Electronic and geometric determinants of adsorption: fundamentals and applications

Adsorption is an essential phenomenon in surface science and is closely related to many applications such as catalysis, sensors, energy storage, biomedical applications and so on. It is widely accepted that the adsorption properties are determined by the electronic and geometric structures of substrates and adsorbates. The d-band model and the generalized coordination number model take the electronic and geometric structures of substrates into consideration respectively, successfully rationalizing the trends of adsorption on transition metals (TMs), TM nanoparticles (NPs) and some TM alloys. The linear scaling relationship (LSR) uncovers the role of the electronic structures of adsorbates in adsorption and allow the ascertainment of the trend of adsorption between different adsorbates. Recently, we develop an effective model to correlate adsorption energy with the easily accessible intrinsic electronic and geometric properties of substrates and adsorbates which holds for TMs, TM NPs, near-surface alloys and oxides. This intrinsic model can naturally derive the LSR and its generalized form, indicates the efficiency and limitation of engineering the adsorption energy and reaction energy, and enables rapid screening of potential candidates and designing of catalysts since all parameters are accessible and predictable. In this comprehensive review, we summarize these models to clarify their development process and uncover their connection and distinction, thereby drawing an explicit and overall physical picture of adsorption. Consequently, we provide a more comprehensive understanding about the broad applications of these models in catalysis. The theoretical part introduces necessary theoretical foundations and several well-built models with respect to the electronic models, the geometric models, the LSR and the intrinsic model. The application section describes their broad scope in catalysis, including oxygen reduction reaction, CO2 reduction reaction and nitrogen reduction reaction. We believe this review will provide necessary and fundamental background knowledge to further understand the underlying mechanism of adsorption and offer beneficial guidance for the rapid screening of catalysts and materials design.


Introduction
Adsorption is one of the most important phenomena on solid surfaces and is often essential in determining the application of materials in many fields, such as catalysis, energy storage, biomedical applications and so on [1][2][3][4]. For instance, a catalytic process consists of several elementary steps: adsorbing the reactants to surface, diffusing on surface, breaking some bonds and forming new bonds, and eventually desorbing from surface [5]. According to the Sabatier principle [6], an 'ideal catalyst' should bind the reactants and products neither too strong nor too weak. The too weak (strong) binding strength gives rise to the difficulty of the dissociative chemisorption of reactants (desorption of products), reducing the activity of catalyst. Furthermore, the Brønsted-Evans-Polanyi relation [7] has verified that the activation energy of a certain reaction depends strongly on the reaction energy, indicating that the reaction kinetics is also related to the properties of adsorption. Hence, determining the adsorption energy on various materials has been one of the central goals in the fields.
Adsorption can occur between numerous adsorbates and substrates and forms a variety of adsorption systems. Small molecules such as species binding via O, C, and N are likely the simplest adsorbates and are the most widely studied ones, since they are the important intermediates for oxygen reduction reaction (ORR), oxygen evolution reaction, N 2 reduction reaction (NRR), CO 2 reduction reaction (CO 2 RR), and the decomposition and/or oxidation of CH 4 , CH 3 OH and HCOOH. The corresponding substrates contain pure transition metals (TMs), nanoparticles (NPs), alloys, oxides, carbides, nitrides, sulfides and single-atom catalysts (SACs). The diverse adsorption systems bring multiple factors to adsorption properties, including the materials-dependent nature, sites and facets selectivity, strain and coverage, not to mention temperature and pressure. Generally, the electronic nature and geometric structure of adsorbates and substrates are the basis of these factors' effects. Therefore, identifying the electronic and geometric determinants of adsorption and the underlying mechanism has been a long-term goal in past decades.
The rapid development of density functional theory (DFT) methods makes it possible to systematically study the general properties of adsorption without complicated experimental costs and external interferences, in particular in providing atomic-scale mechanistic insight for adsorption. Accordingly, several successful models in regard to the electronic and geometric determinants of adsorption are put forward, rationalizing the general trends of adsorption for variant adsorbates and substrates. The d-band model by Nørskov et al [5,8,9] considers a surface chemisorption as a two-step interaction between the states of adsorbates and surface atoms, suggesting that it is the d-states of TMs that determine the variation of adsorption strength from one metal to the next, which holds for a wide range of metal systems including pure TMs (in particular the late TMs), some alloys, some TM surfaces with strains, TM surfaces with promoters and poisons [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. The generalized coordination number (CN) model [25,26], a first-order extension of the usual coordination number, is used to describe the local geometric environment on surface. This simple arithmetical descriptor effectively characterizes the geometric effects of pure TMs and NPs for adsorption [27][28][29][30][31], providing rational theoretical guidance for engineering surface structures to reach optimal catalytic activity. Besides the electronic and geometric descriptors of substrates, Nørskov et al [32] also identified the linear scaling relationship (LSR) for the adsorption energies between atoms and their partially hydrogenated species on TM surfaces, implying that the slope of the LSR strongly depends on the valence of central atoms of adsorbates. Calle-Vallejo et al [33] generalize the LSR by taking the structure sensitivity into account, suggesting that the offset of LSR is dominated by the coordination of active sites. The LSR, established from the electron counting rules and bond-order conservation theory, explicitly elaborates the general trends of adsorption for different adsorbates and uncover some adsorption limitations of reactions on TMs and NPs. Recently, we develop an effective intrinsic model to describe adsorption energy based on three main factors: the valence of adsorbates, the valence and electronegative of surface atoms, and the coordination of active sites, which holds for a variety of materials including pure TMs, NPs, near-surface alloys (NSAs) and oxides [34]. This intrinsic model correlates the adsorption energy with the intrinsic electronic and geometric properties of adsorption systems, deduces the existing LSR naturally, clarifies the efficiency and limitation in modulating the adsorption energy and reaction energy, and enables the rapid materials design, consequently offering a novel physical picture of adsorption.
In this review, we report several models to describe the electronic and geometric determinants of adsorption, clarify their development process and uncover their connection and distinction, thereby drawing an explicit and overall physical picture of adsorption. Consequently, we provide a more comprehensive understanding about the broad applications of these models in catalysis. In section 2, we introduce necessary theoretical foundations for several well-built models such as the electronic models, the geometric models, the LSR and the intrinsic model. Section 3 exhibits their broad scope in catalysis, including ORR, CO 2 RR and NRR. In section 4, we summarize the main conclusions in this review and present the future scopes of adsorption field.

Theoretical foundations
Many attempts have been made to uncover adsorption properties, laying necessary foundations for comprehensively understanding the physical picture of adsorption. So far, many important understandings are based on numerical calculations and theoretical models. The former uses the advanced calculation methods such as DFT to obtain the electronic and geometric information and energetics of adsorption systems, while the latter sets out from the physical background to understand the underlying mechanisms of adsorption such as the crystal orbital overlap population (COOP), the frontier orbital theory and the effective medium theory (EMT). Benefit from these contributions, the subsequent electronic and geometric models to explain the adsorption properties have achieved great success.

DFT calculations
With the substantial enhancement of the performance of computers, quantum theoretical calculations have attracted extensive attention [35][36][37]. DFT calculations can verify the rationality and reliability of experiments and fulfill the simulations with extreme conditions that are not realizable experimentally. Most importantly, they can set out from the view of electronic and geometric structures to elaborate the mechanism of numerous phenomena, thereby providing beneficial guidance for experiments.
In DFT, the energy of a given system is a function of the electron density (n) according to the Kohn-Sham (K-S) single-electron approximation [38,39], which can be expressed as, T(n) is the kinetic energy term, the second and the third term represent the external potential energy (interaction between nucleus and electron) and the Coulomb potential energy, and E xc (n) is the exchange-correlation energy. Note that only the E xc (n) term in equation (1) contains approximations that are necessary for calculations, namely equation (1) provides the possibility of accurately solving the Schrödinger equation as long as the exchange-correlation functionals are sufficiently accurate. Therefore, finding the appropriate exchange-correlation functional is the most crucial issue with respect to DFT calculations, which controls the error of calculations. The commonly used exchange-correlation functionals can be classified into five categories based on the Jacob's ladder [40,41]. The first rung is the local spin density approximation [42], which is obtained with homogenous electron gas model. The second rung is the functional with the various degree of gradient corrections for electron density, namely the generalized gradient approximation (GGA) [43], in which Perdew-Burke-Ernzerhof (PBE) functional [44] yields reasonable accuracy for a large variety of systems and is one of the most widely used functionals. The third rung of the ladder is the meta-GGA that adds the local spin kinetic density into GGA. The fourth rung is hybrid functional (could be either GGA or meta-GGA), which usually admixes a fraction of Hartree-Fork exchange to the GGA-or meta-GGA-based exchange, such as Heyd-Scuseria-Ernzerhof [45], B3LYP [46] and M06-class functionals [47][48][49]. Hybrid functionals can be sufficiently accurate but are computationally very demanding. The double hybrid density functional (DHDF) locates at the fifth rung which mixes the electron correlation corrections to the hybrid functional. For adsorption, GGAs are commonly used functionals and can reach reasonable accuracy, with the error of adsorption energy about ±0.2 eV. For instance, Hammer, Hansen, Nørskov modified PBE functional (rPBE) [50] is often used to calculate adsorption properties. However, GGA does not consider the orbital kinetic energy density, suffers the empirical error of self-interactions and lacks the van der Waals (vdW) interactions, which play crucial roles in many systems. Hence, other functionals such as meta-GGA and DHDF and the accurate vdW interactions must be adopted for describing the specific adsorption systems.
Meta-GGA is one of the most popular functionals at present by simultaneously considering the electron density, the gradient of electron density and the orbital kinetic energy density [51], among which the M06-class functionals [47][48][49] are suitable for a large variety of systems and the strongly constrained appropriately normed (SCAN) functional [52] is expected to be one of the most promising functionals to describe all kinds of interactions within systems. Both of these two kinds of functionals can be used to characterize the properties of adsorption. M06-class functionals belong to hybrid functionals that mix Hartree-Fork exchange and DFT exchange, including M06-L [48], M06-HF [47], M06 and M06-2X [49]. M06-L is a local functional and performs well at TM area. The local constraint makes it inefficient in main-group compounds, transition states and non-covalent interactions. M06-HF, possessing the full Hartree-Fork exchange, is expected to be the only functional that can describe the electronically excited states with useful accuracy, while is not applicable for TMs. M06 functional admixes partial Hartree-Fork exchange to the PBE exchange and is mainly suitable for TMs and nonmetals. M06-2X is similar to M06 but with twice the Hartree-Fork exchange mixing (2X), which performs well only for nonmetals. SCAN is the first nonempirical meta-GGA that fulfills all the 17 known constraints that a meta-GGA can. Some appropriate norms such as rare-gas atoms and nonbonded interactions are also added into SCAN. These characteristics of SCAN lead to the significantly better performance for both energies and geometries than LDA and PBE. Remarkably, SCAN yields comparable accuracy for a large amount of materials (even for soft materials), covering metallic, covalent, ionic, hydrogen and intermediate-vdW bondings [53]. Moreover, the computational cost of SCAN is close to that of GGA, which makes SCAN promising for predicting the electronic and geometric properties of materials.
DHDF combines a standard hybrid function with the non-local electron correlation treated by second-order perturbation theory, which can be sufficiently accurate for thermodynamic data and can also be used to predict important quantities such as reaction energies, barrier heights and isomerization energies. The first DHDF is B2PLYP developed by Grimme [54], containing the B88 exchange and LYP correlation, which are followed with several modifications such as B2GPPLYP [55], DSD-BLYP [56], PTPSS and PWPB95 functionals [57]. B2GPPLYP follows the same strategy as B2PLYP but with different amounts of the Hartree-Fork exchange and perturbative correlation mixing. DSD-BLYP also uses the B88 exchange and LYP correlation, while the perturbative part depends on the spin-component scaling idea. PTPSS and PWPB95 use the different ingredients from B2PLYP: TPSS exchange and correlation for PTPSS, and PW exchange and B95 correlation for PWPB95. Both of these two functionals are sufficiently accurate for main group compounds and show good performances for TMs. Moreover, there exist other kinds of DHDFs such as XYG3 [58]. XYG3 includes the nonlocal orbital-dependent component in the exchange and the unoccupied K-S orbitals' information in the correlation, performing well for thermochemistry, reaction barrier heights, and nonbond interactions of main group compounds. It is noteworthy that one evident shortcoming of DHDFs is the underestimation of the long-range vdW interactions (only 25% semi-local correlation is corrected by perturbation). Hence, Grimme applied the D3 approach (what will be introduced in the following) into DHDFs for capturing the long-range dispersion energy [59]. Furthermore, although DHDFs yield highly accurate results, the computational costs are extremely demanding that are usually N 5 or N 4 (N denotes the size of system), hampering its application to large-scale systems.
The vdW interactions are indispensable for numerous systems especially for large-scale molecules such as DNA, proteins, aggregates, and vdW heterostructures [60]. However, conventional exchange-correlation functional with GGA cannot describe accurately the long-range vdW forces. Many methods have been proposed to solve this issue, of which the most widely accepted methods are the DFT+D approach (such as the D1-D3 approaches by Grimme [61] and the Tkatchenko-Scheffler (TS)-based models by Tkatchenko and Scheffler [62]) and the vdW-DF method [63]. The DFT+D approach considers an additional dispersion correction to the self-consistent K-S energy while the vdW-DF method directly applies the correction to the exchange-correlation function.
Wu et al firstly proposed the idea of dispersion corrected DFT method and tried it on molecular dimers and metal carbonyls [64]. Then Grimme systematically developed a series of DFT+D approaches to consider the dispersion energy. The dispersion corrected total energy is given as, where E DFT is the usual self-consistent K-S energy calculated by chosen functional. E vdW is the dispersion correction which can be expressed as, where R AB denotes the interatomic distance between atom pair A and B, R 0 A and R 0 B are the vdW radii for atoms A and B, and C 6AB represents the corresponding C 6 coefficient. f damp is the damping function that is necessary to avoid the near-singularities for small R. According to equation (3), it is most important to determine the C 6 coefficient for calculating the vdW forces. For D1 approach [65], C 6 coefficient is obtained as, C 6AA and C 6BB are the free atom reference values of C 6 coefficient of A and B, which are cited from experiments. DFT+D1 approach can give a reasonable description of the pairwise vdW interactions while possesses some empirical properties since the C 6 coefficient is taken from experiments. As a successor of D1 approach, D2 approach [66] uses the geometric mean of the homoatomic C 6 coefficient between A and B as, The homoatomic C 6 coefficient is obtained with C 6AA = 0.05NI a p α a , where N is equal to 2, 10, 18, 36 and 54 for atoms corresponding to the period number 1-5 of the periodic table, I p and α are the atomic ionization potentials and static dipole polarizability. D2 approach eliminates some degree of the empiricism of D1 approach and represents one of the first semiclassical methods to describe the vdW interactions which holds for a large amount of elements of the periodic table. However, this approach loses efficiency for molecular environment. Then in 2010, Grimme et al proposed D3 approach [67] to consider the many-body interactions by using the concept of coordination numbers. Overall, D1-D3 series methods provide the effective descriptions of the long-range vdW interactions and are widely used in many systems.
Tkatchenko and Scheffler proposed a density-dependent scheme to consider the vdW interactions (TS-vdW) [68]. The C 6 coefficient is calculated as follows, where α 0 A and α 0 B is the static polarizability of atoms A and B. For free atoms, α 0 and C 6 are extracted from literature. For atoms in molecules or solids, α 0 and C 6 are calculated by the effective volume (V eff ) as, The TS-vdW method effectively describes the pairwise dispersion energy of which the electronic structures are captured by the Hirshfeld partitioning [69,70] and the precomputed parameters ensure the simple calculation of the C 6 coefficient. However, this scheme does not contain the long-range electrostatic screening effects that are important in both short-and long-range effects of the environment. Therefore, the original TS-vdW method was extended by Tkatchenko et al [71] to consider the long-range screening effects in the effective atomic polarizabilities by solving the classical electrodynamics self-consistent screening (SCS) equation, where α TS (r;iω) represents the sum of the TS-vdW effective polarizabilities and τ (r-r ′ ) denotes the dipole-dipole interaction tensor. Thus the screened polarizability α SCS and screened static polarizability α 0,SCS computed by equation (9) are used to calculate the screened C 6 coefficient C 6,SCS . Furthermore, the many-body dispersion (MBD) [71,72] interactions can be well described by TS-based methods using the coupled fluctuating dipole model [73,74]. The MBD correction alleviates the shortcoming of TS method that overestimates the dispersion energy. Particularly, Tkatchenko et al also proposed an effective method (TSsurf) [75] to describe the adsorption on metal surfaces by considering the nonlocal Coulomb screening within the bulk via combining the TS-vdW method with the Lifshitz-Zaremba-Kohn (LZK) theory. The screened heteronuclear C ab 6,LZK interaction coefficient is calculated by the homonuclear C aa 6 and C bb 6,LZK coefficient, where α a 0 and α b LZK denote the static polarizabilities of the adsorbate atom and the effective atom-in-a-solid system. Note that the LZK related parameters α b LZK and C bb 6,LZK are independent of the adsorbate atom, namely they can be considered as intrinsic properties of bulk.
TS-based methods use the electron density-dependent quantity to describe the dispersion energy and adopt relatively less empirical parameters. Remarkably, the TS-based scheme holds for a large variety of systems with the relatively lower computational costs. TS-vdW is mainly suitable for molecular systems in a relatively small scale, such as noble-gas atoms and other small molecules [68], while SCS can effectively describe the large scale molecular systems and solid [71,76], e.g. for understanding the point defects of semiconductors [77]. MBD has made great success in the multiple non-covalent bond systems [71,72], which captures the dispersion interactions of conjugated aromatic molecules, heteroaromatic compounds [78,79], biomolecules [80], high-pressure materials [81,82], and variety nanostructures like graphite, hexagonal boron nitride and nanotubes [62,[83][84][85][86]. For adsorption, TSsurf method [75] is effective for describing the adsorption on metal-based substrates by considering the long-range screening effects [87] and the MBD method is powerful in characterizing the adsorption of nanomaterials where the many-body effects of vdW interactions are pronounced [84,88].
Despite the great success of DFT, it usually describes the static process under 0 K and zero pressure, whereas atoms and molecules are always in movement affected by complex quantities such as temperature and pressure, which are crucial for realistic applications in surface science. The most widely used method to address this issue is ab initial molecular dynamics (AIMD) [89], which aims to simulate the dynamic process of molecules for describing the realistic physical picture of interactions. Combining DFT and molecular dynamics, AIMD replaces the semi-empirical effective potentials with the realistic physical potentials, of which the Car-Parrinello molecular dynamics [90] is one of the main methods to solve the AIMD. In Car-Parrinello molecular dynamics, the equations of motion of the electron ψ i , nuclear R I and external constrains α v imposed on system such as volume and strain are expressed as, where E is the potential energy of the system, M I is the physical ionic masses, µ and µ v are arbitrary parameters of appropriate units. Λ ik is the Lagrange multipliers to maintain the constraints that d 3 rψ * i (r,t) ψ j (r,t) = δ ij . If the velocities vary, the temperature of the system can be reduced and the potential energy E reaches minimum for T → 0. AIMD provides an effective approach for describing the dynamic process and closing up the temperature and pressure gap, e.g. describing the isothermal and isobaric processes.
The effects of temperature and pressure can be also effectively captured with first-principles atomistic thermodynamics [91][92][93][94][95] for surface adsorption in gas phase condition by taking the chemical potentials of systems into account although still focusing on the static process. Scheffler et al [96,97] demonstrate that the effects of temperature and pressure for RuO 2 (110) surface with O 2 and CO environment are directly related to the chemical potentials of gas phases, which can be easily obtained at any temperature and pressure by extrapolating from the conditions under 0 K and zero pressure. Hence, the equilibrium structure and reaction steps including the effects of temperature and pressure can be calculated in terms of first-principles atomistic thermodynamics.

COOP and frontier orbital theory
Hoffmann proposed the concept of COOP [98][99][100] to describe the bonding and antibonding contributions to the chemical bonds and Saillard and Hoffmann applied the frontier orbital theory to study the interactions during adsorption [101], thereby drawing an effective physical picture of bonding. When adsorption occurs on surface, partial bonds between adsorbate and surface atoms are formed while the bonds within adsorbate are weakened [102,103]. Thus identifying the bonding properties upon adsorption is essential for understanding the mechanism of adsorption. Mulliken analysis [104] gives a mathematic understanding of bonding. For a two-center orbital, then the normalized orbital is as follows, Clearly, the overlap term 2c 1 c 2 S 12 represents the bonding characteristics. The overlap integral S 12 is usually taken as positive, so that 2c 1 c 2 S 12 is positive for c 1 and c 2 with the same sigh, while negative for those with the opposite sigh. The magnitude of the overlap term 2c 1 c 2 S 12 is the so-called Mulliken overlap population [104], which depends on c 1 , c 2 and S 12 . Hoffmann proposed an overlap population weighted density of states (DOS) to illustrate the bonding and antibonding contributions to the surface chemical bonds: COOP. Figure 1(a) is the schematic DOS and COOP curve of H-chain [105], where the positive and negative regions of COOP curve correspond to the bonding and antibonding respectively. The integral of the COOP up to the  [105]. (c) Schematic illustration of the interactions between the energy levels of molecules A and B and molecule A and solid surface [105]. Reprinted with permission from [105]. Copyright 1988 American Physical Society.
Fermi level corresponds to the total overlap population of the bond. Note that the strength of the COOP curve depends on the number of states, the magnitude of the coupling overlap and the size of the coefficients of orbitals. Taking CO adsorption on Ni(100) surface as an example, figure 1(b) plots the COOP curve for Ni-C bonding and C-O bonding [105]. It is evident that the 5σ orbital of CO and the d-band of Ni mainly contribute to the Ni-C bonding, while the 2π * -d π interaction dominates the Ni-C and C-O antibonding above the Fermi level. The total overlap population is 0.84 for Ni-C bond and 1.04 for C-O bond, while is 1.21 for the free CO molecule. The weakening of the bond upon adsorption is mainly attributed to the population of 2π * orbital. A chemical interaction can be understood from the energy levels of the interacting atoms or molecules. Consider a simple pairwise interaction as shown in the left side of figure 1(c) [105], in which molecules A and B possess 4 and 6 electrons, respectively. Intuitively, there are three kinds of interactions between the individual energy levels of A and B: interaction x (y), interaction z and interaction {. Interaction x (y) is a typical 2-electrons interaction and the stable true bond. Interaction z is a repulsive one since the antibonding state is filled with two electrons, leading to the increase of energy. Interaction { corresponds to no energy results due to the unoccupied energy levels. It is generally accepted that the highest occupied molecular orbital and the lowest unoccupied molecular orbital control the interactions between two molecules, which are called as the frontier orbitals [106]. The frontier orbitals are the valence states of molecules and are most easily affected by the interactions. Intriguingly, Hoffmann found that the two-orbital four-electron interaction (interaction z in the right side of figure 1(c)) and two-orbital zero-electron interaction (interaction { in the right side of figure 1(c)) between molecule and surface can also form bonding although the interactions x and y still dominate the adsorption properties [105], which is different from the bonding between molecules. This is attributed to the fact that the electrons of antibonding state in interaction z transfer to the states of surface atoms and the bonding state in interaction { is filled with electrons donated by the surface atoms, stabilizing the interactions z and { and improving the bonding of adsorption. Interaction | occurs between surface atoms, which is the second-order energetic and reflects the bonding consequences of shifts of electron density around the Fermi level.

Effective medium theory
It could be very complex to calculate the energies of adsorption systems, especially for those with low symmetry. Hence, finding a simple way, for instance, a reasonable approximation to determine the energy of an adsorption system seems to be meaningful. EMT [107] has offered a far-reaching solution to this issue, and its basic idea is to replace the inhomogeneous environment of host with the homogeneous electron gas. The binding energy of a homogenous adsorption system is a function of the electron gas density as ∆E hom (ρ). If one introduces a perturbation that changes the density of homogeneous electron gas ρ to that of inhomogeneous system as ρ (r) = ρ + δρ (r), then the change in ∆E hom (ρ) should be as [108], where δ and ∆ represent the perturbation-induced changes and atom-induced changes respectively. Z is the charge of adsorption atom. ∆ρ = ρ(r)−ρ 0 is the difference between the density of adsorption system and the substrates alone. δv(r) is the difference between the potential due to the positive background and the potential due to the ion cores of the bounded substrate. Now one chooses the average electron density ρ such thatˆ∆ where ∆ϕ is the atom-induced electrostatic potential in homogeneous substrates. Hence, the zero-order, first-order and second-order term of equation (16) are as follows, where ϕ 0 (r) is the electrostatic potential of substrates, R and R ′ are the parameters related to the static polarizabilities P and P 0 and the dielectric functions ε and ε 0 of the homogenous electron gas with and without adsorbed atom. Note that the second-order term (including equation (20)) needs to be taken into account only when the adsorbate-induced resonance state is close to the Fermi level such as Li and Si that are particularly polarizable. While in most cases, the first-order approximation, namely equations (18) and (19) are accurate enough to describe the energy of adsorption systems as long as the resonance state is well below the Fermi level such as the electronegative adsorbates hydrogen and oxygen that are not particularly polarizable. EMT has profound significance to understand adsorption and has been utilized to explain the mechanism of the LSR [32,33] and the intrinsic model [34].

Electronic models
Adsorption on a surface gives rise to the interaction between the valence states of adsorbates and surface atoms, which involves a study of electronic properties of materials. Over the past decades, several electronic models have been proposed for elucidating the roles of electronic structures in adsorption. In the following, we will introduce the electronic models and descriptors which are suitable for different kinds of materials such as pure TMs, NSAs, TM oxides and SACs.

News-Anderson model for adsorption
Adsorption occurs when the distance between an adsorbate (an atom or molecule) and surface atoms is getting closer, along with the states of adsorbates coupling with all the valence states of surface atoms. This coupling can be illustrated by the News-Anderson (NA) model [109,110]. Here we take TMs as example to explicit the interaction between adsorbates and the different valence states of TM. Figure 2(a) illustrates a typical DOS of a given TM [5]. Clearly, a TM's states include a broad s-band and a narrow d-band, the latter is caused by the small coupling matrix strength between the localized d-states of TM (V dd ), namely the strength of V dd is proportional to the bandwidth. When an adsorbate interacts with the s-states of TM, only a single resonance state is generated below the energy of the original adsorbate state. In contrast, the interaction between an adsorbate state and d-states of TM gives rise to the split-off bonding and antibonding states which locate below and above the original adsorbate state respectively as seen in figure 2(b) [5].
According to the NA model, adsorption on surfaces can be considered as a two-step process. At first, the valence state of a given adsorbate interacts with the metal s-state, generating a resonance state below the original valence state of adsorbate called renormalized state. Then the renormalized state couples with d-states of TMs. Since the d-band is narrow, this interaction gives rise to the state splitting: a bonding state and an antibonding state, which are below and above the renormalized state respectively. This two-step process is well illustrated in figure 2(c) [5]. Therefore, adsorption energy can be expressed as, where ∆E a−s and ∆E a−d are the energy of renormalized state (adsorbates-s hybrid state) and adsorbates-d hybrid state. It is noteworthy that all TMs have similar half-filled s-band. Thus ∆E a−s for all TMs could be considered as a constant, which means that the variation of adsorption energy from one metal to the next is dominated by the interaction between the renormalized state and metal d-state ∆E a−d . When the strength of ∆E a−d is weak, adsorption energy only depends on the position of the d-states relative to the Fermi level to a first approximation, which is exactly the essential of the d-band model [5,8,9]. ∆E a−d can be quantitatively estimated by the NA model as [111], n a and ε a are the degeneracy and energy of the adsorbate energy level.
dε ′ are the adsorbate-induced changes to the metal states multiple the coupling between the adsorbate and metal states (P is the principal value of the integral for the transform). The coupling matrix element V ad , which is strongly dependent on the spatial extent of metal d-states, can be expressed based on the bandwidth and elemental radius [112]: where η is a structure constant in LMTO, and l i (i = a, d) is the angular momentum quantum number of adsorbate and metal. In M l = √ s 2l i + 1 ∆ l i , s is the neutral sphere radius and ∆l i is a LMTO potential with respect to the bandwidth. Equation (22) can be applied to DOS distribution to obtain the interaction energy between the particular DOS distribution and an adsorbate level [112]. [5,8,9], one of the most successful electronic models, has severed as a basis to clarify the mechanism of adsorption, which is proposed initially for TM surfaces especially for the late TMs. This model demonstrates that it is the d-states of TMs that determine the variation of adsorption energy from one TM to the next, namely the adsorption energy on TMs is controlled by the position of d-states relative to the Fermi level. In this section, we will make use of the d-band model to elaborate the materials-dependent effects, the coverage effects and strain effects of adsorption.

Materials-dependent effects for atomic adsorbates
The first moment of the electronic DOS distribution ρ(x), ε d , namely the center of d-band, and the nth moments m c n of ρ(x) centered at ε d are defined as, Note that another crucial electronic property, the d-band width of TM W d , is defined as [112] demonstrate that the semi-empirical fits to the DFT-calculated DOS are effective in representing the d-band properties of TMs. Table 1 summarizes the first and second moments of the semi-empirical fits to the DFT-calculated DOS and the coupling matrix element V 2 ad relative to Cu for 24 TMs.
A more simplified expression of ∆E a−d can be obtained by assuming that both the renormalized adsorbate state and the partially filled metal d-band can be approximated by single energy levels ε a and ε d as [113], Here α and β involve the information of d-band fillings, the level degeneracy, and the spatial extent of the adsorbate level state. The first term represents the attractive interaction between the adsorbate and metal d-states and the second term denotes the Pauli repulsion due to the orthogonal overlapping of states. According to equation (26), one can easily obtain that there exist three main factors that control the adsorption of TMs and can serve as the basic electronic descriptors: d-band center ε d , d-band width W d and d-band filling. Note that these three factors are related with each other. For instance, the variation of W d  [112]. Clearly, ε d shows a linear relation with group numbers except the outliers Ni, Pd and Pt, while W d exhibits a broken-line correlation with group numbers. Both ε d and W d decrease with increasing group number for the late TMs while ε d and W d show opposite trends with group numbers for the early TMs. Figure 3(c) plots the adsorption energies of O for the late TMs as a function of the d-band center ε d [112]. Clearly, the adsorption energies correlate well with ε d except the outlier Pd and Pt, which demonstrates that ε d is indeed a good descriptor in understanding the trends of adsorption of oxygen. This corresponds to the classical d-band theory that has been widely used to understand the materials-dependent effects of adsorption.
There exist two main effects as shown in figure 3(c): (a) the adsorption strength decreases from the left of periodical table to the right and (b) the further down the periodical table, the weaker the adsorption energy. To understand the mechanism of these two effects, one needs to uncover the roles of d-states for different TMs. The d-states of metals increasingly move downward from the left of periodical table to the right as the d-band filling increases, finally all d-bands are occupied and are well below the Fermi level. This means that the antibonding states are gradually filled from the left to the right of periodical table and there are no empty antibonding states for Cu, Ag and Au. Therefore, the further to the right of periodical table, the weaker the attraction interactions and finally turning to the repulsive interactions, leading to the decreasing adsorption strength. In addition, the states of two atoms will be orthogonal to each other when the states overlap, which gives rise to the Pauli repulsion. The strength of Pauli repulsion for adsorption is proportional to the square of the coupling matrix element V 2 ad (see equation (26)). For a given adsorbate and a fixed adsorbate geometry, the strength of V ad only depends on the metal d-states (see equation (23)). Since the d-bands are well below the Fermi level for TMs with essentially filled d-states, it is the Pauli repulsion that determines the variation of adsorption energy. V 2 ad increases down through the groups of the periodical table, rendering the fact that the repulsive interaction of TMs obeys the order that 5d > 4d > 3d metals. Hence, Au is the noblest metal [8]. It should be mentioned that both the attractive and repulsive interactions increase with increasing coupling matrix element (see equation (26)), and it is the competition between attraction and repulsion that determines the net results of coupling strength of adsorbate-d hybrid states.
As ε d gives a good description of O adsorption energy for the late TMs despite the other factors that matter to the variation of adsorption energy such as W d , we now discuss this origin. Actually, for the late TMs, ε d and W d are independent of each other. The hybridization energies (∆E a−d ) are dominated by ε d while depend weakly on W d for the late TMs [112]. The case for the early TMs is different from that for the late TMs, which we will discuss in section 2.4.2.5.

Materials-dependent effects for molecular adsorbates
Molecular adsorbates on TM surfaces obey the same rule as atomic adsorbates, while the former's mechanism is more complicated. First, taking the H 2 molecule adsorbed on TM surfaces as an example [8,9], the two s-states of H 2 overlap to generate bonding state σ g and antibonding state σ u * as shown in figure 4(a) [9]. When the adsorption occurs, both the σ g and σ u * couple with the metal s-states, leading to the two corresponding broad resonance states below the original σ g and σ u * states. After that, the two renormalized states interact with the narrow d-band of TMs and split into bonding and antibonding states seen in figure 4(a). Since the two bonding states σ g −d and σ u * −d are well below the Fermi level, it is the two corresponding antibonding states (σ g −d) * and (σ u * −d) * that control the coupling strength with metal d-band. Figure 4(b) shows that the activation barriers of four TMs (Ni, Pt, Cu and Au) exhibit a linear relation with the d-band center ε d except the outlier Pt, making ε d a good descriptor for the materials-dependent effects of H 2 adsorption [8]. It is clear that the activation barrier of H 2 dissociation varies for different TMs in the order Au > Cu > Pt = Ni, while the dissociation of H 2 on the latter two surfaces occurs spontaneously. This order can be explained by the d-band model. All the (σ u * −d) * states locate well above the Fermi level, leading to the large attraction. The position of (σ g −d) * states varies from one TM to the next, namely the antibonding states are just above the Fermi level for Ni and Pt but are well below the Fermi level for Au and Cu [8]. Hence, there exists attractive interaction on Ni and Pt surfaces and repulsive interaction on Au and Cu surfaces towards H 2 caused by the (σ g −d) * states. Since 5d metals have larger coupling matrix element than 3d metals, the stronger repulsion of Au than Cu makes Au exhibit the largest activation barrier seen in figure 4(b). In fact, the bonding between the H atom and Au is not weak at all due to the fact that the large attraction attributed to the coupling with the metal s-states is only partially compensated by the repulsion with d-bands. However, comparing with the bonding in H 2 molecule, the bond is only marginally stable for Cu and totally unstable for Au [8]. It is noteworthy that Cu and Pt hold the similar ε d (−2.42 eV for Pt and −2.46 eV for Cu) while the activation barriers of H 2 dissociation vary a lot (see figure 4(b)). This is due to the different d-band filling (0.9 for Pt and 1.0 for Cu) and d-band width W d (6.51 eV for Pt and 3.66 eV for Cu) [112] for Pt and Cu, which are also important in determining adsorption (see equation (26)). The larger W d and less-filled d-band filling of Pt make the antibonding state (σ g −d) * is just above the Fermi level, while the (σ g −d) * state of Cu locates below the Fermi level [8,114]. Hence, the empty (σ g −d) * state of Pt and the filled (σ g −d) * state of Cu lead to the attractive and repulsive interaction respectively, accounting for the different activity for the H 2 dissociation.
Then we discuss the adsorption of CO. The correlation between the adsorption energy of CO for TMs and the d-band center ε d is illustrated in figure 5(a) [34], where the linear correlation indicates that ε d serves as a good descriptor in describing the materials-dependent nature of CO adsorption. The widely accepted Blyholder model [106,115] implies that it is the 5σ and 2π * states of CO determine the adsorption. Similar to H 2 , the 5σ and 2π * states of CO couple with metal s-states firstly and then interact with d-states of TMs, leading to the split-off bonding and antibonding states seen in figure 5(b) [5]. One important effect from figure 5(b) is that the 5σ state of CO plays a minor role in the strength of adsorption compared with 2π * state due to the fact that there are only few empty antibonding states of 5σ state and all the antibonding states of 2π * state are well above the Fermi level. Hence, 2π * interaction dominates the variation from one TM to the next. Moreover, from the view of symmetry, the highly directional 5σ state plays more vital roles for atop adsorption while the 2π * state of CO is dominant for hollow sites adsorption [116,117].

The interaction between adsorbates
What we discussed above is the adsorption of a single atom or molecule, neglecting the interaction between adsorbates, namely the coverage effects of adsorbates, which is crucial in practical applications. Generally, both the attractive and repulsive interactions can exist between adsorbates. The former is often due to the surface reconstruction and leads to island formation and clustering, while the repulsive interaction is the most common situation and results in separated overlayers [118][119][120]. Here one introduces the two main effects between adsorbates. One is the direct interaction which is caused by the overlaps of the valence states between adsorbates. If the valence states are close enough to the Fermi level so that the interaction may shift the states through the Fermi level, the attractive interaction will be generated. Otherwise, in more cases this direct interaction is dominated by the Pauli repulsion [5]. The other is the indirect interaction between adsorbates. Adsorption on surface changes the electronic structure of surface atoms, thereby affecting the coupling between another adsorbate (an atom or molecule) and surface atoms. It is normally considered that adsorption gives rise to the downshift of d-states of neighboring surface atoms [121,122], namely the antibonding states are less empty and the attractive interaction is weakened. Therefore, the bonding between the second adsorbate and surface atoms may not be as strong as the first one.

Electronic model for strain effects
Strain effects are common phenomena in catalysis and are widely used to create numerous intriguing properties different from that of strain-free status [123], which can occur either externally by applying mechanical loading or internally by creating the lattice mismatch of surface layers (substituting surface atom with another one of different atomic radius or creating heterogeneous overlayers on surfaces such as alloying). Actually, strain effects are complicated and depend on numerous properties, such as the species of adsorbates and substrates, surface orientations, adsorption sites, and the intensity and direction of strain [124][125][126][127][128][129][130][131]. Both the d-band model [132] and the generalized coordination number model [133] can provide a partial explanation for strain effects. In this section, we focus on the electronic effects of strain, while the geometric effects of strain are discussed in section 2.5.3.
When a surface is subject to tensile or compressive strain, the electronic structures in particular the d-bands of TMs vary a lot. Tensile (compressive) strain on surface leads to the increasing (decreasing) atomic distances and decreasing (increasing) overlaps of d-states. Consequently, the width of d-band decreases (increases) as well. It is a general finding that there is no charge transfer under strain, which means that the d-band of metals would shift upward or downward to maintain a constant filling. Hence, compressive and tensile strain generate downshift and upshift of d-band [132]. These effects are explicitly illustrated in figure 6(a). Moreover, strain effects also depend on the adsorption sites which cannot be well clarified by the d-band model [134]. Figure 6(b) shows the adsorption energy of CO and O as a function of strain (lattice constants) and d-band center ε d [132]. Clearly, the strength of adsorption enhances with increasing strain (the larger (smaller) lattice constant corresponds to the larger tensile (compressive) strain), which could be well understood by the d-band model as discussed above. From compressive strain to tensile strain, d-band moves to higher energy position, making the antibonding states coupled by adsorbate and metal d-states more likely be pushed above the Fermi level. Therefore, the larger the strain, the larger the net attractive interaction and the stronger the adsorption energy. As the shift of d-band can be well described by the center of d-band ε d , ε d is thus a good descriptor to characterize the trends of strain seen in figure 6(b).
Moreover, the strain effects are strongly related to the adsorption sites, namely the different sites may correspond to the different responses of substrates to strain. Figure 6(c) shows the calculated CH 2 binding energy to Cu (110) surface under biaxial strain [134]. For the bridge-site adsorption, the tensile strain makes the binding stronger, whereas the compressive strain accounts for the enhancement of binding strength at four-fold hollow site. This opposite trend generates a crossover of optimal adsorption sites as a function of strain shown in figure 6(c). This conflicting observation cannot be simply explained by the d-band model.  (0001) surface. The data are adapted from [132]. (c) The binding energy of CH2 on Cu (110) surface versus strain at bridge sites and four-fold sites. The data are adapted from [134]. Actually, it is the different response to strain of these two sites that control the variation tendency of binding energy. The fundamental difference between bridge and four-fold sites is that the neighboring metal atoms are put outward at bridge sites but are put inward at four-fold sites [134]. Therefore, the bridge-site adsorption feels the tensile strain to stabilize the interaction between adsorbate and surface atoms, on the contrary, the compressive strain is helpful to enhance the interaction at four-fold site.

Electronic model with the upper-edge of d-bands
It is noteworthy that the d-band center ε d can provide a good description of the adsorption energy only for the late TMs [5,8,9]. This is due to the different contributions of ε d and W d to the adsorption energy for the late and early TMs: ε d dominantly determines the adsorption energy for the late TMs while both ε d and W d contribute to the adsorption energy for the early TMs [112]. Therefore, a refined descriptor that contains both the contributions of ε d and W d is needed.
Vojvodic et al [112] find that the position of the upper-edge of the d-band shifts downwards as the center of d-band shifts down for 4d-series TMs. This implies that the upper-edge of the d-band is also an important quantity that varies from one TM to the next. Actually, bond strength strongly depends on the position and filling of antibonding states, which are pinned at the upper-edge of d-bands, thus it is reasonable to put forward a refined descriptor ε d + W d /2 to introduce a d-band width dependent term, ensuing the correlation between the position of the upper-edge of the d-band and the binding strength. As seen in figure 7, ε d + W d /2 describes the trends of adsorption energy well including the above outliers Pd and Pt [112]. Unfortunately, this refined descriptor is unable to rationalize the trends of TMs with completely filled d-bands such as Cu and Ag, which is due to the fact that the position of d-band for these metals is no longer pinned to the Fermi level.

Electronic model for NSAs
It has been verified both theoretically and experimentally that introducing guest atoms into the subsurface layer of a host to form NSAs can change the electronic properties of atoms on top surface layer [20,[135][136][137], thereby modulating the numerous properties such as adsorption and catalytic activity. Koper et al [138] find that the number of valence electrons for adsorption systems (which are the valence electrons of Pt, the guest atom in the second layer and the adsorbate) provides a good description for the variations of adsorption energy on Pt@M(111) NSAs (M represents the guest atom in subsurface layer from 3d, 4d and 5d TMs) seen in figures 8(a)-(c). One of the most remarkable features of figures 8(a)-(c) is that the minimum of adsorption energies in all cases locates at the total number of valence electrons 24: the minimum is 10 for C and Si (corresponding to Ni, Pd and Pt), 9 for N and P (corresponding to Co, Rh and Ir), and 8 for O and S (corresponding to Fe, Ru and Os). The underlying mechanism of the constant total minimum valence electrons can be understood by the electron-counting rules to form a noble-gas-like electron cloud, following the 8-electrons and 18-electrons for adsorbate and surface atoms respectively [139,140]. Therefore, one obtains that adsorption energies on NSAs are strongly correlated to the number of valence electrons from top surface layer atoms, subsurface layer atoms and adsorbates. Moreover, the constant total valence numbers of minimum adsorption energy make it possible to predict the adsorption energies of other NSAs such as the NSAs of Au.
Linic et al [141] make use of the d-band model to elucidate the trends of adsorption energies on Pt@M(111) NSAs for 3d series TMs. Since the distance between the Pt host and the guest atom is similar for all 3d-series TMs, namely the geometry of Pt@M(111) NSAs is dominated by the contributions of sp-states of TMs, which are approximated as constant for all TMs. Hence, the d-band width projected on geometrically  similar Pt sites should only be a function of the spatial extent of the d-states (r d ) of the 3d-metals directly coupling to the Pt host. Figures 8(d) and (e) exhibit the adsorption energies on NSAs as a function of d-band center ε d and r d , where the linear scaling behaviors exist [141]. The 3/2 power-dependent term between adsorption energies and r d can be explained by tight-bonding (TB) approximation [142][143][144] in which the coupling matrix element V ad is proportional to r d 3/2 .

Electronic model for transition-metal oxides
TM oxides have attracted extensive attention and been applied to numerous fields due to the high catalytic activity, inexpensiveness, versatility and stability [145][146][147][148]. They are expected to reduce or completely replace the utilization of noble metals in many fields such as catalysis, fuel cells and metal-air batteries [149][150][151]. Hence, it is crucial to develop proper models to elucidate the trends of adsorption energy for TM oxides. Calle-Vallejo et al [152] demonstrate that the outer electrons (N oe ) which are defined as the number of remaining valence electrons of the metal atom upon oxidation can describe the variation of adsorption energies for species of O, OH and OOH on monoxides (MO) and perovskite oxides (ABO 3 ) (100) surfaces.
Taking Cr as an example, N oe is equal to 6 for pure Cr, 4 for Cr 2+ in CrO, 3 for Cr 3+ in LaCrO 3 and 2 for Cr 4+ in SrCrO 3 . Figures 9(a)-(c) illustrates the linear relation between the adsorption energies and N oe on MO and ABO 3 [152]. Clearly, the slopes of linear relation for OH and OOH are almost identical in all cases while the slope for O is about twice than that for OH and OOH seen in figures 9(a)-(c). One can easily obtain that the adsorption energies are weakened with the increasing N oe despite the different species and surface structures. Note that different TM oxides share the same coordination and spatial distribution although their oxidation states are different. The e g and t 2g orbitals are normally nonbonding or antibonding states while the O 2p-orbitals are bonding states, therefore, the increase of N oe gives rise to the reductions in empty antibonding states and the weakening of adsorption energies. The outliers deviating from the guiding lines shown in figures 9(a)-(c) such as CaO, LaScO 3 , YScO 3 , SrTiO 3 , CaTiO 3 and BaTiO 3 are attributed to fulfilling the octet or 18-electrons rules.
The p-band center of O or the average O 2p-state energy is initially proposed by Kitchin et al [153] to explain the adsorption energy of O for six 3d-series TMs and their rock-salt oxides. Subsequently, Nørskov et al [154] demonstrate that the average O 2p-state energy (ε 2p ) can serve as a descriptor to clarify the variation of E O −E OH on face-centered cubic (fcc) metals (111), rutile oxides (110) and perovskite oxides (100) surfaces [155] (E O −E OH implies that H atom is adsorbed on O atom rather than metal atom and is thus indicative of the reactivity of O atom).ε 2p can be expressed as the first moment of the DOS projected onto the atomic 2p-states of an adsorbed oxygen atom (after deprotonation) relative to the Fermi level: where ε min and ε max are chosen to −10 eV and 2 eV respectively. Figures 9(d)-(f) exhibits that E O −E OH is strongly linearly correlated toε 2p with the slope close to unity except the case of fcc (111), suggesting that the variation ofε 2p directly corresponds to that of E O −E OH . The linear relation implies that the higher-lying 2p-states lead to the weakening of adsorption energy, which is different from the d-band theory [5,8,9] that the upshift of d-bands gives rise to the receding adsorption strength. Note thatε 2p depends on the calculation of DOS, leading to the cost and qualitative results which is similar to the d-band model.

Electronic model for SACs
As the smallest size metal catalysts, SACs represent the study frontier in catalysis [156][157][158][159][160][161][162][163][164][165][166]. SACs can furthest improve the metal utilization with well-defined and atomically dispersed metal sites on supports. Moreover, as ideal and simplified models, SACs are beneficial to make a direct comparison between experiments and theoretical calculations [167]. Thus, understanding the physical and chemical nature of adsorption on SACs is of great importance. Zeng et al [168] propose an electronic descriptor in terms of valence electrons of occupied d-orbital of metal element (θ d ) and electronegativity of surface atoms to explain the trends of adsorption energies of OH and H on graphene-based SACs (see figures 10(a) and (b)). The electronic descriptor takes the local environment of single TM atom and its nearest neighbors into consideration and is expressed as follows, E M , E N and E C represent the electronegativity of metal element, nitrogen element and carbon element respectively. E O/H is the electronegativity of O element or H element which is determined by the adsorbate OH or H. n N and n C are the number of nearest-neighbor N and C atoms, and α is coefficient correction. α is  (111), rutile (110) and perovskite oxides (100) surfaces. The data are adapted from [154] and [155]. equal to 1.25 for single-TM-atom-pyrrole-N 4 and is equal to 1 for other active centers. It could be figured out that the adsorption free energy of OH ∆G OH scales linearly with increasing φ on graphene-based SACs, namely the adsorption strength weakens with φ increases. This trend can be understood by the d-band model [5,8,9], which demonstrates that the lower d-band center gives rise to lower antibonding states and higher occupations, leading to the weak adsorption strength. Figure 10(c) shows a negative correlation between the band center of the antibonding orbital and φ. Therefore, the increase of φ corresponds to the downshift of d-band, thereby weakening the adsorption energy. Figure 10(b) illustrates the relationship between the adsorption free energy of H ∆G H and φ, exhibiting a broken-line behavior. For SACs with φ > 27, ∆G H increases linearly with φ, which is consistent with the trends of ∆G OH . In contrast, the opposite trend can be observed for SACs with φ < 27 that ∆G H decreases with increasing φ. This is due to the fact that the bonding states locate near the Fermi level instead of the deep-lying level when φ is too small so that the bonding states upshift with the upward shifting of d-band (corresponding to the decrease of φ), leading to the weakening of adsorption energy [169]. Remarkably, this electronic model can be easily generalized to describe the trends of adsorption energies for macrocyclic-molecule based SACs, implying the portability of this model. Unfortunately, this model cannot describe the trends of adsorption energies for TMs.

Geometric models
We now attempt to uncover the geometric determinants of adsorption. It is widely acknowledged that adsorption on surfaces is strongly dependent on the geometric structures of surface atoms. For example, atoms of step surfaces or kink sites of NPs are normally more reactive than those of flat surfaces and bind to adsorbates stronger. This geometric effect is generally understood with the concept of coordination number (CN). Herein we introduce several geometric models in regard to CN, unravel their mechanisms in understanding adsorption on surfaces, discuss the geometric effects of strain, and clarify the correlation between the geometric and electronic models.

Usual coordination number model
One of the most intuitive descriptions of geometric structure for surface atoms is the usual coordination number (CN u ), which is defined by the number of nearest atoms bonded to the central atom, for instance, CN u is equal to 9 for atoms on fcc (111) surfaces. This descriptor provides a good description for the geometric effects of adsorption energy as seen in figures 11(a)-(c) [33,170], where the adsorption energies of OH, OOH, OCH 3 and H 2 O increase linearly with CN u for difference extended surface sites. The mechanism of this trend can be understood by the bond-order conservation theory [171,172]: the larger CN u of surface atoms such as (111) facet implies the less broken bonds, consequently the tendency for surface atoms to form new bonds and the strength of adsorption energy become weaker as seen in figures 11(a)-(c).

Generalized coordination number model
CN u successfully describes the geometric effects of adsorption energy despite its simple arithmetical form. However, it indeed has some limitations. For surface sites with similar coordination distribution, it is too hard for CN u to effectively distinguish their geometric effects. As seen in figure 11(d) [25], several surface sites of different NPs possess the same CN u , while the adsorption energies of these sites vary up to 0.5 eV, which is far beyond ±0.2 eV, the error of DFT with (semi-) local approximation. Therefore, finding a more general expression of the geometric structure is necessary. A successful model is the generalized coordination number (CN) [25], a first-order extension of CN u , which takes the second nearest neighbors of active sites into consideration. For a given atom i with n i nearest neighbors on surface, CN is expressed as follows, where CN max is the CN u for the corresponding crystals, for instance, CN max is 12 and 8 for fcc [or hexagonal close-packed (hcp)] crystals and body-centered cubic (bcc) crystals respectively. Moreover, equation (29) can be extended to bridge sites with CN max = 18 for fcc crystals, which is the number of nearest neighbors that surround the two bridge-site atoms in bulk, and to hollow sites with CN max = 22 for threefold sites and CN max = 26 for four-fold sites. Figure 11(e) exhibits the adsorption energies of OH against CN, where an evident linear relationship is observed [25]. More importantly, the adsorption energies for the similar sites of different NPs are separated apart and follow the guiding line. Since CN involves more general geometry of surfaces and is more sensitive to the geometric variations of active sites, it is no wonder that CN provides a more accurate description for the geometric effects of adsorption than CN u . Likewise, the linear increase of adsorption energy with CN can also be explained by the fact that the decrease in coordination corresponds to the increase in the bonding strength, following the bond-order conservation theory [171,172].

Geometric model for strain effects CN
can also be extended to explain strain effects by slightly modifying its definition [133]. In the case of strain, the usual coordination number CN u can be expressed as, where d bulk and d i −d k=1 are the interatomic distance in the bulk and in the surface respectively. The simplest approximation of equation (30) is that the difference between bulk and surface interatomic distance is negligible, leading to CN u = n i ∑ k=1 1 = n i . Then one can consider the second-nearest neighbors for an atom by the sum of weights of its nearest neighbors as, Note that equation (31) will transform to equation (29) automatically when d bulk is approximated equal to d i −d k=1 . In contrast, if the difference between bulk and surface interatomic distance is not negligible while the surface interatomic distance is approximated as an average valued, CN * can be expressed as, Recalling that strain (S) is related to the interatomic-distance difference that obeys the relation S =d −d bulk d bulk =d d bulk − 1 [132]. Accordingly, one obtains that: Equation (33) demonstrates that strain effects can be involved in the concept of coordination. Note that tensile strain and compressive strain correspond to the positive and negative value of S respectively, namely tensile strain (compressive strain) gives rise to the decrease (increase) of CN * shown in equation (33). cavity B surfaces with strain ranging from −3% to +3% [133]. Clearly, the adsorption energies of OH and OOH scale linearly with CN * for all surfaces, demonstrating that CN * indeed captures the strain effects of adsorption. Since both the generalized coordination number model and the d-band model describe a linear relationship with adsorption energy under strain, there may exist some links between these two models.

The relationship and distinction between the coordination number model and the d-band model
Generally, the variation of geometric structures may also give rise to the change of electronic structures [5]. Therefore, it is necessary to re-recognize the coordination number model from the view of electronic structures such as the d-band model and uncover the links and distinctions between the two models. The change in coordination numbers of surface atoms generates the change in the overlaps of electronic states, leading to the movement of states especially that of d-bands for TMs. When the coordination number for a given atom on TM surface decreases, the overlap of d-band also declines, thereby the d-band width W d decreases and d-band center ε d moves upshift towards Fermi level [5,173,174]. Hence, more empty antibonding states interact with adsorbates, leading to the enhancement of adsorption strength. From this Calle-Vallejo et al [25] derive an approximately linear correlation between ε d and CN based on broken-bond model [175][176][177] (which is a semi-empirical model to calculate surface energies of elemental crystals in solid phase) as follows, where ε surf d and ε bulk d are the d-band center of atoms on surface and in the bulk respectively, E is cohesive energy of materials, θ d is the average occupations of d-states and f is an additional ratio factor (θ d = 1.8 and f = 1.125 for Pt). Equation (34) demonstrates that the d-band center of surface atoms is mainly dominated by four factors: the d-band center of bulk atoms, the cohesive energy of materials, the electronic occupations of d-bands and the geometric environments around atoms. It should be noted that for a given kind of material, ε bulk d , E, θ d , f and CN bulk are all constants, which means that ε surf d scales linearly with CN surf . Equation (34) provides a quantitative relationship between ε d and CN and proves again the variation of geometric structures leads to the variation of electronic structures.
Although the underlying mechanism of the coordination number model in adsorption can be explained by the d-band model, it does not mean that the former can be replaced by the latter. Actually, both the two models and their descriptors have strengths and limitations. Figure 13 illustrates the adsorption energies of OH and OOH on Pt 201 NPs and extended surfaces against ε d and CN respectively [25]. Although two linear scalings can be easily established, there are about ten different sites with variable adsorption energies possessing the similar ε d at around −2.34 eV, indicating that ε d is not a good descriptor in characterizing the trends of adsorption energy for these sites such as kinks and edges. In contrast, CN can well describe the variation of adsorption energies for these sites. On the other hand, ε d is more effective to clarify the variation of adsorption energies from one TM to the next compared to CN. Particularly, for a fixed adsorption site on given crystal surfaces, such as the top site on fcc(111) surfaces, CN is the same for all kinds of fcc metals, while the adsorption energies of different metals vary a lot. Namely, CN becomes invalid under this circumstance.
Overall, there indeed exist the links between the d-band model (electronic model) and the coordination number model (geometric model), whereas these two different models are not identical in many cases.

Orbitalwise coordination number
CN captures relatively accurate geometric information of a given active center and effectively describes the geometric effects of adsorption. Although it is a general concept and can be easily extended to the third-or more neighboring atoms, CN is unable to describe some specific systems such as alloys. A series of alloys may  have the same value of CN but the ability of adsorption varies a lot. Xin et al [178] consider both the electronic and geometric structure of surface atoms, quantify the degree of coordinative saturation of metal atoms and their inclination to form new bond, and thus propose a descriptor based on the moments theorem [179] called orbitalwise coordination number as, M α 2,i is the second moment of the distribution {ε α i } ({ε α i } is the moment eigenspectrum of the projected DOS onto the TM or noble metal atom i), which determines the variation in the local chemical reactivity of an adsorption site perturbed by neighboring atoms. t α,∞ nn is the sum of the α-electron hopping integrals to relevant valence orbitals of a neighboring atom in the bulk, which depends strongly on the orbital size or shape and internucleus distance. Figures 14(a) and (b) plot the adsorption energies of CO on NPs with various size and shape (cuboctahedron, truncated octahedron, tetrahedron and cube) as a function of CN and CN α [178]. Clearly, CN α provides a better description for the trends of adsorption energy of various-shape NPs due to the larger regression coefficient and smaller standard deviation. More importantly, CN α can easily capture the properties of alloys as seen in figure 14(c) since it considers the electronic structure of surface atoms [178]. The small deviations between the predicted and DFT-calculated adsorption energies of CO and O on a series of core-shell NPs demonstrate the validity and reliability of CN α in describing alloys. Unfortunately, this model requires complex DFT-calculations to obtain the basic parameters, resulting in the expensive costs and inconveniences.

The linear scaling relationship
The electronic models and the geometric models discussed above are mainly used to explain the trends of adsorption energy on different substrates, while rarely uncover the correlation between the adsorption energies of variant adsorbates. The LSR [32,33] provides a profound solution to this issue, and it propose the relation between the adsorption energies for atoms and their partially hydrogenated species. In this section we summarize a simple derivation of the LSR, including its slope and intercept, and uncover its underlying mechanisms and connotations as well as the limitations.
Nørskov et al [32] first discover the LSR and ascertain the determinants of its slope for atoms and their partially hydrogenated species on TM surfaces. As shown in figures 15(a)-(d), the adsorption energies of AH x is linearly related to adsorption energies of atom A as, The slope in equation (36) is equal to γ(x) = (X m − X)/X m , where X m and X are the maximum and actual numbers that the central atom of a given adsorbate bond to the specific coordinating group. The underlying mechanism of equation (36) can be understood by the d-band model [5,8,9]. As mentioned above, the adsorption energy can be divided into the coupling of adsorbate states with the sp-states and d-states of TMs, seen in equation (21). The coupling between adsorbates and metal s-states can be considered as a constant for different metals, and the main contribution to the variation in bond strength from one TM to the next comes from the coupling to the d-states. Assuming that E AHx d = γ (x) E A d , one can rewrite equation (21) as, This is exactly equation (36). Hence, in order to ensure the validity of equation (36), it is crucial to estimate whether the above assumption that the d-coupling for AH x is proportional to γ(x) is correct, which requires the establishment of the expression between adsorption energy and the electronic properties of adsorbates and substrates.
EMT [107] can be utilized to understand and obtain the expression of γ(x). In EMT, adsorption can be viewed as the adsorbates interacting with a homogeneous electron gas. Generally, there exists an equilibrium position of adsorption with electron density n 0 , where the adsorption energy reaches minimum. Nørskov et al [32] adopt the zero-order approximation in EMT (equation (18)) to rationalize the slope term of the LSR, obtaining the expression for AH x : The d-band model suggests that the coupling to d-states scales with the square of the coupling matrix element V 2 ad . The decay length of n surf is related to work function, which is the energy of the Fermi level relative to the vacuum. Since d-states also has energy close to the Fermi level, the decay length of d-bands may approximately the same as that of n surf , namely V 2 ad ∝ n surf [32]. Therefore, one can approximate that as follows, Equation (39) demonstrates the reasonability of equation (36) and unravels an approximate physical picture about the dependence of the LSR' slope. However, an obvious deficiency of these derivations is the neglect of the first-order term of EMT, which had been demonstrated to be essential for describing the adsorption energy of adsorbates that are not particularly polarizable such as hydrogen-containing molecules [107].
Now we turn to discuss the intercept of the LSR. Calle-Vallejo et al [33] find a linear correlation between the offsets of the adsorption-scaling for oxygenates versus oxygen and the usual coordination number CN u (see figure 15(e)), namely the offset of the LSR depends on the geometric structures of surfaces. To rationalize this finding, they made a key assumption, by assuming that the electronic and geometric determinants of adsorption energy can be separated independently. Given species at facet i, one can obtain that: where { ω j } is a set of electronic-structure parameters, g (i) is a function of { ω j } , and β is a parameter of surface geometry. Thus, the general expression for the offset of the LSR is as, To obtain the dependence of the offset of the LSR on the surface geometries, it immediately requires , which is exactly the assumption that made by Nørskov et al [32] and Koper et al [138] for explaining the slopes of LSR. Considering a special case that species 2 and 3 scale with species 1 with the same slope such as OH and OOH versus O, one obtains, Thus, the difference of adsorption energy is only determined by the geometric parameter β. The fitted correlations in figure 15(e) [33] further indicate that although β (i) 3 and β (i) 2 depend on the coordination numbers, their difference is constant for all considered surfaces. Namely, the difference between adsorption energies for any pair of oxygenates that binding to surface with oxygen is constant despite the electronic nature and geometric structure of surface, which reflects the experimental thermodynamics limitation for ORR [33,180]. Now a complete expression of the LSR and its generalized form can be provided for atoms and their partially hydrogenated species as, Equation (43) represents the entire LSR and its generalized form. However, the coefficient of CN u term µ and the constant term ϑ 1,2 remain unclear and this expression has only been verified to be effective for oxygen and its partially hydrogenated species such as OH x (x = 0, 1), OOH and OCH 3 . Despite the great significance, the LSR and its generalized form still need further evidence and support from a more basic perspective. First, it is crucial to verify whether the assumptions that the d-coupling for AH x is proportional to γ(x) and the electronic and geometric determinants of adsorption energy can be separated independently are correct. Secondly, the generalized LSR does not provide a general expression for determining the geometric effect, with the prefactor of the coordination term and the constant term in equation (43) ambiguous. Thirdly, the perturbation induced by the adsorbed atom to the electron density, namely the contribution of the first-order term of EMT to the adsorption energy, has to be identified. Fourth, the slope of LSR is only suitable for the atoms and their partially hydrogenated species and has not been applied to more complicated adsorbates like CO, CHO and COOH, which are crucial intermediates in numerous reactions. Last but not least, the LSR is the relationship between the adsorption energies of two species, thus requiring DFT-calculations to obtain the adsorption energy for at least one species on all considered substrates which are still expensive. To solve these issues, it is necessary to study the relation between the adsorption energy and the easily accessible intrinsic electronic and geometric properties of adsorbates and substrates.

The intrinsic model
Recently we propose a novel model [34] to correlate the adsorption energy with the intrinsic properties of adsorbates and substrates. This intrinsic model can naturally deduce the well-established LSR and its generalized form, generalizes the efficiency and limitation of engineering the adsorption energy and reaction energy, and enables rapid screening of potential candidates and designing of catalysts since all parameters are accessible and predictable. In this section, we introduce the framework of this intrinsic model and unravel its underlying theoretical basis and mechanistic insights.

Framework and performance
We identify the three main factors that control the adsorption on surfaces are the valence and electronegativity of surface atoms, the coordination of active sites and the valence of adsorbates, which correspond to the electronic descriptor ψ and the geometric descriptor CN. The entire expression of the adsorption energy on TMs and NPs can be expressed in terms of ψ and CN as, where X m and X are the maximum and actual numbers that the central atom of a given adsorbate bond to the specific coordinating group. θ is constant for a given adsorbate and is the only pending parameter that needs to be determined, e.g. through DFT calculations, while the rest parameters are intrinsic and readily accessible. Clearly, this scheme allows a rapid estimation of the adsorption trend on different surfaces (without calculation cost). Note that the determination of θ only needs the calculations of a given adsorbate on one substrate. One of the most important features of the intrinsic model is that we introduce a brand-new descriptor ψ to effectively consider the local-environment effect of the active sites based on the valence and electronegativity of surface atoms as: where S vi and χ i are the numbers of outer electrons and the electronegativity of the ith atom around the surface active center, and N is the sum number of neighboring atoms. Normally, the minimum repeat units are utilized to consider the effects of the local environment around the active centers. Taking the top-site adsorption on Pt-X@Pt(111) as an example, the local active centers contain one Pt atom at the adsorption site with N adsorption (Pt) = 1, and six Pt atoms with N 1st (Pt) = 6 and three X atom with N 1st (X) = 3 at the first-nearest neighboring sites so that the sum number of N is equal to 10. By considering the geometric mean of the valence and electronegativity for these atoms, one can obtain the value of the descriptor ψ. Note that equation (45) is a universal expression which is suitable for pure TMs, NPs, NSAs, oxides and SACs. Since the active center and its local coordinations are the same in pure TMs and NPs, equation (45) automatically decays to a simpler form in calculating pure TMs and NPs that:   Figure 16 exhibits the adsorption energies against the electronic descriptor ψ for various adsorbates on TMs, NPs, NSAs and oxides [34]. The adsorbates we considered cover CH x (x = 0, 1, 2, 3), CO, COH, CHO, CHOH, COOH, NH x (x = 0, 1, 2), NNH 2 , OH x (x = 0, 1), OOH, OCH 3 , P, F, and Cl and the surfaces include the TM extended surfaces, NPs, NSAs with Pt and Pd hosts, monoxides, dioxides and perovskite oxides [13,18,32,138,141,152,154,181,182]. Clearly, the adsorption energies on all surfaces scale linearly with ψ and the predicted coefficients of the ψ term by equation (44) are in good agreement with the direct fitted ones of DFT calculations (see tables 2 and 3 and figure 16) [13,18,32,138,141,152,154,181,182], which strongly supports the reliability and generality of the scheme. Specially, for oxides, the adsorption energies obey the relation as E ad = 2 × Xm− X Xm + 1 ×ψ + b. Note that the constant prefactor of oxides is 2 rather Table 2. Comparison between the predicted prefactors of ψ [34] in equation (44) and the fitted DFT-calculated ones for different adsorbates on pure transition-metal (TM) close-packed surfaces (CPS), (211) surfaces and (100) surfaces.
NSAs: |k| = 0.1× (Xm − X)/(Xm + 1) i [141] ii [181] iii [138] iv [182] Species  (44), ensuring that the efficiency of engineering the adsorption energies for pure TMs, NSAs and oxides is roughly the same from one surface to the next. The linear relationship between the offset b (b = E ad − 0.1× Xm− X Xm + 1 ×ψ) and CN is illustrated in figure 17 [34], where the DFT-fitted slopes are also consistent with the predictions by equation (44) (see table 4) [26][27][28][29][30][31]33]. The wide choice of the species and surfaces [the species involve CO, CHO, COOH, COH, NH x (x = 0, 1), OH x (x = 0, 1), OOH and OCH 3 , and surfaces cover Au, Cu, Ag, Pt, Co and Ni extended surfaces and NPs] ensures the generality and reliability of this model. All these findings demonstrate that the intrinsic model captures the inherent electronic and geometric determinants of adsorption systems in adsorbate-substrate bonding.

Theoretical origin and basis
Having exhibited the high performance of the intrinsic model, we now show the underlying theoretical mechanisms of this scheme. Since the coupling between adsorbates and surfaces is determined by the valence properties of adsorption systems, it is reasonable for the electronic descriptor ψ to characterize the electronic nature of adsorption, as ψ is a function of the valence-electron number and electronegativity of surface atoms. Figure 19 compares the electronic descriptor ψ with several established descriptors, revealing the theoretical origin and basis of ψ [34]. ψ scales linearly with the d-band center ε d for 17 TMs but quadratically with that of the other 7 TMs (see figure 19(a)). It is noteworthy that these 7 TMs are the ones possessing hybridization energies that depend strongly on ε d but weakly on the d-band width W d , namely those can be well described by the d-band center model [112]. Hence, we introduce a W d -dependent term to test this effect. By introducing the W d effect on the top of ε d with the formula of ε d +W d /8 for the 7 outlying metals, these 7 TMs immediately shift to the queue established for the other 17 TMs that only uses ε d (see figure 19(b)). By introducing the W d effect on the top of ε d with the formula of ε d −W d /8 for the 17 TMs (but only using ε d for 7 outlying metals), the 24 TMs consistently exhibit a quadratic relationship with ψ (see figure 19(c)). Hence, one can speculate that ψ reflects both the properties of ε d and W d , thus capturing the intrinsic electronic features of adsorption. Moreover, ψ exhibits an evident linear relationship with the upper edge of the d-states [112], namely the refined descriptor ε d + W d /2 (see figure 19(d)), which accounts for the effects of both the average energy of d-bands and their spread energy on the position of adsorbate-metal antibonding states. Actually, ψ performs even better than ε d + W d /2 due to the fact that ψ describes a single scaling line with the adsorption energy for TMs while ε d + W d /2 displays three different lines that are divided by the periods of TMs (see figure 7).
However, to the best of our knowledge, the descriptors established from the NA model are mainly suitable for pure TMs and TMs with few intermetallics, but not for oxides. Since the intrinsic model is general for the adsorption on TMs, NPs, NSAs and oxides, in particular for the adsorption of OH, F and Cl on NSAs and the adsorption of OH x on oxides that cannot be accurately described by the d-band model [5,8,9], ψ indeed captures the electronic nature of NSAs and oxides. The reason of the inefficiency of the d-band model is most likely due to the fact that the d-band model cannot accurately characterize the effect of the local environment around the active sites. For the NSAs, ψ likely reflects not only the d-band center projected on the adsorption site but also the interaction between the local d-states of the substrate and the adsorbate state [141]. In the case of oxides, we correlate ψ with the number of outer electrons and the p-band center (or the O-2p average energy), which have been demonstrated as good descriptors in describing the trends of adsorption energy for oxides [152,154]. Notably, ψ is also associated linearly with the number of outer electrons and the p-band center (see figures 19(e) and (f)), implying that ψ well manifests the outer-electron characteristics of surface atoms for oxides that cannot be depicted by the d-band model.
The prefactor of the electronic term Xm−X Xm + 1 can be understood and deduced by EMT [107]. As we have discussed above, in EMT, the inhomogeneous adsorption system is approximated as the interaction between the adsorbate atom and the homogeneous electron gas, so that the adsorbate atom exhibits a noble gas behavior at the optimal adsorption structures. Importantly, the first-order approximation of EMT [equations (18) and (19)] is essential for describing the adsorption of atoms that are not particularly polarizable like oxygen. One can derive the zero-order term based on the homogeneous electron gas model  [112]. (e) The outer-electrons (Noe) versus ψ for monoxides (MO) and perovskite oxides (ABO3) [152]. (f) The p-band center (p-band SO) versus ψ for ABO3 [155]. Reproduced from [34]. CC BY 4.0.
that E (0) ∝ Xm−X Xm n 0 (n 0 is denoted as the homogenous electron density). Note that Nørskov et al [32] utilized the zero-order approximation to rationalize the slope term of the LSR as discussed above, although the slope of the LSR is not simply determined by the zero-order approximation. Since the bond number of adsorbates can only be integer, we introduce a perturbation effect induced by the adsorbed atom to the electron density, as ∆n = n 0 /(X m + 1). Then the first-order term is as follows, Therefore, as long as the considered adsorbates are not particularly polarizable, the adsorption energy E ad should obey the relation as, Equation (48) rationalizes the prefactor of the electronic term in equation (44). Actually, it is the determination of equation (48) that generates the slope of the LSR which is expressed by Xm−X Xm , as the results in [32], which we will derive in the following.
The prefactor of the coordination term X + 1 Xm + 1 in equation (44) can be explained by the bond-order conservation theory [171,172]. From the view of substrate, the adsorption energy is proportional to the coordination of active sites, which means that the saturated-bond number of active center controls the binding strength to adsorbates. This scheme also holds for adsorbates so that the adsorption energy is proportional to the saturated-bond number of adsorbates.
The constant θ term in equation (44) is a parameter related to adsorbates and is most likely attributed to the coupling between the valence of adsorbates and the sp-states of the metals in terms of the d-band model. We also find that θ is strongly related to the bond energy between the atom binding to surface and its coordination atoms in adsorbates (see figure 20).

Implications
The intrinsic model incorporates the effects of electronic and geometric structures of substrates and adsorbates together to determine adsorption, thereby providing a novel physical picture for adsorption. This model identifies the previous assumption that the electronic and geometric structures for substrates are independent of each other in determining adsorption and gives an intrinsic expression of adsorption energy [33,70]. Hence, the intrinsic model can automatically derive the LSR and its generalized form [32,33], uncovers the efficiency in modulating the adsorption energy and reaction energy, and deduces naturally the thermodynamic limitations of catalysis. All these implications pave the ways for applying the intrinsic model in catalysis. Remarkably, the intrinsic model can derive the LSR and its generalized form automatically. For atoms (A) and their partially hydrogenated species (AH x ) on TM surfaces, one obtains that in terms of equation (44): The actual bonding number for atoms (A) and their partially hydrogenated species (AH x ) are 0 and x respectively. Combining equations (49) and (50), one can naturally derive the correlation between the two species as, this is exactly the LSR, as equation (36). Moreover, with equations (44) and (51), the offset of the LSR is as follows, It is noteworthy that CN u and CN are almost identical in describing the extended surfaces. Clearly, the offset ξ should scale linearly with the coordination of surface atoms, namely we theoretically derive the generalized form of the LSR [33]. Moreover, the generalized LSR only demonstrates that the offset is related to the coordination of surface atoms but cannot give the accurate expression for the prefactor of the coordination term and the constant term. Our scheme remedies this shortcoming and provides a general expression for the offset of the LSR, proposing that the prefactor of the coordination term is determined by the bonding numbers of the central atoms of the two species (see equation (52)). The prefactors of CN is predicted to be ∼0.10 for the LSR of OH vs. O, OOH vs. O and OCH 3 vs. O since X 1 = 0, X 2 = 1 and X m = 2 OH radical, OOH radical, OCH3 radical, OCHCH2 radical, OCHCH3 radical, or OCH2CH3 radical. Reproduced from [34]. CC BY 4.0. in equation (52). This is in good agreement with the direct DFT-fitted results (0.08-0.11) [33]. We also identify that the constant term of the coordination-determined relation is governed by the θ term of the two adsorbates and is affected by their actual bonding numbers. Overall, the intrinsic model can derive the LSR and its generalized form naturally by providing a more general expression, reflecting the solid physical-chemical foundations.
With the intrinsic model, one can engineer the adsorption energy for a given adsorbate by modulating the electronic and geometric structures of substrates respectively. According to equation (44), the prefactors of electronic and geometric terms dominate the efficiency of the electronic and geometric determinants in engineering the adsorption energy, namely it is the adsorbates that control the efficiency in modulating the adsorption energy. Figure 21 shows the efficiency in modulating the adsorption energy for species binding via C, N and O. Clearly that the electronic descriptor ψ is more effective in unsaturated atoms like C and N, but the geometric descriptor CN becomes more important in saturated atoms like CH 3 , COOH and NH 2 . This insight paves the way for effectively engineering the adsorption energy to obtain the optimal properties. Since the ψ of TMs and NPs are distributed discretely from 0 to 100, thus alloying allows one to fill in the gap between different TMs and NPs.
This scheme also uncovers the efficiency for modulating the reaction energy. The adsorption energy difference of any pair of adsorbates with the same central atom is as follows, clearly, the bonding number difference of the central atom between two species determines the adsorption energy difference. To maximize the modulating efficiency, ψ and CN should vary oppositely. We find that the predicted prefactors of ψ and CN of equation (53) are in good agreement with the direct DFT calculations for CO 2 RR (see table 5) [18]. For instance, the predicted prefactors of ψ and CN of equation (53) for CO → HCO are merely 0.01 and 0.02, corresponding to a small variation for the energetics of CO → HCO as the nature or structure of surfaces changes. This is consistent with the literature findings [14,18], where the limiting potential of CO protonation exhibits a nearly horizontal line over TMs. A special case is that with X 1 = X 2 , the adsorption energy difference is always constant regardless of the electronic and geometric structures of substrates, leading to a limitation that it is impossible to engineer the reaction energy through changing the nature or the morphology of surface sites for any pathways involving a pair of intermediates with the same bonding numbers of the central atom. This exactly corresponds to the known thermodynamic limitations of ORR [33,180]. Moreover, our scheme suggests that the associative pathway of NRR is Table 5. Comparison between the predicted and DFT-calculated prefactors (ψ and CN) [34] in equation (53) for five reaction steps of CO2RR on transition-metal (TM) surfaces. The DFT-calculated results are extracted from [18]. These outliers could be used to break the inherent adsorption thermodynamics to reach optimal performances. Overall, with the intrinsic model, one can naturally deduce the LSR and its generalized form, recognize the d-band model and the generalized coordination number model, and uncover the efficiency and limitations in modulating the adsorption energy and reaction energy, thereby enabling the potential applications in catalysis.

Applications
We have systematically discussed the electronic and geometric determinants of adsorption and enumerated several crucial models, which have profound significance for applications in many fields especially in catalysis. In the following, we specifically introduce some typical applications of these models in catalysis, such as ORR, CO 2 RR and NRR, compare their performances and indicate the direction for screen catalysts with optimal properties.

Oxygen reduction reaction
ORR, an important electrochemical reaction that plays a key role in alternative energy conversion and storage devices such as fuel cells and metal-air batteries [195][196][197][198][199][200]. It has been identified that a variety of materials exhibit high activity towards ORR including pure TMs and NPs [193,194], TM oxides [149,150], alloys [16] and SACs [168]. Therefore, finding the proper descriptors to explain the activity trends of ORR for these materials has been a long-term goal in the fields.
There have been some descriptors for describing the activity trends of ORR for TMs, Pt 5 M-series alloys and NSAs. Norskov et al [193,194,201] demonstrate that the binding energy of O (E b [O]) can well describe the activity trends of ORR for TMs as shown in figure 22(a). Clearly, pure Pt is the best catalyst for ORR among TMs which is in agreement with experiments. Pt 5 M, in which M represents the alloying elements with La-series metals and Ca, is one of crucial members in Pt-based catalysts [16,20]. The previous work by Escudero-Escribano et al [202] has correlated the activity of ORR for Pt 5 M with the lattice parameter, obtaining a volcano-shaped relation shown in figure 22(b). They propose that Pt 5 Tb and Pt 5 Gd are the best catalysts for ORR among considered substrates, which bind OH neither too weak nor too strong. Actually, the lattice constant reflects the strain effects of Pt overlayer on Pt 5 M: compressive strain generates the downward shift of the d-band center of Pt, resulting in the reduction of the binding strength of O-species and the enhancement of the activity of ORR [203,204]. NSAs represent an important family of alloy catalysts, where the guest atoms on top or second layer significantly change the electronic properties of surfaces [20,[135][136][137]. The activity of ORR as a function of adsorption energy of OH (E OH ) for NSAs is illustrated in figure 22(c), where the substrate is Pt (111) surface and the A x B 3−x /A y B 3−y formulation represents the thin film of the first and second atomic layer of the NSAs [205]. The volcano curve indicates that Pt 3 /PtAu 2 performs best among considered NSAs since it locates near to the peak of the volcano.
We find that the electronic descriptor ψ can give a good description for the activity towards ORR among all the above discussed substrates [206]. Figure 22(d) plots the volcano curve of the activity of ORR against ψ for pure TMs [194]. Remarkably, the left leg of the volcano exhibits two branches, dividing the TMs into two groups. The first group (the upper scaling line) includes the metals that all comply with the d-band model with ε d independent on W d , while the second group (the lower scaling line) represents the metals with ε d correlated strongly to W d [112]. As has been mentioned above, the electronic descriptor ψ can well capture the different role of W d for TMs (see figure 19(a)). The volcano behavior implies that Pt and Pd are the for pure TMs [194]. (b) The volcano curve of the activity of ORR versus the lattice parameter a for Pt5M-alloys [202]. (c) The volcano curve of the activity of ORR versus the binding energy of OH relative to pure Pt E ad [OH] − E ad [OH](Pt) for NSAs [205]. (d)-(f) The volcano curve of the activity of ORR versus the electronic descriptor ψ for pure TMs [194] (d), Pt5M-alloys [202] (e) and NSAs [205] (f). (a), (c), (d)-(f) are reprinted with permission from [206]. Copyright (2020) American Chemical Society. optimal catalysts towards ORR among considered TMs. There is still room to design catalysts to further reach the volcano peak. These results are consistent with the traditional adsorption-energy based volcano curve in figure 22(a). One of the advantages of the intrinsic model is that it can effectively screen alloy catalysts by only considering the first-group metals since catalysts always obey the electronic properties of the dominant metal in alloys, leading to a limitation in enhancing the activity for the second-group metals expect Cu, Ag and Au. In fact, the experiments and DFT calculations have proved this point: the activity of Ir alloys approaches that of pure Pt after alloying Ir with Co [207]; Pt 5 M family alloys are known to be more reactive than pure Pt [208]. Moreover, there are no alloys whose dominant metal comes from the second-group metals that are as active as pure Pt for ORR. Figure 22(e) plots the volcano curve between the activity of ORR and electronic descriptor ψ for Pt 5 M [202]. Note that the f -electrons of La-series metals are taken into consideration, namely the S v term in equation (45) including 4f -, 5d-and 6s-electrons. The reason of the deviation from the guiding line for Ca is attributed to the different properties of Ca compared with La-series metals: there is no f -electron for Ca. Clearly, Pt 5 Tb stands close to the peak of the volcano curve, serving as an optimal catalyst among Pt 5 M for ORR. It is noteworthy that the activity of Pt 5 Dy is substantially less than that of Pt 5 Tb despite the similar value of ψ (53.46 for Pt 5 Dy vs 52.71 for Pt 5 Tb), which is due to the fact that the right leg of the volcano curve decreases dramatically compared to the left one. This implies that it is more convenient to choose the metals belonging to the volcano's left leg to further design advanced Pt 5 M-based catalysts for ORR. Moreover, this result also implies that ψ captures the strain effects of Pt 5 M alloys, namely it is the intrinsic valence and electronegativity of Pt and La-series metals that control the strain effects of Pt overlayer and the activity of ORR. The activity of ORR as a function of ψ for NSAs is illustrated in figure 22(f) [205], where the volcano curve exists. Remarkably, ψ divides the NSAs into three classes compared to the adsorption-energy based descriptors in figure 22(c), exhibiting three different volcano-type behaviors. It is intriguing that the classification is based on the components of top surface layer atoms, and the efficiency of modulating the activity of alloying is in the order Pt 3 > Pt 2 M > PtM 2 for NSAs in the left leg of the volcano due to the different slope. Therefore, one obtains a strategy to design optimal NSA catalysts: pure Pt is selected as the first-layer metal of NSA and adjusting the components of the second layer with a bit smaller ψ than that of Pt 3 /PtAu 2 on Pt(111) surface. Overall, the electronic descriptor ψ not only characterizes the activity trends of ORR for pure TMs, Pt 5 M-series alloys and NSAs but also indicates the strategy to screen optimal catalysts for ORR.
In addition to the determination of the optimal materials for catalysts, it is also necessary to find out which structure or site is more prior, namely the geometric effects must be taken into account. In the case of ORR, the first proton-electron transfer (O 2 is transferred into * OOH) and the last proton-electron transfer ( * OH is transferred to H 2 O) are the two potential-determining steps [194]. Figure 23 plots the reaction energies of these two steps as a function of the generalized coordination number CN [26]. The typical volcano curve indicates that the optimal activity should have geometric sites with CN ≈ 8.3. Clearly, the optimal sites should possess more neighbors than (111) terrace whose CN is equal to 7.5 in extended surfaces and sufficiently large NPs. Although some sites possess larger CN u than (111) terrace such as the bottom of (100) step sites (CN u = 10) and the bottom of (111) step sites (CN u = 11), their CN are 8.75 and 9.5 respectively, exceeding the predicted optimal value. In addition, these sites normally have weak activity due to the steric hindrance and adsorbates on these sites are inclined to diffuse to the neighboring uncoordinated sites. Thus, an effective method to reach the optimal activity is to artificially create some defective sites with CN close to 8.3. In figure 23, Calle-Vallejo et al [26] propose two kinds of cavities whose CN are 8.0 and 8.17 respectively. Theoretically, these two cavities should possess good catalytic properties. This prediction has been confirmed by experiments [26]. The defective Pt electrodes can increase the activity 3.5 times compared to Pt (111) surface, demonstrating that the geometric-descriptor based volcano curve is indeed reasonable to screen catalysts with optimal sites. Actually, this geometric effect to design optimal facets is involved in the intrinsic model [34] as seen in equation (44).
We also identify that the electronic descriptor ψ can be used to elucidate the activity of ORR for SACs [206]. where Sc, Ti and V systems stand at the left branch and the rest SACs locate at the right-higher or right-lower branch. This classification is related to the element period of the involved TMs in SACs. The third period TMs (Cr, Mn, Fe, Co, Ni and Cu) constantly locate at the right-lower branch while the fourth period TMs (Mo, Ru, Rh and Pd) are in the range of the right-higher branch. Intriguingly, N-doping affects the behavior of the fifth period TMs such as Pt, Ir and Os: Pt and Ir (Os) belong to the right lower (higher) branch for M-C 4 -G, while they are classified into the right higher (lower) branch for M-N 4 -G. The special case for Au and Ag is due to the deep-lying d-bands relative to the Fermi level. Moreover, the crystal structures of doped TM atoms on SACs in bulk also contribute to distributions of the SACs on the relation of E ad vs ψ. The adsorption energies of O on SACs increase as the bulk structures of the doped TM atoms change with the order of bcc < hcp < fcc. Figures 24(c)-(f) shows the onset potential of ORR against ψ on SACs. One remarkable feature is that the left branch TMs are integrated into the right-lower one and the whole trends show two volcano-type relations. Rh and Pt for M-C 4 -G and Co and Fe for M-N 4 -G locate near the peak of the volcano curve, thus are the optimal catalysts for ORR among the considered SACs. The volcano curve provides guidance to design catalysts for SACs: one should choose the right branch candidates in figures 24(a) and (b) to serve as the catalysts of ORR.  [168]. Reprinted with permission from [206]. Copyright (2020) American Chemical Society.
Overall, the electronic descriptor ψ can elucidate the volcano-type trend of activity of ORR for pure TMs, NPs, Pt 5 M-alloys, NSAs and SACs. Namely ψ is a universal descriptor in describing the catalytic properties of ORR.

CO 2 reduction reaction
As a key reaction for producing fuels and chemicals that matter to the renewable energy, CO 2 RR has attracted extensive attention. Normally, CO 2 RR contains four essential pathways which generate CO, HCOOH, CH 4 /CH 3 OH and H 2 respectively [209,210]. Numerous intermediates are included in these four pathways such as C-terminated species, O-terminated species and H atoms, which are also important in understanding many other reactions. Moreover, CO 2 RR has been performed both experimentally and theoretically on TMs and NPs [18,27,28], TM oxides [211,212], alloys [213,214] and SACs [189,215,216]. Thus, it is crucial to identify effective descriptors to characterize the selectivity of these pathways for CO 2 RR on these substrates.
The previous works have proposed some descriptors to explain the trends of activity towards CO 2 RR. Nørskov et al [14] use the binding energy of CO as a descriptor to elucidate the activity trend of CH 4 production since the protonation of CO is the rate-determining step, identifying that Cu serves as the optimal catalyst among considered TMs (see figure 25(a)). The adsorption energy of H * is utilized to clarify the trends of hydrogen reduction reaction (HER) over TM surfaces [18] (see figure 25(b)). Moreover, the adsorption energy of COOH * is considered as the descriptor to explain the trend of CO production by Jaramillo et al [217] since the adsorbed COOH * is the key intermediates for CO production, proposing that Au should be the optimal catalyst since it stands at the peak of the volcano (see figure 25(c)). They also find that the * HCOO binding energy could serve as a descriptor for the activity of CO 2 RR to HCOO − . Sn appears close to the top of the volcano and acts as the optimal catalyst (see figure 25(d)). Although these descriptors rationalize the activity trends of the corresponding pathways, they can hardly reveal the entire selectivity of CO 2 RR since the four pathways are competitive and coupled. Note that the binding energy of CO has been considered as a universal descriptor to characterize the activity towards the entire CO 2 RR, unfortunately, the underlying basis is unclear.
The intrinsic model provides a new picture in describing the activity of CO 2 RR [218]. Figures 26(a) and (b) exhibit the theoretical onset potentials for four competitive pathways (CO, HCOOH, CH 4 /CH 3 OH and H 2 formation) versus the electronic descriptor ψ on both close-packed and stepped surfaces [18,219]. Clearly, the volcano-shaped trends can be observed for all four competitive pathways. This is encouraging, since ψ could compare the activities for multiple pathways simultaneously which is inaccessible or controversial for the traditional adsorption-energy based descriptors. The intrinsic model suggests that Cd is inclined to produce HCOOH, while Pt, Pd and Ti favor H 2 production. As Cu locates at the turning point or the peak of the volcano for the CH 4 /CH 3 OH production, Cu serves as the optimal catalyst among considered TMs. Figure 26(c) plots the correlation between the experimental onset potential and ψ [220]. The volcano-shaped behavior of the experimental onset potential further confirms the applicability of this Figure 25. The traditional models in describing the activity of CO2RR for transition metals (TMs). (a) Limiting potential (UL) of the CH4 production versus the binding energy of CO for TMs. The data are adapted from [14]. (b) UL of the HER versus the adsorption free energy of * H for TMs. The data are adapted from [18]. (c) The CO partial current at −0.9 V vs RHE versus the binding energy of COOH * for TMs. (d) The HCOOpartial current at −0.9 V vs RHE versus the binding energy of * HCOO for TMs. The data are adapted from [217]. scheme, where Cu is inclined to produce CH 4 /CH 3 OH as it stands at the turning point of the volcano and Au shows the optimal performance towards the overall CO 2 RR. By comparing the experimental trends with theoretical results [18], one can easily obtain that the experiments in [220] are performed on close-packed surfaces of TMs since the trend of CH 4 /CH 3 OH production of the experiments is close to the theoretical prediction on close-packed surfaces. The trends in figure 26(c) can also be utilized to guide the selectivity of CO 2 RR. For TMs with ψ < 76, the selectivity of CH 4 /CH 3 OH is enhanced with the decreasing ψ; for TMs with ψ > 76, the increase of ψ from 76 improves the selectivity of CO and other products like HCOOH. Experimental partial current densities for the overall CO 2 RR at −0.8 V and for the CO and HCOOH production at −0.9 V are correlated with ψ shown in figures 26(d) and (e) [217,220], where the volcano curves emerge again. Au has the highest activity for the overall CO 2 RR at −0.8 V. For the CO and HCOOH production at −0.9 V, Au and Cu possess the highest activity respectively. Similar to the relation of onset potential and ψ, ψ can characterize the selectivity of the CO and HCOOH production at −0.9 V in experiments: TMs with ψ < 64 are prone to produce HCOOH while TMs with ψ > 64 are inclined to generate CO. The experimental exchange currents of HER are correlated with ψ in figure 26(f). The clear volcano curve can be observed and Pt locates at the peak of the volcano, serving as the optimal catalyst for HER among considered TMs [221,222]. The unreactive TMs such as Cu, Au and Ag stand at the right leg of the volcano, while the reactive TMs locate at the left leg. Note that the left leg of the volcano is divided into two branches, which is consistent with the conventional adsorption-energy based descriptors. This is due to the difference of the transfer coefficient in one electron transfer reaction. All these findings demonstrate that the electronic descriptor ψ can well clarify the activity and selectivity of CO 2 RR. Figure 26(g) exhibits the theoretical onset potential for CO, CH 4 /CH 3 OH and H 2 production as a function of ψ on M-N 4 -G SACs [215]. The clear volcano curves for all three competitive reduction pathways verify that ψ is indeed reasonable to elucidate the activity trend of CO 2 RR for M-N 4 -G SACs. Note that Rh-N 4 -G and Ir-N 4 -G possess the highest activity for all three pathways and serve as the optimal catalysts. The left leg of these volcano curves shows clear selectivity towards CO 2 RR, while the selectivity towards CO 2 RR for the right leg is comparable. It is clear that Rh-N 4 -G and Ir-N 4 -G are inclined to produce CO at a low onset potential, Fe-N 4 -G and Co-N 4 -G produce CO and/or H 2 with an overpotential of −0.4 V, while a high overpotential is required for Ni-N 4 -G to CO production. The experimental activity volcano curve of The theoretical onset potential versus ψ for CO, CH4/CH3OH, HCOOH, and H2 production on close-packed (a) and stepped TM surfaces (b) [18,219]. (c) The experimental onset potential versus ψ for the total CO2RR (to any product) and CH4/CH3OH production [220], along with the theoretical onset potential versus ψ for CH4/CH3OH production on TM close-packed surfaces [18]. (d) The current density versus ψ for the total CO2RR on TM surfaces [220]. (e) The experimental current density versus ψ for CO and HCOOH production on TM surfaces [217]. (f) The experimental current density versus ψ for HER on TM surfaces [221]. (g) The theoretical onset potential versus ψ for CO, CH4/CH3OH, and H2 production on M-N4-G SACs [215]. (h) The experimental onset potential versus ψ for CO and H2 production and turnover frequency (TOF) versus ψ for CO on M-N4-G SACs [223]. Reprinted with permission from [218]. Copyright (2020) American Chemical Society. CO 2 RR against ψ for M-N 4 -G SACs is illustrated in figure 26(h) [223], which is in good agreement with the theoretical predictions. The experimental order of catalytic activity also fulfills the theoretical prediction: Co-N 4 -G > Fe-N 4 -G > Mn-N 4 -G > Cu-N 4 -G > Ni-N 4 -G. Moreover, figure 26(h) demonstrates that both the volcano relationship between the activity of H 2 production and ψ and the order of the catalytic activity for H 2 production on M-N 4 -G SACs exhibit excellent consistency between theoretical predictions and experimental measurements. Hence, ψ can be used to clarify the catalytic properties of CO 2 RR for both TMs and SACs, exhibiting good generality [218].

Nitrogen reduction reaction
NH 3 is one of the most promising carbon-free alternative energy storage materials with a high weight fraction of hydrogen [224,225]. Traditional Haber-Bosch process to produce NH 3 is incomplete and requires harsh conditions [226], making NRR an alternative strategy for green and sustainable production of NH 3 due to the environmental-friendly character and mild reaction conditions. The previous studies have demonstrated that TMs [13], TM oxides and sulfides [227,228], SACs and bi-atom catalysts (BACs) [229,230] can serve as the potential catalysts for NRR. Hence, effective descriptors should be made to elucidate the activity trends towards NRR for these catalysts.
For NRR, the volcano curve has been found as a function of binding energy of nitrogen to the TM close-packed and stepped surfaces seen in figures 27(a) and (b) [231]. The volcano curve illustrates that Ru is the optimal catalyst towards NRR. The intrinsic model can also be utilized to elucidate the activity of NRR. Figures 27(c) and (d) exhibit the volcano curve that is obtained by plotting the limiting potential for NRR versus the electronic descriptor ψ for TMs on both close-packed and stepped surfaces [231]. Remarkably, the peak of the volcano curve for stepped surfaces shifts rightwards relative to that for close-packed surfaces, which is attributed to the different determining steps for the left leg of the volcano. The left leg on close-packed surfaces corresponds to the NH * protonation to form NH 2 * while that on stepped surfaces represents the proton transfer from NH 2 * to NH 3 . Overall, Ru should be the optimal catalyst for NRR since it stands close to the top of the volcano curve, which is in good agreement with the traditional adsorption-energy based descriptors. It is noteworthy that all considered TMs exhibit higher activity on stepped surfaces than on close-packed surfaces except Re, following the bond-order conservation rule that the surfaces with smaller coordination number are more inclined to form new bonds. The reason for the abnormality of Re is due to the fact that the adsorption energies of * NH 2 and * NH for Re are too strong, leading to the lower limiting potential. Similar result has been made by Wang et al [232] that Mo (110) preferred orientation and NP-clustering morphology were beneficial for NRR compared with Mo foil. This scheme also allows one to uncover the surface selectivity for NRR: close-packed surfaces possess higher activity for TMs with 25 < ψ < 36 while stepped surfaces are more favorable for TMs with 29 < ψ < 53. This provides beneficial guidance to design optimal catalysts for NRR. Note that the onset potential of HER is always larger than that of NRR (see figure 27), namely the activity of HER is higher than NRR. Hence, it is crucial to suppress HER in order to improve the NRR.

Conclusions and remarks
In summary, we introduce several typical models with respect to the electronic and geometric structures of substrates and adsorbates for describing and predicting adsorption energy, including the d-band model, the coordination-based models, the LSR and the intrinsic model, and uncover the inherent connection of these models. Among these models, the d-band model is original and can serve as a reasonable basis for understanding and rationalizing the other subsequent models, the generalized coordination model is prominent in describing the geometric effect of substrates, and the LSR is effective in correlating the adsorption energies between different adsorbates, while the intrinsic model bridges the gap between the adsorption energies and the readily accessible intrinsic properties of adsorption systems and provides an effective way to capture the local environment effects of active centers. The intrinsic model reflects a relatively comprehensive physical picture for adsorption on solids and is more effective for rapid screening materials with optimal adsorption properties, since it combines the nature of the d-band model, the generalized coordination model and the LSR and its parameters can be obtained by table lookup. In particular, the easily accessible parameters of the intrinsic model are convenient for machine learning to do material design. By further applying these models into reactions, such as ORR, CO 2 RR and NRR, one can see that the adsorption energy is universal in describing reactivity because of the Sabatier principle, while the intrinsic model is also effective in describing reactivity and selectivity across multiple scale materials. We expect all these models to continue to play an important role in catalytic reactions in the future.
Although the introduced models have been useful and successful in describing the adsorption properties, there are still many challenges that need to be further explored: (a) There still lacks a complete picture for understanding the strain effects [132,134] on adsorption. The underlying mechanism of strain effects has to be identified based on the electronic and geometric structures of adsorption systems. (b) The adsorption properties are not yet quantitatively correlated to the stability of materials, which is crucial in applications, although many attempts have been made in this regard [233,234]. A model that bridges the gap between the stability and activity of catalysts needs to be uncovered in the future. (c) The reaction barriers and diffusion barriers of adsorbates are essential for catalysis but are rarely correlated with the properties of adsorbates and substrates. One needs to find out the applicability of Brønsted-Evans-Polanyi relation and seeks a more general picture for correlating thermodynamics and kinetics of reaction [7,[235][236][237]. Moreover, it is also crucial to build a simple model for describing the friction and energy dissipation during reactions [238][239][240][241][242][243].