Augmented Pairwise Additive Interaction Model for Lateral Adsorbate

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Interfaces: Adsorption, Reactions, Films, Forces, Measurement Techniques, Charge Transfer, Electrochemistry, Electrocatalysis, Energy Production and Storage

Augmented pairwise additive interaction model for lateral adsorbate interactions: the NO-CO reaction system on Rh(100) and Rh(111) Lu Tan, Liangliang Huang, Yingchun Liu, and Qi Wang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b04383 • Publication Date (Web): 05 Apr 2018 Downloaded from http://pubs.acs.org on April 5, 2018

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Augmented pairwise additive interaction model for lateral adsorbate interactions: the NO–CO reaction system on Rh(100) and Rh(111) Lu Tan,† Liangliang Huang,‡ Yingchun Liu,∗,† and Qi Wang∗,† †Department of Chemistry, Zhejiang University, Hangzhou 310027, People’s Republic of China ‡School of Chemical, Biological & Materials Engineering, University of Oklahoma, Norman, Oklahoma 73019, United States E-mail: [email protected]; [email protected] Abstract Lateral adsorbate interactions have been acknowledged to play an important role in heterogeneous catalytic kinetics. To quantify such energies efficiently and accurately, a lattice-gas-based augmented pairwise additive interaction model was proposed. The model Hamiltonian is defined as the summation over all isolated binding energies and all pairwise interactions, with the pairwise interaction used for each adsorbate pair consisting of the corresponding isolated pairwise interaction and some needed modification terms if the surroundings meet specific conditions. The parameters used in this augmented pairwise additive interaction model for the NO–CO reaction system on Rh(100) and Rh(111) were collected based on DFT calculations. The reliability of the model was examined by the reproduction of some DFT-based predictions and

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experimental observations. Results suggest that the lateral interactions on Rh(100) are relatively short-range and greatly susceptible to the lateral displacements of adsorbates, whereas on Rh(111), the lateral interactions are relatively long-range and the lateral displacements of adsorbates are more limited. In addition, the Brønsted–Evans–Polanyi relations for adsorbate diffusion were constructed, and a modification strategy was proposed for certain diffusion processes. This work opens up the possibility of accurate microkinetic modeling for this reaction system with a faithful account of lateral interactions, and motivates the extensions to other complex surface systems.

Introduction The purification of hazardous emissions from industrial, commercial and personal activities is one of the biggest challenges in our modern society. Among them, the aftertreatment for automobile exhaust gases has received widespread attention. 1–4 To date, the so-called threeway catalyst has had a great achievement on reducing emissions of nitrogen oxides (NOx ) along with carbon monoxide (CO) and uncombusted hydrocarbons (HCs) for the conventional stoichiometric gasoline engine. Rh, though more expensive than other components, is widely used in this catalyst due to its outstanding NOx reduction activity. 2 However, due to the increasingly stringent regulations and endless needs for clean atmosphere across the world, a more effective strategy is extremely necessary. Therefore, a in-depth knowledge of the NO–CO reaction over Rh catalyst is of great importance either to improve the performance of current three-way catalyst or to develop more efficient and economical alternatives. Interactions among adsorbates, or lateral interactions, have long been known at least since ordered adlayer structures were discovered at low temperature. They have more recently been acknowledged to play an important role in heterogeneous catalytic kinetics. 5–8 Such interactions have a major impact on both binding energies and spatial arrangements of adsorbates on catalyst surfaces, and consequently affect activation energies and rate constants of surface chemical reactions. This is also true for the NO–CO reaction over Rh 2


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catalyst. For a better understanding and prediction of the intrinsic kinetics and concomitantly the reactivity and selectivity of the catalyst, it is necessary to quantify the lateral interactions in this reaction system. Because of the extreme complexity, lateral interactions on catalyst surfaces are quite suitable to be studied at the atomic scale by quantum chemical calculations, especially using density functional theory (DFT). Such calculations have an advantage over experiments because one knows precisely the surface structures and adsorbate arrangements being dealt with. However, due to their great computational cost, periodic DFT methods are limited in the sizes of supercells and thus in the range of adsorbate configurations that can be probed. To this end, a computationally efficient alternative is needed. One promising method is developing an analytical model Hamiltonian and parameterizing it by affordable DFT calculations. Once developed and parameterized, the model Hamiltonian can be used to rapidly calculate the energy of any arbitrary adsorbate configuration. Combined with microkinetic modeling methods like kinetic Monte Carlo simulation, one can obtain the statistical properties of surface chemical reactions much more precisely than through the traditional mean-field approach. Given the discrete nature of surface adsorption sites, lattice-gas model, which makes a coarse-grained description of the surface by mapping it into a well-defined lattice, forms now an active area of research. The cluster-expansion (CE) methodology has attracted lots of attentions recently in the field of constructing such Hamiltonians based on DFT calculations. 9–13 In the CE Hamiltonian formalism, the total interaction energy is given by a linear combination of the clusters, i.e., pair, trio, and higher-order interactions among neighbor adsorbates. By fitting to a training dataset containing limited adsorbate configurations derived from DFT calculations, a set of effective cluster interactions for the chemical system one is interested in can be obtained. Because an arbitrarily large number of clusters may be included in the expansion, a systematic improvement of accuracy is possible. The crossvalidation method and the more recently introduced Bayesian statistics 14–16 are often used

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to select the most relevant clusters for the CE Hamiltonian built. Despite its universality in principle, the construction of this CE Hamiltonian can be tedious in practice. This approach is fundamentally limited by the number of calculations needed to generate the required training dataset for the targeted system. The configuration space which needs to be covered may easily surpass the number of configurations that can be calculated directly by DFT method. Three or more adsorbate species is nowadays a nontrivial problem size for the modern supercomputing capacity, not to mention that the problem is exacerbated if the catalyst surface contains several inequivalent adsorption sites. Therefore, the generation of a CE model is impractical for the NO–CO reaction system we are targeting at. Recently, more and more scholars devote themselves to developing more applicable methods for complex heterogeneous catalytic systems. 17–21 Some of the methods are truly inspiring. However, there are few suitable ones for the problem we are facing so far. In this paper, we parameterize the various pairwise interactions (PIs) in the NO–CO reaction system based on DFT calculations, and develop a description of lattice-gas-based augmented pairwise additive interaction (APAI) model to determine the energy of any adsorbate configuration efficiently and accurately. Four kinds of adsorbates are taken into account, namely, N, O, NO, and CO. These are the main surface species in the NO–CO reaction system. Meanwhile, the properties on both Rh(100) and Rh(111) are investigated. These two termination facets of Rh crystal are of particular importance because Rh(111) is most compact thus thermodynamically most stable, and Rh(100) is slightly more open and thus believed to have better catalytic capability. Some model Hamiltonians relevant to the NO–CO reaction system on these two surfaces have been constructed previously. Most of them include only pairwise interactions focusing on different subsets of the four species, 22–26 and others with higher-order interactions only involve one kind of surface species. 14,27 Our model improves on these previous models both from the number of species considered and the response of lateral interaction to local configuration. In addition, the diffusion behavior of

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the four species is also studied, aiming at providing fundamental information for integrating the APAI model into further kinetic modeling.

Computational details Spin-polarized DFT calculations employing the revised form of Perdew–Burke–Ernzerhof (RPBE) 28,29 generalized gradient approximation (GGA) functional for the exchange-correlation energy are carried out in the Vienna ab initio simulation package (VASP). 30,31 The projector augmented wave (PAW) method 32,33 is adopted for calculating the interaction between ionic cores and valence electrons, while the 4d and 5s electrons of Rh with the 2s and 2p electrons of C, N, O are treated as valence electrons, and the wave functions are expanded in the plane-wave basis up to a kinetic energy of 400 eV. For both Rh(100) and Rh(111), the surface is modeled as a five-layer slab in a 4×4 unit cell along with a 15 Å thick vacuum region while adsorbates are introduced on one side of the slab. During the optimization, the bottom three atomic layers are kept fixed to mimic bulk structure with a lattice constant of 3.842 Å obtained in our prior study 34 (the experimental value is 3.80 Å 35 ), and the top two atomic layers are allowed to relax.. The reciprocal space is sampled with a 3×3×1 k -point grid automatically generated using the Monkhorst-Pack method, 36 and the first-order Methfessel-Paxton scheme 37 with a smearing parameter of 0.1 eV for partial occupancies are used. Meanwhile, the isolated atoms and molecules are simulated in 14 Å×15 Å×16 Å non-cubic cells with k point sampling using the Γ point only and Gaussian smearing 37 using the 0.002 eV smearing parameter. Two kinds of convergence criteria for the electronic-selfconsistency iterations and the ionic-relaxation loops are used in structure optimization: 10−6 eV with 0.01 eV/Å and 10−5 eV with 10−4 eV. The former is used for most configurations, except some unstable ones and the ones in multiple-adsorbate-coadsorption testing dataset. Dimer method 38 is used to locate the transition-state structures and the corresponding energies. The initial structure is constructed by placing the targeted adsorbate at its near-transition-state structure and

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depositing surrounding coadsorbates. In this study, the binding energies derived from DFT calculations use the following expression: EbDFT = Esys − Esur −

ad Egas ,

(1)

ad ad where Esys , Esur and Egas represent the energies of the substrate with adsorbates, the corre-

sponding clean surface and a certain gas-phase adsorbate, respectively. The model Hamiltonian for such binding energies considering additive PIs can be described as Ebmodel =

ad Eb,iso +

Eipair ,

(2)

pair

ad

ad where Eb,iso is the binding energy of a certain adsorbate adsorbed on a certain surface site in

isolation and Eipair represents a certain PI energy. The summations are over all adsorbates and all adsorbate pairs, respectively. The binding energy error for a specific configuration is defined as Eberror = Ebmodel − EbDFT /Nad ,

(3)

where Nad is the number of adsorbates in the configuration. Meanwhile, the activation energies are calculated by Ea = EbTS − EbIS ,

(4)

where EbTS and EbIS are the binding energies of transition state and initial state, respectively. All the codes we use here to implement the APAI model which we will describe later can be acquired in Supporting Information. We construct them as a relatively universal utility. On the one hand, accompanied with the parameters provided, binding energies for structures in the NO–CO reaction system on Rh(100) and Rh(111) can be calculated directly. On the other hand, by modifying some constants and providing one’s own parameters, it may be reused to other surface adsorption systems with similar lattice geometry. Refer to README.md in the package file for more information. 6


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In order to make an elementary comparison with experimentally observed phenomena based on the codes developed and the model Hamiltonian built, ground-state-searching calculations are performed. The following algorithm is adopted to get a local minimum energy and the corresponding structure: 1. Randomly deposit adsorbates on the surface lattice with the restrictive conditions fully satisfied to get an initial configuration. 2. For a certain adsorbate, calculate the binding energies of configurations where the adsorbate is moved to one of the available nearby sites within 1-surface-basis-vector distance. 3. Move the adsorbate to the site leading to the lowest binding energy if the energy is lower than the current one. Skip such movement otherwise. 4. Repeat step 2 and 3 until there is no more adjustment needed for all adsorbates. Total of 10000 runs are performed to find out the global ground state of a certain species combination. For structure visualization, VESTA 39 program is used.

Results and discussion Isolated adsorption and isolated pairwise interaction First, we investigate the adsorption of N, O, NO, and CO on the Rh(100) and Rh(111) surfaces, with NO and CO perpendicularly adsorbed via N or C atom respectively. The calculations are carried out by placing only one adsorbate on the surface in a 4×4 unit cell, and the adsorbate is allowed to fully relax during the structure optimization. The following high-symmetry sites on these two surfaces are considered: top, brg, hol for Rh(100) and top, fcc, hcp for Rh(111). The adsorption of all four species at the brg site on Rh(111) and N and O at the top site on both surfaces are not considered, since they are relatively unstable. 7


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The binding energies for stable adsorption states can be found in Table S1 in Supporting ad Information. These binding energies will be used as Eb,iso , i.e., the isolated binding energy

of a certain adsorbate adsorbed on a certain surface site, in the model Hamiltonian. It is worth noting that all current GGA functionals tend to overestimate the chemisorption bond strength for the sites with high coordination. This is widely known as the CO/metal puzzle, since the preferential adsorption site of CO on a number of metal surfaces is incorrectly predicted. 40 Lots of attempts have been made to solve this problem, 41–43 but there is still not a universally accepted efficient solution yet. Thus no special treatment is adopted in this work, and the influence of this overestimation will be discussed later. Next, we investigate the isolated pairwise interaction (IPI). It is achieved by calculating the binding energies for configurations where there are only two adsorbates coadsorbed on the surface in a 4×4 unit cell, and the adsorbates are allowed to fully relax during the structure optimization. According to eq (2), a specific IPI energy is then derived by subtracting the isolated binding energies of two adsorbates from the overall binding energy obtained, and then dividing the value by the multiplicity of such PI in the configuration if necessary. Meanwhile, considering that the highest expected coverage is 1 monolayer (ML) for most surface species, the coadsorption of adsorbates separating from each other within 0.7-latticeconstant distance, i.e., 1-surface-basis-vector distance, is regarded as impossible in current study to reduce the model complexity. On Rh(100), the PIs up to 1.5-lattice-constant separation are considered. All the possible inequivalent site pairs for a species pair are illustrated in Figure 1. The notation for these site pairs is defined as: the first letter represents the site one adsorbate located at and the second for the other, with the letter used indicates the surface site that has the identical first letter; the third number is the separation rank; and the fourth letter (if it exists) defines the first encountered site when linearly connecting the two binding sites. For example, the label “bb2h” means that two adsorbates locate at two next-nearest-neighbor brg sites with a hol site on the connecting line. The energy values of these PIs for all species pairs can be found

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in Table S2 in Supporting Information. It is worth noting that when two atomic adsorbates locate at nearby hol sites, local surface reconstruction may be induced. Namely, N and O atoms may reside on pseudo-threefold hollow sites that form from the distortion of original fourfold hollow sites. Therefore, the hh1 and hh2 interactions between atomic adsorbates are calculated under two circumstances: one with surface reconstruction and the other without. The one with lower energy value will be accepted. As a result, the coadsorption of two N atoms or one N atom and one O atom will cause surface reconstruction at hh1 separation, and the coadsorption of two O atoms will cause surface reconstruction at hh1 and hh2 separation.

Figure 1: Schematic view of inequivalent site pairs on Rh(100). Large green spheres are substrate Rh atoms, and medium red spheres and small blue spheres are adsorbates, with the red one representing the first participant in a pairwise interaction and the blue one representing the other. Yellow dashed lines separate the space for three kinds of first participants. The label on the red sphere identifies the surface site where the first participant resides. And the label attached to the blue sphere identifies the surface site where the other participant resides and its separation relation with the first participant. The combination of these two labels identifies a particular site pair. These notations also apply to the subsequent figure for Rh(111). On Rh(111), we take into account all the PIs which can be obtained in the 4×4 unit cell, which leads to a maximum range of 1.63-lattice-constant separation. All the possible inequivalent site pairs for a species pair are illustrated in Figure 2. The notation for these site pairs is similar to that used for Rh(100). The energy values of these IPIs for all species pairs can be found in Table S3 in Supporting Information. Broadly speaking, the IPIs vary from several meV to several hundred meV. On Rh(100), 9


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Figure 2: Schematic view of inequivalent site pairs on Rh(111). the IPIs within the 0.8-lattice-constant range are all repulsive and have relatively large energy values. The ones beyond that range have significantly smaller energy values and some of them are attractive. On Rh(111), almost all the IPIs are repulsive, with the ones up to 0.8lattice-constant separation showing relatively large energy values, and the others showing smaller but non-negligible energy values.

Augmented pairwise additive interaction model The aforementioned IPIs reflect the lattice-gas-based PIs under low surface coverage. In this circumstance, the lateral displacements of adsorbates are quite substantial when the two adsorbates are coadsorbed at relatively close surface sites. Since such lateral displacements will be greatly affected under high surface coverage, the PIs for site pairs with small separation may differ greatly in diverse local configurations. This is also part of the origin of many-body interactions in traditional CE models. Obviously, directly using the aforementioned IPI energies as Eipair in the model Hamiltonian will cause significant error under high surface coverage. Therefore, we propose an APAI model to rectify the deviation. Broadly speaking, the rectification is mainly achieved

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by modifying the PIs according to the surroundings where the two adsorbates are immersed. For each adsorbate pair, the Eipair used in the model Hamiltonian equals the corresponding IPI energy plus some needed modification interaction (MI) energies if the surroundings meet specific conditions. Namely, the following equation is used for the calculation of each Eipair : pair + Eipair = Ei,iso

type Ei,mod ,

(5)

type

pair type is the corresponding IPI energy, Ei,mod represents a kind of MI energy for the where Ei,iso

targeted PI, and the summation is over all needed modification strategies. The classification of these MIs, the strategies to use them, and the methods to estimate the corresponding energy values will be described in the following. On Rh(100), there are two kinds of modifications taken into account. One is for the restricted departure of adsorbates from each other, and the other is for surface reconstruction. In the calculations for IPIs, two adsorbates coadsorbed at relatively close surface sites may depart from each other slightly to reduce the repulsive interaction. However, such lateral displacements will be restricted under high surface coverage, leading to an increase in the repulsive PI. To reflect such effect, restriction-type MIs are proposed for the hh1, bb2h, bb2t, tt1, hb2, bh2, bt2, tb2, and hh2 site pairs of all species pairs. The notation for these MIs is the label of the corresponding site pair with a terminal “f”, e.g. “hh1f”. The strategy to use them can be described as: each surrounding adsorbate adsorbed at the site (1) near the extension line of the targeted adsorbate pair and (2) close to any of the two adsorbates no farther than the distance between the two will introduce one copy of the corresponding restriction-type MI energy for that PI. To be more specific, the surrounding adsorbates which will introduce MIs for these PIs are illustrated in Figure 3. To estimate the energy values of these restriction-type MIs, we perform calculations similar to those for IPIs except that the bottom atoms of the two adsorbates are fixed at ideal adsorption sites in xy plane and only allowed to relax in z direction. The MI energy for each pair is

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defined as half of the difference between such partly-fixed-type interaction energy and the corresponding fully-relaxed-type IPI energy. The factor of half is introduced because the movements of both adsorbates are restricted, which leads to a multiplicity of two for the MI in such partly-fixed-type interaction being more appropriate. The energy values of these MIs can be found in Table S4 in Supporting Information.

Figure 3: Schematic view of the restriction-type modification strategy on Rh(100). Large green spheres are substrate Rh atoms, medium red spheres are the adsorbates participating the targeted pairwise interaction, and small blue spheres represent the surrounding adsorbates each of which may introduce one copy of the corresponding restriction-type modification interaction energy. Yellow dashed lines separate the space for the cases of (a) hh1 (bb2h, bb2t, and tt1 use identical relative spacial relation), (b) hb2 (bh2, bt2, and tb2 use identical relative spacial relation), and (c) hh2 site pairs. As we have mentioned in the last section, the coadsorption of N-N, N-O, and O-O at hh1 separation and O-O at hh2 separation may induce local surface reconstruction. Since such reconstruction varies at different local configurations, reconstruction-type MIs are proposed, denoted as “hh1c” for hh1 interactions of the three species pairs and “hh2c” for hh2 interaction of the O-O species pair. Inspired by some calculations under high surface coverage, the following strategy is proposed to indicate how different surroundings enhance or reduce such reconstruction and reflect such effect in the model Hamiltonian: (1) if the two nearest 12


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hol sites on the extension line of the targeted hh1 interaction are both occupied, or the ten neighboring hol sites of the targeted hh1 interaction or the six nearest hol sites of the targeted hh2 interaction are occupied larger 30%, the surface reconstruction will be restricted, i.e., the opposite value of the corresponding hh1c MI energy needs to be introduced; (2) if it is not the former case and at least one neighboring brg site of the targeted hh1 or hh2 interaction is occupied, neither enhancement nor reduction of the reconstruction will occur; (3) if it is not the former two cases and at least one of the two adsorbates is involved in another PI which may induce surface reconstruction, an enhancement of the reconstruction will occur, i.e., the corresponding hh1c or hh2c MI energy for the targeted PI needs to be introduced. The energies of these MIs are computed as the difference between the corresponding IPI energies with and without surface reconstruction. These energy values can be found in Table S4 in Supporting Information. On Rh(111), the modifications for ff1, hh1, and tt1 interactions of all species pairs are taken into account. Since no surface reconstruction is found on Rh(111), the deviation is mainly caused by affected lateral displacements of adsorbates. The lateral moving behavior of adsorbates on Rh(111), a hexagonal lattice system, is quite different from that on Rh(100), which is a tetragonal lattice system. Therefore, the calculations for the MI energies and the strategies to use them vary a little. On the one hand, similar to the situation on Rh(100), surrounding adsorbates adsorbed at some specific surface sites may restrict the lateral departure of the two adsorbates at ff1, hh1, or tt1 separation and thus enhance their repulsive interaction. Accordingly, restrictiontype MIs labeled as “ff1r”, “hh1r”, and “tt1r” are proposed for the corresponding ff1, hh1, and tt1 interactions of all species pairs. The strategy to use them is also that each surrounding adsorbate adsorbed at some specific sites will introduce one copy of the corresponding MI energy for the targeted PI, and such sites for the ff1, hh1, and tt1 interactions are illustrated in Figure 4. The quantification for these MI energies is carried out by calculating the binding energies of configurations where there are three adsorbates coadsorbed in line in a

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4×4 unit cell. The MI energies are then deduced from these binding energies according to the APAI model Hamiltonian in reverse. A more detailed supplementary explanation of such quantification and the values of these MI energies can be found in Table S5 in Supporting Information.

Figure 4: Schematic view of the modification strategies on Rh(111). Large green spheres are substrate Rh atoms, medium red spheres are the adsorbates participating the targeted pairwise interactions, small blue spheres represent the surrounding adsorbates each of which may introduce one copy of the corresponding restriction-type modification interaction, and small brown spheres represent the surrounding adsorbates each of which may introduce one copy of the corresponding pushing-type modification interaction. Yellow dashed lines separate the space for the cases of (a) hh1, (b) ff1, and (c) tt1 site pairs.

On the other hand, the coadsorption of three adsorbates at three nearest-neighbor fcc or hcp sites surrounding a top site will enhance the lateral departure of these adsorbates and thus reduce their repulsive interactions. Hence we propose pushing-type MIs labeled as “ff1p” and “hh1p” for the corresponding ff1 and hh1 interactions of all species pairs to reflect such effect. Specifically, if there is an adsorbate located at the site which forms the aforementioned triangle along with the targeted adsorbate pair, such ff1p or hh1p MI energy needs to be introduced for the targeted ff1 or hh1 interaction. The sites which will induce such modification are also illustrated in Figure 4. To quantify these MI energies, all the configurations where there are three adsorbates coadsorbed in a 4×4 unit cell forming such a 14


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triangle are taken into account. The binding energy of each stable configuration is calculated and disassembled according to the APAI model Hamiltonian. The values of these MI energies are then obtained by solving the formed linear least-squares problem, and the result can be found in Table S5 in Supporting Information.

Prediction capability of the model To check out how the APAI model works on Rh(100), two kinds of testing datasets are collected. One consists of 135 configurations related to the modification strategy for surface reconstruction. It mainly involves the coadsorption of O atoms with different spatial arrangements in the 4×4 unit cell up to 0.5 ML. The distribution of the binding energy errors in this testing dataset is illustrated in Figure 5. The root-mean-squared error (RMSE) for the testing dataset reduces from 0.057 to 0.028 eV per adsorbate when only considering the modification for restricted departure, and it further reduces to 0.018 eV per adsorbate when including the modification for surface reconstruction. The other consists of 400 permutations of the four species with the overall coverage ranging from 0.19 to 0.63 ML uniformly. The random selecting of species combination and spatial arrangement is adopted. The distribution of the binding energy errors in this testing dataset is illustrated in Figure 6(a). The RMSE value reduces from 0.031 to 0.012 eV per adsorbate when considering the modification strategies. Such improved behavior justifies the modification strategies used, and reflects the prediction capability of the proposed APAI model. To find out whether a further simplification by reducing the needed parameters is acceptable, we test the prediction capability of the model with respect to the PIs included based on the second testing dataset. The result is shown in Figure 6(b). As we can see, the interactions up to 0.8-lattice-constant separation are of extreme importance, and the ones within a further extension to 1.1-lattice-constant length also induce considerable decrease in prediction errors. Meanwhile, the others with farther separation are not quite necessary. Accordingly, the exclusion of some PIs can be adopted in accordance with the targeted 15


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Figure 6: Prediction results for the multiple-adsorbate-coadsorption testing dataset on Rh(100): (a) error distribution; (b) prediction capability of the model with respect to the cutoff distance of included pairwise interactions.

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precision. Similar to that on Rh(100), a testing dataset including 400 permutations of the four species with the overall coverage ranging from 0.19 to 0.63 ML uniformly is constructed on Rh(111). The distribution of the binding energy errors in this testing dataset is illustrated in Figure 7(a). Unlike that on Rh(100), the RMSE value without modification is intrinsically small, and it reduces from 0.018 to 0.013 eV per adsorbate when the modification is added. The result verifies the usability of the APAI model on Rh(111). d3dd dddddddddd dddddddd

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Figure 7: Prediction results for the multiple-adsorbate-coadsorption testing dataset on Rh(111): (a) error distribution; (b) prediction capability of the model with respect to the cutoff distance of included pairwise interactions. The test for the cutoff distance of included PIs is also performed, and the result is shown in Figure 7(b). As we can see, the interactions all the way up to 1.63-lattice-constant separation 17


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are of considerable importance, with the importance slightly decreasing accompanied by the increase of separation distance. Therefore, the exclusion of any of these PIs is not favorable. Based on the difference between the behavior of the APAI model on Rh(100) and that on Rh(111), it can be deduced that the lateral interactions on Rh(100) are relatively short-range and greatly susceptible to the lateral displacements of adsorbates, whereas on Rh(111), the lateral interactions are relatively long-range and the lateral displacements of adsorbates are more limited. Furthermore, to see how the model reproduces experimental results, ground-state-searching simulations are performed using the model Hamiltonian with all parameters included. On account of that current available experimental discoveries are mainly about homogeneous adlayers, we focus on the respective adsorption of the four species in the 4×4 unit cell. The simulated ground-state information on Rh(100) and Rh(111) can be found in Figure S1 and S2 in Supporting Information, respectively. The following experimentally observed ordered structures are successfully captured: Rh(100)-c(2×2)-N, 44,45 Rh(100)-p(2×2)-O, 46,47 Rh(111)p(2×2)-O, Rh(111)-p(2×1)-O, 48–50 Rh(111)-p(2×2)-3O, 51 Rh(111)-c(4×2)-4NO, 52,53 Rh(111)p(2×2)-CO, and Rh(111)-p(2×2)-3CO. 54 The (2×2) reconstruction structure of 0.5 ML O on Rh(100) is also captured, though we can not identify it as (2×2)p4g 46,47 structure concretely. Meanwhile, it has been found out experimentally that N does not form a well ordered p(2×2) structure as O does on Rh(100) 45 in the low coverage range. This is also observed in our simulation. Moreover, the conventional preparation methods provide a maximum coverage of 0.5 ML for the adsorption of N or O atoms on Rh(100) and Rh(111), 44,45,50,55 which is in good accordance with the simulated abrupt increase of differential binding energy once exceeding this surface coverage. The saturation coverage for NO on Rh(111) has been revealed to be over 0.7 ML around room temperature, 56 and that for CO has been identified as 0.78 ML at low temperature. 57 These two values are in reasonable agreement with the simulated coverage values where the differential binding energy becomes positive for the first time, namely, less than 0.9 ML for NO and around 0.8 ML for CO. The aforementioned

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conformance confirms the reliability of the model to a certain extent. Though quite a lot experimental observations are satisfactorily reproduced, there is still some improving space for the current model. On the one hand, the ordered structures like Rh(111)-p(2×2)-3NO 53,58 and Rh(100)-c(2×2)-CO 59,60 are not captured as ground states based on the model Hamiltonian. The adsorbates in these structures are inclined to occupy the top site more compared with those in the simulated ground states. It is probably due to the so-called CO/metal puzzle mentioned earlier, and this problem happens to NO too, as also suggested by Huang and Mason. 43 Thus, a more efficient solution for this CO/metal puzzle in DFT calculations is needed to refine the parameters used in the current model. On the other hand, it has been found that NO may be adsorbed on Rh(100) highly inclined at low surface coverage, 61,62 and NO and CO may occupy brg sites on Rh(111) under certain situations to release the repulsive lateral interactions among each other. To reflect these phenomena, a more sophisticated description of the surface lattice and the species on it is necessary.

BEP relation for diffusion As an extensional work aiming at providing fundamental information for further kinetic modeling, we investigate the Brønsted–Evans–Polanyi (BEP) relations 63,64 for the diffusion of N, O, NO, and CO on Rh(100) and Rh(111). Such BEP relation relates the energy shifts in the initial and final states affected by lateral interactions to a change in the activation energy: Ea = Ea(0) + α Er − Er(0) , (0)

(6) (0)

where Ea (Ea ) is the activation energy with (without) lateral interactions, Er (Er ) is the reaction energy, i.e. the energy difference between final and initial states, with (without) lateral interactions, and the coefficient α measures how much the effect of lateral interactions on the initial and final states influences the activation energy. Conventionally, the coefficient

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α is simply assumed as 0.5 for the surface-diffusion processes in most cases. However, such assumption may be invalid especially when the adsorption properties on different surface sites exhibit enormous difference, which is the case we are facing. Thus, the BEP relations for the diffusion processes in the NO–CO reaction system are studied in detail deliberately. On Rh(100), the hops between nearest-neighbor brg and hol sites and the hops between nearest-neighbor brg and top sites are taken into account. The activation energies of such hops are calculated at different O-precovered surface configurations, with these surrounding O atoms successfully providing the diverse lateral-interaction backgrounds. By fitting the activation energies with respect to the reaction-energy changes according to eq (6), the BEP relations are constructed for the four species. The obtained parameters are listed in Table 1. The fitting plots and configurations used can be found in Figure S3 to S8 in Supporting Information. Table 1: BEP-Relation Parameters for the Diffusion of N, O, NO, and CO on Rh(100) adsorbate

a b

process

0 Ea,fwd (eV)a

αfwd

0 Ea,rev (eV)b

N brg↔hol 0.01 0.06 0.50 O brg↔hol 0.03 0.34 0.12 NO brg↔hol 0.29 0.55 0.03 NO brg↔top 0.46 0.79 0.13 CO brg↔hol 0.20 0.88 0.02 CO brg↔top 0.13 0.53 0.07 The subscript “fwd” denotes the forward step in the reversible event. The subscript “rev” denotes the reverse step in the reversible event.

αrev 0.94 0.66 0.45 0.21 0.12 0.47

On Rh(111), a similar procedure is carried out. The hops between nearest-neighbor fcc and hcp, fcc and top, and hcp and top sites are taken into account. Unlike those on Rh(100), the correlations between activation energy and reaction-energy change are not quite satisfactory. We tentatively accept the simple BEP relations for the hops between fcc and top sites and the hops between hcp and top sites because the residual error for each configuration is relatively small. However, the deviation is quite significant for the hops between fcc and hcp sites. To find out a solution, all one-O-atom-surrounded and lots

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of two-O-atom-surrounded local environments are tested for the diffusion of O atom. By analyzing the obtained results, we propose a modification strategy to improve the original BEP relation: Ea = Ea(0) + α Er − Er(0) + Nh Eh + Nl El .

(7)

Nh (Nl ) is the number of adsorbates located at some specific sites which will induce the increase (decrease) of the activation energy. These sites are illustrated in Figure 8(a). Eh (El ) is the corresponding increased (decreased) energy value, which can be acquired through leastsquares polynomial fit. To check whether the type of coadsorbate relates to the modification energy values, we perform an identical procedure but changing the coadsorbate to NO. The obtained Eh and El values are very similar to those obtained from using O as coadsorbate. Therefore, it can be supposed that the modification energy values are site-dependent and coadsorbate-type-irrelevant. Such modification strategy also applies to N, NO and CO. The effectiveness of this modification on the four species are demonstrated in Figure 8(b). The fitted parameters for all these diffusion processes are listed in Table 2. The fitting plots and configurations used can be found in Figure S9 to S16 in Supporting Information. Table 2: BEP-Relation Parameters for the Diffusion of N, O, NO, and CO on Rh(111) adsorbate

process

0 Ea,fwd (eV)

αfwd

0 Ea,rev (eV)

αrev

Eh (eV)

El (eV)

N O NO NO NO CO CO CO

fcc↔hcp fcc↔hcp fcc↔hcp fcc↔top hcp↔top fcc↔hcp fcc↔top hcp↔top

0.53 0.44 0.14 0.57 0.63 0.04 0.17 0.22

0.39 0.43 0.48 0.74 0.70 0.40 0.43 0.58

0.65 0.39 0.19 0.14 0.16 0.10 0.17 0.17

0.61 0.57 0.52 0.26 0.30 0.60 0.57 0.42

0.13 0.07 0.04 / / 0.01 / /

−0.05 −0.04 −0.02 / / −0.02 / /

Briefly speaking, the conventional assumption that the coefficient α equals to 0.5 is not always acceptable for surface diffusion processes. Moreover, some modification terms may need to be added to the original BEP relation under certain circumstances, particularly for the processes on hexagonal surfaces like Rh(111). 21


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Figure 8: Modification of the original BEP relation for the adsorbate diffusion between nearest-neighbor fcc and hcp sites on Rh(111). (a) Schematic view of the modification strategy. Large green spheres are substrate Rh atoms, the medium red sphere is the targeted adsorbate, small blue spheres represent the surrounding adsorbates which may introduce El , and small brown spheres represent the surrounding adsorbates which may introduce Eh . (b) Activation energy versus reaction-energy change. The modification term Nh Eh + Nl El is subtracted from the original Ea for the modified values demonstrated here.

Conclusions In summary, we have proposed a lattice-gas-based APAI model to determine the energy of any adsorbate configuration efficiently and accurately. The model Hamiltonian is defined as the summation over all isolated binding energies and all PIs. Considering that the lateral displacements of adsorbates may vary significantly in different local environments, the PI used for each adsorbate pair consists of the corresponding IPI energy and some needed MI energies if the surroundings meet specific conditions. The parameters used in this APAI model for the NO–CO reaction system on Rh(100) and Rh(111) have been quantified based on DFT calculations. The binding-energy RMSE values derived from the comparison between modeland DFT-based predictions for testing datasets have been identified to be less than 0.02 eV per adsorbate. It verifies the reliability of the methodology proposed. Meanwhile, a number

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of experimental observations were successfully captured through ground-state-searching simulations based on the model Hamiltonian, though some properties were faultily estimated. This confirms the usability of the model Hamiltonian for this reaction system to a certain extent and provides the further improving directions. Through the calculations, it can be found that the lateral interactions on Rh(100) are relatively short-range and greatly susceptible to the lateral displacements of adsorbates, whereas on Rh(111), the lateral interactions are relatively long-range and the lateral displacements of adsorbates are more limited. In addition, the BEP relations for adsorbate diffusion have been constructed, with a modification strategy being proposed to improve the performance for the hops between fcc and hcp sites on Rh(111). This work opens up the possibility of accurate microkinetic modeling for the NO–CO reaction system on Rh(100) and Rh(111) with a faithful account of lateral interactions. It also motivates the construction of model Hamiltonian involving accurate lateral interactions for other complex surface adsorption systems with an affordable amount of underlying DFT calculations for parameterization.

Acknowledgement This work was supported by the Program for the National Natural Science Foundation of China (Grant 21676232, 21673206) and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No.U1501501.

Supporting Information Available • SI-file.pdf: isolated binding energy values, IPI energy values, MI energy values, simulated ground states, and BEP-relation fitting plots. • SI-code.zip: parameter dataset files, testing dataset files, and python scripts used to

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implement the APAI model.

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Augmented Pairwise Additive Interaction Model for Lateral Adsorbate

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