate descriptors into MLR research proves beneficial for achieving more precise predictive results. This is because CHO&CH₃OH represents the energy-linear relationships within the entire reaction system, thereby enriching the feature space and enhancing both robustness and generalization capabilities. The combination with multiple linear regression imparts potent predictive capabilities. This comprehensive approach will provide deeper insights and more reliable predictive results for thermodynamic studies focusing on copper-based catalysts.

3.5. Connections with previous work

The most significant point of interest throughout this work is the proposition of dual intermediate descriptor CHO&CH₃OH, tailored for CO₂ hydrogenation on Cu-based bimetallic alloys. Although such a descriptor excels in this specific reaction system, its applicability to other systems remains uncertain. It is therefore crucial to further examine its accuracy and suitability, and the CO&O descriptor proposed in ref. 21 is selected as a reference for comparative analysis. Fig. 5(c) and (d) show the scaling relationships between the CO&O descriptor and various reaction intermediates. Compared with Fig. 5(a) and (b), CO&O is found to exhibit weaker linear fits with all intermediates, except for bi-H₂CO and cis-COOH, that is, despite its weaker performance than CO&O on individual intermediates, CHO&CH₃OH still overwhelmingly dominates in the overall reaction. This linear distribution pattern can be partially explained by the molecular structure characteristics. Unlike the CO&O descriptor that lacks the hydrogen element, both CHO and CH₃OH components in CHO&CH₃OH comprise all elements involved in methanol synthesis, including C, H and O, making it formally closer to the reaction intermediates. Additionally, CH₃OH can be regarded as the hydrogenated product of CHO species, and from this point of view, CHO&CH₃OH is essentially the composite form of CHO and thus features better linear behavior. To ascertain whether this pattern holds in different model systems, a weight analysis is conducted using the energy data from ref. 21. The results demonstrate that among the ten reported intermediates, CHO&CH₃OH reveals superior linear relationships with the adsorption energies of H, CO, CO₂, OH, CHO, CH₃O and CH₃OH than CO&O. As for O, H₂O and CH₂O species, the linear difference between CO&O and CHO&CH₃OH is minimal, as detailed in Table S6 (ESI†). In addition, another comparison with CO&O on Cu(1 1 1) substrates modified with Ga, Mg and Ti is recorded in Table S7 (ESI†).⁶⁸ It is evident that, for all intermediates presented in the table, the CHO&CH₃OH descriptor consistently exhibits a superior linear performance compared to CO&O. This observation reaffirms the reliability of CHO&CH₃OH in depicting the thermodynamic properties. In light of this, the CHO&CH₃OH descriptor retains its superiority in fitting adsorption energy, even in different model systems, such as the cluster structures, different facets and the bimetallic structures. This study underscores the pivotal role of linear descriptors in the investigation of reaction performance, as summarized in ref. 69, simultaneously providing a prime example of the superiority of linear descriptors over SISSO descriptors.

4. Conclusion

In this work, the role of the CHO&CH₃OH descriptor in CO₂ conversion towards methanol catalyzed by Cu, Au–Cu, Ni–Cu, Pd–Cu, Pt–Cu and Zn–Cu was studied. A complete dataset consisting of 19 intermediates and three reaction pathways tailored for predicting the thermodynamic energy in CO₂ reduction is established, in accordance with the adsorption characteristics of reaction species on the surface of copper-based substrates. By examining the scaling relationships, it is found that all adsorption energies of reaction species are highly correlated with the dual intermediate pairing of CHO&CH₃OH. Moreover, in comparison to the previously reported CO&O descriptor, CHO&CH₃OH outperforms in terms of linear behaviors, and it is therefore refined as an updated descriptor for energy representation. Furthermore, the fundamental linear correlation between the CHO&CH₃OH descriptor and reaction energies during methanol synthesis is also identified. In MLR analysis, the integration of the CHO&CH₃OH descriptor into the dataset is proved to yield accurate predictive performance comparable to the DFT results and outperforms the SISSO method, embodied in the remarkable MAE of 0.14 and R² of 0.99. Such amalgamation of MLR with linear descriptors may provide insightful inspiration for the construction of foundational frameworks in subsequent energy research. Overall, the proposed linear scaling relationship based on the CHO&CH₃OH descriptor provides valuable insight for designing improved copper-based bimetallic catalysts, meanwhile illuminating the development directions of linear relationships in CO₂ reduction research.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 52276127).

References

1. S. Dang, H. Yang, P. Gao, H. Wang, X. Li, W. Wei and Y. Sun, A review of research progress on heterogeneous catalysts for methanol synthesis from carbon dioxide hydrogenation, Catal. Today, 2019, 330, 61–75.
2. X. Jiang, X. Nie, X. Guo, C. Song and J. G. Chen, Recent advances in carbon dioxide hydrogenation to methanol via heterogeneous catalysis, Chem. Rev., 2020, 120, 7984–8034.
3. G. A. Olah, Beyond oil and gas: the methanol economy, Angew. Chem., Int. Ed., 2005, 44, 2636–2639.
4. N. Pasupulety, H. Driss, Y. A. Alhamed, A. A. Alzahrani, M. A. Daous and L. Petrov, Studies on Au/Cu–Zn–Al catalyst for methanol synthesis from CO₂, Appl. Catal., A, 2015, 504, 308–318.
5. M. Cheng, E. L. Clark, H. H. Pham, A. T. Bell and M. Head-Gordon, Quantum mechanical screening of single-atom bimetallic alloys for the selective reduction of CO₂ to C1 hydrocarbons, ACS Catal., 2016, 6, 7769–7777.
6. Q. Tan, Z. Shi and D. Wu, CO₂ hydrogenation to methanol over a highly active Cu–Ni/CeO₂-Nanotube Catalyst, Ind. Eng. Chem. Res., 2018, 57, 10148–10158.
7. A. Alvarez-Garcia, E. Flórez, A. Moreno and C. Jimenez-Orozco, CO₂ activation on small Cu-Ni and Cu-Pd bimetallic clusters, Mol. Catal., 2020, 484, 110733.
8. Y. Xu, Z. Ding, R. Qiu and R. Hou, Effect of support and reduction temperature in the hydrogenation of CO₂ over the Cu–Pd bimetallic catalyst with high Cu/Pd ratio, Int. J. Hydrogen Energy, 2022, 47, 27973–27985.
9. A. G. Nabi, R. Aman Ur, A. Hussain, G. A. Chass and D. Di Tommaso, Optimal icosahedral copper-based bimetallic clusters for the selective electrocatalytic CO₂ conversion to one carbon products, Nanomaterials, 2022, 13, 87.
10. A. C. Garcia, J. Moral-Vico, A. Abo Markeb and A. Sanchez, Conversion of carbon dioxide into methanol using Cu–Zn nanostructured materials as catalysts, Nanomaterials, 2022, 12, 999.
11. V. Deerattrakul, P. Dittanet, M. Sawangphruk and P. Kongkachuichay, CO₂ hydrogenation to methanol using Cu–Zn catalyst supported on reduced graphene oxide nanosheets, J. CO₂ Util., 2016, 16, 104–113.
12. H. Qin, K. Wang, H. Zhang, X. Wang and J. Wu, Understanding the effects of cluster restructuring on water gas shift reaction over Cu/TM (TM = Cu, Ni and Pt) under high concentration of CO, Appl. Surf. Sci., 2023, 613, 155954.
13. F. Liu, C. Wu and S. Yang, Strain and ligand effects on CO₂ reduction reactions over Cu-metal heterostructure catalysts, J. Phys. Chem. C, 2017, 121, 22139–22146.
14. A. Bagger, W. Ju, A. S. Varela, P. Strasser and J. Rossmeisl, Electrochemical CO₂ reduction: a classification problem, ChemPhysChem, 2017, 18, 3266–3273.
15. D. Li, F. Xu, X. Tang, S. Dai, T. Pu, X. Liu, P. Tian, F. Xuan, Z. Xu, I. E. Wachs and M. Zhu, Induced activation of the commercial Cu/ZnO/Al₂O₃ catalyst for the steam reforming of methanol, Nat. Catal., 2022, 5, 99–108.
16. Z.-j Wang, H. Song, H. Pang, Y. Ning, T. D. Dao, Z. Wang, H. Chen, Y. Weng, Q. Fu, T. Nagao, Y. Fang and J. Ye, Photo-assisted methanol synthesis via CO₂ reduction under ambient pressure over plasmonic Cu/ZnO catalysts, Appl. Catal., B, 2019, 250, 10–16.
17. S. A. Akhade, R. M. Nidzyn, G. Rostamikia and M. J. Janik, Using Brønsted-evans-polanyi relations to predict electrode potential-dependent activation energies, Catal. Today, 2018, 312, 82–91.
18. D. Y. Jo, M. W. Lee, H. C. Ham and K.-Y. Lee, Role of the Zn atomic arrangements in enhancing the activity and stability of the kinked Cu(2 1 1) site in CH₃OH production by CO₂ hydrogenation and dissociation: first-principles microkinetic modeling study, J. Catal., 2019, 373, 336–350.
19. L. Dietz, S. Piccinin and M. Maestri, Mechanistic insights into CO₂ activation via reverse water-gas shift on metal surfaces, J. Phys. Chem. C, 2015, 119, 4959–4966.
20. M. M. Montemore and J. W. Medlin, Scaling relations between adsorption energies for computational screening and design of catalysts, Catal. Sci. Technol., 2014, 4, 3748–3761.
21. X. Zhang, J. Liu, B. Zijlstra, I. A. W. Filot, Z. Zhou, S. Sun and E. J. M. Hensen, Optimum Cu nanoparticle catalysts for CO₂ hydrogenation towards methanol, Nano Energy, 2018, 43, 200–209.
22. E. M. Fernandez, P. G. Moses, A. Toftelund, H. A. Hansen, J. I. Martinez, F. Abild-Pedersen, J. Kleis, B. Hinnemann, J. Rossmeisl, T. Bligaard and J. K. Norskov, Scaling relationships for adsorption energies on transition metal oxide, sulfide, and nitride surfaces, Angew. Chem., Int. Ed., 2008, 47, 4683–4686.
23. G. Jones, F. Studt, F. Abild-Pedersen, J. K. Nørskov and T. Bligaard, Scaling relationships for adsorption energies of C2 hydrocarbons on transition metal surfaces, Chem. Eng. Sci., 2011, 66, 6318–6323.
24. B. Xing, X. Pang and G. Wang, C–H bond activation of methane on clean and oxygen pre-covered metals: a systematic theoretical study, J. Catal., 2011, 282, 74–82.
25. J. L. C. Fajín, M. N. D. S. Cordeiro and J. R. B. Gomes, Methanol dissociation on bimetallic surfaces: validity of the general Brønsted–Evans–Polanyi relationship for O–H bond cleavage, RSC Adv., 2016, 6, 18695–18702.
26. W. D. Williams, J. P. Greeley, W. N. Delgass and F. H. Ribeiro, Water activation and carbon monoxide coverage effects on maximum rates for low temperature water–gas shift catalysis, J. Catal., 2017, 347, 197–204.
27. J. R. B. Gomes, F. Vines, F. Illas and J. L. C. Fajin, Implicit solvent effects in the determination of Brønsted–Evans–Polanyi relationships for heterogeneously catalyzed reactions, Phys. Chem. Chem. Phys., 2019, 21, 17687–17695.
28. S. Roy and A. K. Tiwari, Efficient water–gas shift catalysts for H₂O and CO dissociation using Cu–Ni step alloy surfaces, J. Phys. Chem. C, 2021, 125, 13819–13835.
29. C. Jia, Q. Wang, J. Yang, K. Ye, X. Li, W. Zhong, H. Shen, E. Sharman, Y. Luo and J. Jiang, Toward rational design of dual-metal-site catalysts: catalytic descriptor exploration, ACS Catal., 2022, 12, 3420–3429.
30. B. Huang, G. F. v Rudorff and O. A. v Lilienfeld, The central role of density functional theory in the AI age, Science, 2023, 381, 170–175.
31. D. Roy, S. C. Mandal and B. Pathak, Machine learning-driven high-throughput screening of alloy-based catalysts for selective CO₂ hydrogenation to methanol, ACS Appl. Mater. Interfaces, 2021, 13, 56151–56163.
32. A. Chen, X. Zhang, L. Chen, S. Yao and Z. Zhou, A machine learning model on simple features for CO₂ reduction electrocatalysts, J. Phys. Chem. C, 2020, 124, 22471–22478.
33. F. Liu, P. F. Gao, C. Wu, S. Yang and X. Ding, DFT-based machine learning for ensemble effect of Pd@Au electrocatalysts on CO₂ reduction reaction, ChemPhysChem, 2023, 24, e202200642.
34. R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler and L. M. Ghiringhelli, SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates, Phys. Rev. Mater., 2018, 2, 083802.
35. R. Ouyang, E. Ahmetcik, C. Carbogno, M. Scheffler and L. M. Ghiringhelli, Simultaneous learning of several materials properties from incomplete databases with multi-task SISSO, J. Phys.: Mater., 2019, 2, 024002.
36. G. Xie, R. Jin, P. Ren, Y. Fang, R. Zhang and Z.-J. Wang, Boosting CO₂ hydrogenation to methanol by adding trace amount of Au into Cu/ZnO catalysts, Appl. Catal., B, 2023, 324, 122233.
37. J. Mosrati, T. Ishida, H. Mac, M. Al-Yusufi, T. Honma, M. Parliniska-Wojtan, Y. Kobayashi, A. Klyushin, T. Murayama and A. M. Abdel-Mageed, Low-temperature hydrogenation of CO₂ to methanol in water on ZnO-supported CuAu nanoalloys, Angew. Chem., Int. Ed., 2023, 62, e202311340.
38. H. Zhang, J. Yang, S. Wang, N. Zhao, F. Xiao and Y. Wang, Effect of Ni content on Cu–Mn/ZrO₂ catalysts for methanol synthesis from CO₂ hydrogenation, Catal. Sci. Technol., 2024, 14, 2153–2165.
39. Y. Ren, C. Xin, Z. Hao, H. Sun, S. L. Bernasek, W. Chen and G. Q. Xu, Probing the reaction mechanism in CO₂ hydrogenation on bimetallic Ni/Cu(100) with near-ambient pressure X-Ray photoelectron spectroscopy, ACS Appl. Mater. Interfaces, 2020, 12, 2548–2554.
40. I. MeliaÂn-Cabrera, M. L. Granados and J. L. G. F. P. Terreros, CO₂ hydrogenation over Pd-modified methanol synthesis catalysts, Catal. Today, 1998, 45, 251–256.
41. X. Nie, X. Jiang, H. Wang, W. Luo, M. J. Janik, Y. Chen, X. Guo and C. Song, Mechanistic understanding of alloy effect and water promotion for Pd–Cu bimetallic catalysts in CO₂ hydrogenation to methanol, ACS Catal., 2018, 8, 4873–4892.
42. F. Lin, X. Jiang, N. Boreriboon, Z. Wang, C. Song and K. Cen, Effects of supports on bimetallic Pd–Cu catalysts for CO₂ hydrogenation to methanol, Appl. Catal., A, 2019, 585, 117210.
43. B. A. Fayisa, Y. Xi, Y. Yang, Y. Gao, A. Li, M.-Y. Wang, J. Lv, S. Huang, Y. Wang and X. Ma, Pt-modulated Cu/SiO₂ catalysts for efficient hydrogenation of CO₂-derived ethylene carbonate to methanol and ethylene glycol, Chin. J. Chem. Eng., 2022, 41, 366–373.
44. X. Wang, H. Zhang, H. Qin, K. Wu, K. Wang, J. Ma and W. Fan, A DFT study of methanol synthesis from CO₂ hydrogenation on Cu/ZnO catalyst, Fuel, 2023, 346, 128381.
45. Y. Sun, H. Hu, Y. Wang, J. Gao, Y. Tang, P. Wan, Q. Hu, J. Lv, T. Zhang and X. J. Yang, In situ hydrogenation of CO₂ by Al/Fe and Zn/Cu alloy catalysts under mild conditions, Chem. Eng. Technol., 2019, 42, 1223–1231.
46. P. Amann, B. Klötzer, D. Degerman, N. Köpfle, T. Götsch, P. Lömker, C. Rameshan, K. Ploner, D. Bikaljevic, H.-Y. Wang, M. Soldemo, M. Shipilin, C. M. Goodwin, J. Gladh, J. H. Stenlid, M. Börner, C. Schlueter and A. Nilsson, The state of zinc in methanol synthesis over a Zn/ZnO/Cu(211) model catalyst, Science, 2022, 376, 603–608.
47. H. Zhou, S. R. Docherty, N. Phongprueksathat, Z. Chen, A. V. Bukhtiyarov, I. P. Prosvirin, O. V. Safonova, A. Urakawa, C. Coperet, C. R. Muller and A. Fedorov, Combining atomic layer deposition with surface organometallic chemistry to enhance atomic-scale interactions and improve the activity and selectivity of Cu–Zn/SiO₂ catalysts for the hydrogenation of CO₂ to methanol, JACS Au, 2023, 3, 2536–2549.
48. R. Wang, B. Zhu, G. Zhang and Y. Gao, Theoretical study of CO₂ hydrogenation on Cu surfaces, J. Mol. Model., 2020, 26, 202.
49. Q. Xue, X. Qi, K. Li, Y. Zeng, F. Xu, K. Zhang, T. Yang, X. Qi and J. Jiang, DFT study of CO₂ reduction reaction to CH₃OH on low-index Cu surfaces, Catalysts, 2023, 13, 722.
50. T. Yang, T. Gu, Y. Han, W. Wang, Y. Yu, Y. Zang, H. Zhang, B. Mao, Y. Li, B. Yang and Z. Liu, Surface orientation and pressure dependence of CO₂ activation on Cu surfaces, J. Phys. Chem. C, 2020, 124, 27511–27518.
51. M. D. Higham, M. G. Quesne and C. R. A. Catlow, Mechanism of CO₂ conversion to methanol over Cu(110) and Cu(100) surfaces, Dalton Trans., 2020, 49, 8478–8497.
52. L. Pielsticker, I. Zegkinoglou, Z.-K. Han, J. J. Navarro, S. Kunze, O. Karslıoğlu, S. V. Levchenko and B. Roldan Cuenya, Crystallographic orientation dependence of surface segregation and alloying on PdCu catalysts for CO₂ hydrogenation, J. Phys. Chem. Lett., 2021, 12, 2570–2575.
53. J. Yoshihara and C. T. Campbell, Methanol synthesis and reverse water–gas shift kinetics over Cu(110) model catalysts: structural sensitivity, J. Catal., 1996, 161, 776–782.
54. G. Kresse and J. Furthmiiller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 1996, 6, 15–50.
55. J. P. Perdew, K. Burke and Y. Wang, Generalized gradient approximation for the exchange-correlation hole of a many-electron system, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 16533–16539.
56. G. Henkelman et al. https://theory.cm.utexas.edu/code/vtstscripts.tgz .
57. V. Wang, N. Xu, J. Liu, G. Tang and W.-T. Geng, VASPKIT: a user-friendly interface facilitating high-throughput computing and analysis using VASP code, Comput. Phys. Commun., 2021, 267, 108033.
58. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, A. Muller, J. Nothman, G. Louppe, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot and E. Duchesnay, Scikit-learn: machine learning in python, J. Mach. Learn. Res., 2011, 12, 2825–2830.
59. L. Liu, H. Yao, Z. Jiang and T. Fang, Theoretical study of methanol synthesis from CO₂ hydrogenation on PdCu₃(111) surface, Appl. Surf. Sci., 2018, 451, 333–345.
60. A. Bansode, B. Tidona, P. R. von Rohr and A. Urakawa, Impact of K and Ba promoters on CO₂ hydrogenation over Cu/Al₂O₃ catalysts at high pressure, Catal. Sci. Technol., 2013, 3, 767–778.
61. Z. Ou, C. Qin, J. Niu, L. Zhang and J. Ran, A comprehensive DFT study of CO₂ catalytic conversion by H₂ over Pt-doped Ni catalysts, Int. J. Hydrogen Energy, 2019, 44, 819–834.
62. X. Jiang, N. Koizumi, X. Guo and C. Song, Bimetallic Pd–Cu catalysts for selective CO₂ hydrogenation to methanol, Appl. Catal., B, 2015, 170–171, 173–185.
63. X. Jiang, X. Nie, X. Wang, H. Wang, N. Koizumi, Y. Chen, X. Guo and C. Song, Origin of Pd–Cu bimetallic effect for synergetic promotion of methanol formation from CO₂ hydrogenation, J. Catal., 2019, 369, 21–32.
64. H. Zhang, H. Qin, X. Wang, Y. Pan, P. He, J. Wu and W. Fan, Revealing the influence of oxygen-containing functional groups on mercury adsorption via density functional theory and multiple linear regression analysis, Fuel, 2023, 335, 127040.
65. P. Singh, T. D. Rose, G. Vazquez, R. Arroyave and Y. Mudryk, Machine-learning enabled thermodynamic model for the design of new rare-earth compounds, Acta Mater., 2022, 229, 117759.
66. G. Vazquez, P. Singh, D. Sauceda, R. Couperthwaite, N. Britt, K. Youssef, D. D. Johnson and R. Arróyave, Efficient machine-learning model for fast assessment of elastic properties of high-entropy alloys, Acta Mater., 2022, 232, 117924.
67. M. Andersen and K. Reuter, Adsorption enthalpies for catalysis modeling through machine-learned descriptors, Acc. Chem. Res., 2021, 54, 2741–2749.
68. Y. Santiago-Rodríguez, E. Barreto-Rodríguez and M. C. Curet-Arana, Quantum mechanical study of CO2 and CO hydrogenation on Cu(111) surfaces doped with Ga, Mg, and Ti, J. Mol. Catal. A: Chem., 2016, 423, 319–332.
69. L. H. Mou, T. Han, P. E. S. Smith, E. Sharman and J. Jiang, Machine learning descriptors for data-driven catalysis study, Adv. Sci., 2023, 10, e2301020.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp01009d

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