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Binary Approach to Ternary Cluster Expansions: NO−O−Vacancy System on Pt(111) Anshumaan Bajpai,† Kurt Frey,† and William F. Schneider*,†,‡ †

Chemical and Biomolecular Engineering and ‡Chemistry and Biochemistry, University of Notre Dame, Notre Dame, Indiana 46556, United States S Supporting Information *

ABSTRACT: Cluster expansions (CEs) provide an exact framework for representing the configurational energy of interacting adsorbates at a surface. Coupled with Monte Carlo methods, they can be used to predict both equilibrium and dynamic processes at surfaces. In this work, we propose a threebinary-to-single-ternary (TBST) fitting procedure, in which a ternary CE is approximated as a linear combination of the three binary CEs (O−vac, NO− vac, and NO−O) obtained by fitting to the three binary legs. We first construct a full ternary CE by fitting to a database of density functional theory (DFT) computed energies of configurations across a full range of adsorbate configurations and then construct a second ternary using the TBST approach. We compare two approaches for the NO−O− vacancy system on the (111) surface of Pt, a system of relevance to the catalytic oxidation of NO. We find that the TBST model matches the ternary CE to within 0.018 eV/site across a wide range of configurations. Further, surface coverages and NO oxidation rates extracted from Monte Carlo simulations show that the two models are qualitatively consistent over the range of conditions of practical interest.

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INTRODUCTION Lattice models combined with kinetic Monte Carlo methods are widely applied today to simulate the rate of catalytic reactions at surfaces.1−11 These models are particularly useful when the energies and rates of the elementary steps are sensitive to interactions between adsorbates on nearby lattice sites.12,13 Density functional theory (DFT) methods are commonly used to parametrize the elementary adsorption and reaction steps on the model lattice.14,15 A representative lattice might contain on the order of 1000 sites, and a kinetic simulation might involve computation of many millions of realizations of adsorbates on those sites. Direct DFT computations of all these energies is impractical. An alternative is to map these energies onto a lattice gas Hamiltonian, parametrized by adsorbate−adsorbate interactions.12,13,16−19 These adsorbate−adsorbate interactions are manifested in coverage-dependent adsorption energies,20−22 in adsorbate ordering at surfaces,13,23 and in coverage-dependent barriers to reactions at surfaces.14,24,25 In early work in the field, adsorbate−adsorbate interactions were captured in empirical through-space and through-surface models. Through-space models typically include steric and distance-dependent electrostatic terms,26 while bond order conservation (BOC)27,28 and principles of least-metal-atomsharing11,29 are employed to model through-surface interactions. These through-surface interactions between atomic adsorbates at a surface were later explained in terms of a d-band model.30,31 Given the discrete nature of surface adsorption sites, these models gave way to site-occupancy-based adsorbate interaction © XXXX American Chemical Society

models. Site occupancy models build on the work of Sanchez et al.32 to express the energy of an array of sites in terms of polynomials consisting of products of discrete occupancy variables. The cluster expansion (CE) is a general approach for representing the configurational energy of objects on a lattice,19,32−36 including adsorbates on a two-dimensional lattice of surface sites.13,18,21,31,37−42 The energy ECE of a given σ configuration σ is expanded in a sum of on-site, two-body, and higher-order contributions: EσCE = Jo +

∑ Ji σi + ∑ Jij σσi j + ∑ i

i>j

i>j>k

Jijk σσσ i j k + ... (1)

where the sums run over sites and the Ji are interaction parameters that can be fit to a small set of DFT-derived energies. Typically the fitting is done iteratively, by fitting a CE to the energies of a small number of configurations, predicting the energies of new configurations, comparing the predictions with new DFT calculations, and iterating to convergence.18,19,21,35 The first application of CEs to adsorption at metal surfaces focused on single adsorbates and single-component, high symmetry surfaces.13 Tang et al.13 parametrized a CE for the Pt(111)−O face-centered-cubic (fcc) site system and showed that it was able to recover the experimentally observed p(2 × 2) ordering and, when coupled with Monte Carlo simulations, the order−disorder transition. Subsequent work on this system Received: January 28, 2017 Revised: March 16, 2017 Published: March 20, 2017 A

DOI: 10.1021/acs.jpcc.7b00914 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

NO oxidation rates, and show that the two models are qualitatively consistent over conditions of practical interest.

explored the relationship between CE size and predictive quality21 and used the CE as a Hamiltonian for Monte Carlo based predictions of O2 temperature-programmed-desorption (TPD) spectra43,44 and catalytic NO oxidation rates and rate derivatives.41,45 CEs have been developed for adsorbates on the (111)22,46 and other lower symmetry facets37,47,48 of fcc metals, for metal surface alloys,49 and for surfaces with multiple adsorption sites.17,23,31,42,50 Multiple adsorption site CEs are essential in modeling the systems where site preference for the adsorbate depends on the total coverage of the surface51−56 and have been found useful in simulating processes like TPD.23,43,44 Despite the versatility of multiadsorbate CE models, their application in catalysis23,40,42,47 is still in the incipient stages. A key challenge in building any CE model is to identify and parametrize the clusters required to model the system. For oxygen on Pt(111), for instance, a reasonably converged CE includes approximately 10 terms up to three-body interactions.21 This fitting becomes exponentially more complex as the number of adsorbates increases. Previous applications of both single and dual adsorbate CE models have, in general, relied on either empirical knowledge of the system or some training set error cutoff as a metric to establish the number and nature of interactions essential for the system. Some applications,19,21,37,45 however, have also employed, out of the sample, leave one out cross validation (LOOCV) to limit the overfitting in the model. In this work, we report the results of a DFT-based fitting of the NO−O−vacancy ternary adsorbate system on Pt(111) (Figure 1). This ternary system is relevant to Pt-catalyzed NO

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COMPUTATIONAL DETAILS Plane wave, supercell DFT calculations were performed with Vienna Ab initio Simulation Package58−60 (VASP 5.3.3). Electron exchange and correlation were described with the Perdew−Wang 91 (PW91) implementation of the generalized gradient approximation61 (GGA), and plane waves were included to a cutoff of 520 eV. Core−valence interactions were described with the projector augmented wave58,60,62 (PAW) method. An order 1 Methfessel−Paxton electron smearing of width 0.15 eV was used to set the partial occupancies. Electronic energies were considered converged when the energy difference between subsequent self-consistent iterations was less than 10−5 eV, and atomic positions were relaxed until the force on each atom was less than 0.03 eV/Å. The Pt bulk lattice constant within these approximations is 3.986 Å. The Pt(111) surface was modeled using five metal layers (Figure 2) plus 14 Å of vacuum. The bottom two layers were

Figure 2. Side and top views of the smallest (1 × 1) periodic supercell and unit vectors for a Pt(111) surface.

fixed at the bulk position, and the top three layers were allowed to relax. A 19 × 19 × 1, Γ-centered k-point mesh was used for the primitive (single metal atom per layer) supercell (Figure 2) and scaled for other supercells to maintain the product of the number of atoms in the unit cell and the total number of kpoints close to 2000. The obtained total number of k-points were distributed as a k1 × k2 × 1, Γ-centered mesh with k1 and k2 being inversely proportional to the length of the unit vectors along the surface. O and N down and upright NO were relaxed starting in fcc sites at combinations of coverages from 0 to 1 ML in supercells containing up to 10 fcc sites. Both NO53,56,63,64 and O65,66 are known to populate other sites and even to reconstruct the Pt surface at higher coverages; the limitation to fcc-only sites simplifies the comparison between CE models and is appropriate for NO oxidation conditions. High coverage structures in which the surface was found to reconstruct or either NO or O or both were found to move from fcc sites were excluded from our analysis. A visual inspection of ∼60 excluded structures reveals that the surface reconstructions occurred for total coverages as low as 0.5 and moving of NO to bridge sites occurred for total coverages as low as 0.44. An unsymmetrical presence of more than one nearest neighbor (NN) around an adsorbed fcc NO is a common feature of the structures that reconstructed. A detailed analysis of reconstruction and surface diffusion is beyond the

Figure 1. Schematic of the reaction model for irreversible dissociative adsorption of O2 on an NO−O equilibrated Pt(111) surface.

oxidation and builds on the O−vacancy system that has been applied previously to NO oxidation on Pt(111).41,45 We create a large database of DFT energies, calculated for structures having only NO, only O, and mixed NO−O surface coverages ranging from 0 to 1 monolayer (ML), and fit them to a ternary CE57 of multibody O−O, NO−O, and NO−NO interactions. To avoid overfitting, we apply a 5-fold cross validation approach, common in machine learning domain. We find that the system is well described by a ternary CE of 15 parameters with interactions of up to three bodies. We compare these results with a simplified, three-binary-to-single-ternary (TBST) fitting procedure, in which the NO−O−vacancy (vac) CE is constructed as a linear combination of three binary CEs (O− vac, NO−vac, and NO−O). We find that the TBST model matches the comprehensive ternary CE to within 0.018 eV/site across a wide range of configurations. Further, we perform Monte Carlo simulations on the comprehensive ternary and TBST Hamiltonians, compare predicted surface coverages and B


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Figure 3. Depiction of a two-body second-nearest-neighbor (2NN) ternary feature and the corresponding set of nine distinct clusters. The two NO− O, NO−vac, and O−vac clusters shown in the set are symmetry distinct.

metric.68 To obtain an LOOCV score for a given set of features, a CE model is built by performing a least-squares fit to all but one of the DFT formation energies and the prediction error in the one excluded configuration is calculated. The process is repeated for every configuration in the DFT database, and the root-mean-square (RMS) value of all the prediction errors is the LOOCV score. We predicted the energies of all members of the SD and identified the predicted structures that broke the low-energy hull of the known DFT structures. These structures were then added to the DD and the procedure was iterated until we were satisfied that the DD had covered most of the NO/O ground state structures (GSSs). The self-consistent procedure required 22 iterations. Grand canonical Monte Carlo (GCMC) simulations were performed using the 15-parameter CE developed using ternary and TBST models as Hamiltonians. The Pt(111) surface was represented with a 42 × 42 fcc lattice. Calculations were performed in 50 K temperature increments from 373 to 773 K, NO partial pressures from 1 to 10 000 ppm in increments of powers of 10, and the ln(PNO2/PNO) as {−4, −2, 0, 2, 4}. Surfaces were initialized at zero coverage, and the sampling for oxygen−oxygen binding energies (OOBE) was done after there were on average seven successful MC moves per site or 140 attempted moves per site. For each condition 1500 such simulations were run, and the reaction rates reported here are the overall average of these 1500 simulations.

scope of the current work. Energies are referenced to gas-phase, spin-polarized O2 and NO calculated in a 12 × 12 × 12 supercell. Computed O2 and NO bond lengths were 1.228 and 1.169 Å, respectively, and GGA energies were −9.78 eV/O2 and −12.25 eV/NO. The full NO−O−vac/Pt(111) ternary CE was constructed using the multicomponent Alloy Theoretic Automated Toolkit34,35 (ATAT). Within this formalism, each ternary feature convolutes all individual adsorbate−adsorbate clusters. For example, in Figure 3, a 2NN pairwise feature incorporates nine distinct clusters. If a ternary CE has this feature, it includes contributions from all nine clusters depicted in Figure 3.67 Pairwise features were included up to eighth nearest neighbor (NN), three-body features up to a maximum side length of 6NN, and four-body features up to 4NN. The same set of features were included in the TBST fitting of the O−vac, NO− vac, and O−NO systems. Further details are available under Results. We selected the structures in the DFT database (DD) iteratively,18 guided by the CE machinery, from a pool of 30 687 distinct ternary configurations termed the structure database (SD). First, we performed DFT calculations on a handful of known, stable configurations from the SD and computed formation energies (EF(σ)) relative to O2 and NO (eq 3): Pt +

θO O2 + θNO NO → PtOθONOθ NO 2

E F (σ ) =

(σ )

E0(σ ) − E0(0) θ − O E0,O2 − θNOE0,NO N 2

(2)

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RESULTS DFT Database. The DD consisted of 560 permutations of NO and O in fcc sites on Pt(111) and included multiple calculations with the same [θNO, θO] but different permutations of NO and O. Figure 4 shows the distribution of DD structures with respect to [θNO, θO]. The self-consistent buildup procedure biased the configurations to the coverage range from 0.2 to 0.6 ML, also observed in previous computational39,41,45 and experimental24,69−71 studies that suggest that the total coverage for practical cases typically fall within the specified range. The 1 ML NO and 1 ML O energies were computed in multiple supercells of different sizes, and the standard deviation in the formation energies (eq 3) was used as a measure of the numerical precision of all the DFT energies. Figure 5a plots the computed formation energies against adsorbate coverages, and Figure 5b shows the projection of the resultant convex hull, colored by the formation energy. Twentyfive configurations break the hull by more than 4 meV/site,

(3)

Here, E0(σ) is the DFT calculated energy of configuration σ (eq 2). We neglected the minor changes in the vibrational spectrum of Pt atoms as a result of adsorption and included only the zero point energy (ZPE) of adsorbates in E0(σ). E0(0) is the energy of an identically sized adsorbate-free Pt(111) supercell. E0,NO and E0,O2 are the DFT calculated zero point corrected energies of gas phase molecular NO and O2, respectively. N is the number of fcc sites available on the surface, while θO and θNO are the surface coverages in configuration σ. We fit the formation energies to a ternary ATAT based CE including pair, triple, and quadruple features, using the previously described steepest descent algorithm,18 modified to allow addition or deletion of a feature at each step, and a leaveone-out cross validation (LOOCV) score as the accuracy C


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state configurations in the DD are reported in the Supporting Information (SI), section S1. Ternary Cluster Expansion. We used a 5-fold crossvalidation approach68 to develop the full ternary O−NO− vacancy CE (Figure S4). Figure 6 reports the averaged training

Figure 4. Coverage distribution of DFT calculated structures in the training database. The relative size of a marker is proportional to the number of structures for a given [θNO, θO] having different permutations of NO and O on the Pt(111) surface. The direction of the axis ticks represents the direction of the gridlines. Coverages are expressed in monolayers (ML). Figure 6. Leave one out cross validation (LOOCV) score, training, and testing root-mean-squared errors (RMSE) for the 5-fold cross validation versus the number of features used in the ternary model. The data for the full feature domain is presented in the inset.

including six NO-only, five O-only, and 14 mixed configurations. The negative initial slope of the hull reflects exothermic adsorption of the gas phase molecules, the positive curvature reflects diminishing adsorption with increasing coverage, and the formation energy minima correspond to saturation coverages. Formation energies along the NO− vacancy leg are more negative than those along the O−vacancy leg, consistent with the low coverage binding energies.56 The NO−vacancy leg has a minimum at 3/4 ML, close to the experimentally reported saturation coverage of 0.77 ML observed following NO dosing to Pt(111) at 110 K,51 and the O−vacancy leg has a minimum at 2/3 ML, the same as previously reported for the fcc-only model.21 The NO−O leg has a slight positive curvature, reflecting similarities in the O− O, NO−NO, and NO−O interactions. As a result, the energy minimum along any straight path from 0 to 1 ML occurs near ≈0.7 ML. A constant total coverage is in agreement with experimental observations made at 110 and 250 K from in situ, high resolution X-ray photoelectron spectroscopic (XPS) and TPD51 measurements of NO−O dosed Pt(111) surface prepared using molecular beam impingement. All the ground-

RMSE, testing RMSE, and LOOCV as a function of the number of features. Initially, LOOCV decreases sharply as the number of features increases but then the decrease becomes gradual. Training RMSE follows a trend similar to LOOCV and decreases as the number of features increase because more features allow more flexibility to fit the training set. Test RMSE decreases with an increase in the number of features initially, but beyond 15 features, no noticeable improvement in the model is observed. In fact, the test error gradually starts to increase as more features lead to overfitting in the model. We consider the 15 feature model for the rest of this work. As mentioned earlier, a single ternary CE feature incorporates contributions from every possible combination of species. The 15 features and the corresponding parameters obtained from ternary fitting are reported in fit.eci and clusters.out ATAT files available in the SI. TBST Cluster Expansion. We next explored the ability to describe the ternary system by combining cluster expansions

Figure 5. DFT formation energies (EF) for the fcc NO−O−vac system on Pt(111). (a) Three-dimensional (3D) view of the formation energy convex hull with all (560) DFT calculated energies depicted as a scatter plot. (b) Top view of 3D formation energy convex hull. The vertices of the triangular facets represent ground state configurations. The direction of the axis ticks represents the direction of the gridlines. D


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The Journal of Physical Chemistry C fitted to the three legs of the ternary diagram (Figure 5b). To facilitate this construction, we apply a lattice gas convention in which vacancies do not contribute to the energy and write the formation energy of an NO−O−vac configuration (σ) as a sum over O−O, NO−NO, and NO−O contributions: E FTBST(σ ) = E FLG(σO) + E FLG(σNO) + E FResid(σNO − O)

deviation of its components. As an example, the training error cutoff for NO−vac CE is estimated to be the geometric mean of 4.49 and 5.82, calculated to be 5.57 meV (Figure 7). The clusters were added one at a time using the steepest decent approach. The important single adsorbate clusters included in the TBST model are presented in Figure S7, and the effective interactions are reported in Table S1b. The important cross interaction NO−O clusters are presented in Figure S8, and the corresponding effective interaction values are reported in Table S2b. The 15 features used in the ternary model can also be expanded to a set of adsorbate-specific terms for comparison by evaluating the features for all given sets of occupancies,67 and the details are available in SI, section S3. A set of Matlab scripts used for resolving ATAT features into individual clusters and their interaction values are available in the SI. The single adsorbate (NO−NO and O−O) and cross adsorbate (NO−O) terms obtained for the ternary model are presented in Figures S7 and S8 of the SI, respectively. The expanded interaction values for these clusters are reported in Tables S1a and S2a of the SI, respectively. We find short-range, few-body interactions to be more important as the steepest descent approach leads to a model with two-body clusters up to 5 NN and three-body interactions up to a maximum side length of 4NN. The model did not contain any of the four-body interactions from the feature domain. Comparison: Ternary vs TBST Models. 0 K DFT. We performed DFT calculations on 138 different structures with random distributions of NO−O−vac on Pt(111) surface and compared the accuracies of ternary and TBST models in predicting the formation energy of this independent data set. Parity plots comparing the two models are presented in Figure 8. The full ternary model (Figure 8a) predicts the formation energies of all 138 structures with a standard error of 0.006 eV/ site, comparable to the intrinsic DFT uncertainty. The TBST model has a similar 0.008 eV/site error when restricted to structures with formation energies greater than −0.6 eV/site or total coverages less than 0.5 ML (Figure 8b). TBST errors are greater for configurations with formation energies less than −0.6 eV/site. Across the entire domain, the TBST model has a standard error of 0.018 eV/site. The trends in the errors are given in the residual plots in Figure 9. The errors in the ternary model are uniformly distributed across the entire energy domain (Figure 9a), indicating an absence of systematic error in the model. In contrast, the errors in the TBST model are overall larger, not random, and tend to increase in absolute value with decreasing formation energy. We used the ternary and TBST models to predict the formation energies of 30 687 different NO−O−vac configurations in the SD and looked at a small subset of these predictions to understand the basic adsorption properties of NO and O. Figure 10a−c presents the DFT calculated, ternary, and TBST predictions for the NO−vac leg in Figure 5b. Similar comparisons for the O−vac and NO−O legs are in Figure 10d−f and Figure 10g−i, respectively. All models capture the correct qualitative features, including the shape of the hulls and the energy minima in O and NO coverages. Both models predict formation energy minima at 0.66 and 0.77 ML for O and NO, respectively. For the O−vac system on Pt(111), 1/4, 1/2, and 2/3 ML are the important ground states reported previously using a single adsorbate model21 and we find both the ternary and TBST models are able to predict these ground states. For the NO−vac system on Pt(111), scanning tunneling

(4)

ELG F (σO)

where is calculated from structures that contain only O and vacancies, ELG F (σNO) is calculated from structures that contain only NO and vacancies, and EResid (σNO−O) is calculated F LG using ELG F (σO), EF (σNO), and structures that contain only O and NO. Further details are provided in the SI, section S3. Each binary leg used the same feature domain as the ternary approach, including pairwise interactions up to 8NN, triplet interactions up to a maximum side length of 6NN, and quadruplets up to 4NN. The O−vac and NO−vac used lattice gas CE models, trained on a DD subset comprising of 82 O− vac and 94 NO−vac structures, respectively. The NO−O residual CE was trained on a 1 ML (NO−O) DD subset comprising 54 structures to obtain the interaction values for multibody NO−O clusters. Refer to the SI, section S3, for details. For each of the binary fits, the number of data points was relatively small compared to the ternary CE fitting. Therefore, instead of performing a 5-fold cross validation, we relied on training error cutoff to obtain the appropriate number of clusters for the model (Figure 7). Training error based cutoff is

Figure 7. RMSE versus number of clusters for binary CEs in TBST model. Horizontal dashed lines represent error cutoffs used to decide the number of clusters to include in TBST model.

an alternate method to minimize the extent of overfitting in a predictive model. A typical predictive model has a reducible error, the component that can be reduced by incorporating more features in the model, and an irreducible error, the result of inherent noise in the training data set that cannot be rectified during fitting. One measure of overfitting is when the training error is less than the irreducible error.68 In fitting each leg, we used the inherent imprecision in the VASP calculations as an estimate of the irreducible error, the training error cutoff for number of clusters to be included in the model. To estimate the irreducible error, we performed VASP calculations using the same parameter set on physically identical but differently sized supercells. We performed 27 calculations each for the bare Pt, 1 ML NO, and 1 ML O surfaces, using supercells of various sizes, to obtain standard deviations (in formation energy) of 4.49, 5.82, and 5.32 meV, respectively. The training error cutoff for a binary fit is estimated as the geometric mean of the standard E


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states are mild and do not break the hull significantly. Along the O−NO leg (Figure 10g−i), the formation energy monotonically increases as the NO molecules are substituted by O. An overall comparison with the DFT values in Figure 10a,d,g suggests that DD is comprehensive and contains stable low formation energy structures for each coverage value. Grand Canonical Monte Carlo. We performed GCMC simulations to model coadsorption (Figure 1) of O and NO on Pt(111) fcc sites using the ternary and TBST models as Hamiltonians. Simulations were performed for a wide range of T and chemical potentials (μNO and μO). We take gas-phase NO and NO2 as the reservoirs for for NO and O adsorbates, respectively, as motivated by the NO oxidation model (Figure 1): NO(g) + ∗fcc ⇌ NO*fcc

(5a)

NO2(g) + ∗fcc ⇌ O*fcc + NO(g)

(5b)

The NO and O chemical potentials are related to the NO and NO2 partial pressures and temperature as described in the SI, section S4. Figure 11 presents a parity plot for θNO and θO obtained using the two models. The TBST model has a tendency to predict more NO on the surface (Figure 11a), while the ternary model tends to favor O (Figure 11b). Higher NO coverage for TBST models is consistent with lower formation energy values observed in Figure 10 and their tendency to predict lower formation energies for NO dominated surfaces. However, the total coverages as estimated by the two models (Figure 11c) are more consistent as indicated by a simple calculation of the root mean square of the difference (RMSD) in the coverage values predicted by two models. θNO and θO have RMSD of 0.03 ML while θtotal has RMSD of 0.01 ML. The total coverage for the conditions explored lies between a minimum of 0.28 ML and a maximum of 0.51 ML. On the other hand, θNO and θO range from 0 to ≈0.5 ML indicating a dominance of either NO or O, depending on the practical condition. Figure 12 compares surface coverages as a function of temperature as predicted by the two models for PNO = 100 ppm and three different values of ln(PNO2/PNO). GCMC simulations to obtain the coverages were converged to within 0.003 ML. For the values of ln(PNO2/PNO) presented in Figure 12, maximum θNO is observed to be 0.46 ML at T = 373 K. Both NO and O compete for adsorption sites on the surface and NO dominates at low temperature, while O occupies the surface completely beyond 573 K. The maximum θO is 0.42 ML for ln(PNO2/PNO) = 2. The relative coverage trends for NO and O estimated here are in agreement with the results from the empirical model parametrized to experiments by Bhatia et al.72 for NO oxidation on Pt supported on alumina. We observe that, once there is no NO on the surface, the coverage of oxygen does not decrease as the temperature goes up. We find this behavior for all the conditions of ln(PNO2/PNO) presented in Figure 12. Experimental studies73 for NO oxidation on single crystal Pt(111) have reported a lack of change in oxygen surface coverage with respect to T, PNO2/PNO, and PO2. However, the range of PNO2/PNO values considered by Smeltz et al.73 were ∼1, and therefore we restrict ourselves only to comparison with respect to T. The equilibrated NO−O surfaces can be used to estimate NO oxidation rates using the basis site model described by Bray

Figure 8. Parity plots to compare CE predicted formation energies with DFT calculated formation energies for a validation data set using (a) ternary model and (b) TBST model. Marker color represents the total coverage of the structures.

Figure 9. Plots of residuals versus CE predicted formation energies for the validation data set using (a) ternary model and (b) TBST model. Marker color represents the total coverage of the structures.

microscopy52 (STM) and low energy electron diffraction53 (LEED) studies suggest p(2 × 2) to be a dominant surface configuration and both our models are able to predict a key ground state at 1/4 ML. The models also predict ground states at 2/5 and 1/2 ML for the NO−vac system, but these ground F


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Figure 10. Formation energy convex hulls from DFT calculated and CE predicted formation energies as a function of coverage for three binary (NO−vac, O−vac, and NO−O) subsets of fcc NO−O−vac system on Pt(111) using the ternary model ((b), (e), and (h)) and the TBST model ((c), (f), and (i)). (a), (d), and (e) are the hulls obtained based on the DFT calculated structures in DD. The binary subsets represent the edges of the ternary diagram presented in Figure 5b.

Figure 11. Parity comparison of surface coverage of (a) NO, (b) O, and (c) total as estimated using TBST vs ternary model. GCMC simulations were performed for various conditions of T {773 K, 723 K, ..., 373 K}, PNO {1 ppm, 10 ppm, 100 ppm, 1000 ppm, 10 000 ppm}, and ln(PNO2/PNO) {−4, −2, 0, 2, 4}.

Here μx and Px refer to the chemical potential and partial pressure of species x, respectively, APt is the area of a 1 × 1 Pt(111) unit cell, and T, kB, and mO2 are the temperature, Boltzmann constant, and O2 molecular weight, respectively. The site-dependent O2 dissociative activation energy, Eai , is described using a previously reported Brønsted−Evans− Polanyi (BEP) relationship:56

et al.41 The equilibrium for NO* and O* is governed by eqs 5a and 5b, and the dissociation of O2 on adjacent Pt(111) fcc sites, referred to as basis sites, is assumed to be rate limiting:73−77 O2 + 2∗fcc → 2O*fcc

(6)

The steady-state NO oxidation reaction rate is expressed as a sum over rates at basis sites i weighted by the number density s̃i/s̃max of those sites:44

Eia = max(0, BEi + 2.12)

sĩ(T , μO , μ NO) ⎛ ∑j sj̃ ⎞ ⎜ ⎟ r(T , μO , μ NO , PO2) = ∑ ri(̃ T , PO2) ⎜ s̃ ⎟ ∑j sj̃ ⎝ max ⎠ i

where BEi is the energy of eq 6 evaluated at a candidate 1NN pair vacancy site i, extracted either from the full ternary or TBST CE. Due to the differences in local distribution of O and NO around i, {BEi} is a distribution of energy values. Figure 13 presents a sample distribution at T = 573 K for PNO = 100 ppm and ln(PNO2/PNO) = −2, where BEi values have been categorized into 10 meV bins. The number density s̃i/s̃max is

(7)

where the individual site rates r̃i are given as ri(̃ T , PO2) =

PO2APt 2πmO2kBT

(9)

a

e−Ei / kBT (8) G


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the frequency of energy bins normalized by the total number of vacant pair sites on clean 42 × 42 surface. We have included the sum of bin frequencies ∑j s̃j in eq 7 to highlight the contributions of energetic and configurational aspects to the total reaction rate. ⟨ΔE⟩ values reported in Figure 13 are the Boltzmann weighted22 {BEi} and are a measure of an average binding energy value for all the sites i that will provide the same reaction rates as the distribution. Figure 14 compares the reaction rates obtained using the two models. When color coded with NO coverage (Figure 14a), we find that the data lies on the parity line for reaction conditions when θNO < 0.15 ML and diverges significantly for θNO > 0.3 ML. The scenario is reversed when the parity plot is color coded with θO (Figure 14b), and the models are found to be consistent for θO > 0.2 ML. On comparing the reaction rates with respect to aggregate coverages (Figure 14c), we find that the reaction rates increase as the total coverage decreases. This is a manifestation of our reaction model that requires vacant sites for the O2 adsorption. An increase in temperature leads to more available vacant sites and also higher reaction rates per basis site (eq 8).

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DISCUSSION A multiadsorbate CE can accurately represent complicated, coupled interactions between unlike adsorbates at a surface. One of the major issues in building such a CE system is the curse of dimensionality, which refers to the fact that as the dimensionality of any model increases, the volume of sample space increases exponentially.68 The available data becomes increasingly sparse. If we use n sample DFT calculations to parametrize a model for a single adsorbate, then we will ideally need n2 DFT calculations to build an equally reliable twoadsorbate model. Here, we developed an approximate dual adsorbate ternary CE step-by-step using binary configurations and compared the results to a full ternary CE. The development of this TBST CE model in principle reduces the required DFT data points to ∼3n. We found that two- and three-body clusters are necessary to model NO−O coadsorption on Pt(111). We apply both CEs to predict catalytic NO oxidation rates on Pt(111), in which NO and O are the important surface species. As we find in Figure 9, the TBST residuals are more significant at the low binding energies corresponding to high NO coverages. The TBST CE predicts NO to bind more strongly to the surface than DFT-calculated values; this bias is evident in Figure 11. NO coverages predicted from GCMC

Figure 12. Variation of NO and O surface coverage with temperature for three different values of ln(PNO2/PNO) estimated using (a) ternary and (b) TBST models. PNO = 100 pm.

Figure 13. Sample distribution of using (a) ternary and (b) TBST models at T = 573 K, PNO = 100 ppm, and ln(PNO2/PNO) = −2. Expectation values ⟨ΔE⟩ were calculated using a Boltzmann weighting of BEi distribution.22

Figure 14. Parity comparison of turnover rates as estimated using the TBST vs ternary model for 225 different reaction conditions mentioned in Figure 11, color coded by (a) θNO, (b) θO, and (c) θTotal. PO2 = 0.1 bar. H



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simulations on the TBST CE are higher than those from the full ternary CE, and predicted NO oxidation rates are in the worst agreement (and lowest absolute value) at high NO coverage (Figure 14a). Both CEs predict the surface to be NOcovered at low temperature, evolving to O-dominated at temperatures T > 573 K. The exact transition temperature depends on PNO and PNO2. In this higher temperature range, results are equivalent to a single adsorbate CE.41,45 The type of CE necessary to model a particular catalysis is thus a function of the reaction conditions of interest. The TBST approach thus works acceptably well for the NO−O on Pt(111) system. The ability to represent the fully interacting system with the combination of three binary CEs rests on two key characteristics of the system. First, effective interactions extracted from the NO−vac and O−vac legs should be transferable to the NO−O leg, and second, the O− NO effective interactions extracted at full coverage should be representative of those at lower coverage. Neither of these properties implies any necessary similarity between interactions, and both conditions appear to be satisified well in the system here. If the latter condition was problematic, the NO−O effective interactions could be extracted at lower total coverage, with some increase in complexity of the model. The extensions to other adsorbate combinations and to other metal surfaces are worth further investigation.

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Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b00914. Adsorption ground states, details for translating ATAT features into adsorbate specific interactions, figures illustrating many-bodied cluster interactions, detailed description of TBST model, derivations for GCMC chemical potentials (PDF) Data files for ternary model and Matlab scripts used for translating ATAT features into lattice gas clusters (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 574-631-8754. ORCID

Anshumaan Bajpai: 0000-0003-3567-7217 William F. Schneider: 0000-0003-0664-2138 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the funding provided by the U.S. Department of Energy, Office of Basic Energy Sciences, under Grant DEFG02-06ER15839. Computing resources and technical support were provided by the Notre Dame Center for Research Computing.

CONCLUSIONS

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Multiadsorbate CE models are useful for quantitatively modeling catalytic reactions that involve more than one surface species. Even for simple reactions, the nature of an adsorbed entity can vary depending on the reaction conditions. In this work, we employ two different methodologies to develop dual adsorbate CE models for NO/O adsorption on Pt(111), considering only fcc sites for adsorption. The first model (ternary) uses ATAT’s multicomponent version to build a large database of DFT calculated structures and optimizes the number of parameters needed in the model using 5-fold cross validation. A major issue in developing a reliable CE with more adsorbates is the size of training database needed which increases exponentially as the number of adsorbates increases. We propose a TBST model that uses only a small subset (NO− vac, O−vac, NO−O structures) of the large DFT database to generate three binary (NO−vac, O−vac, NO−O) CEs and merges them for application to ternary NO−O−vac systems. Both models do a similar job in identifying the important cluster interactions. The ternary model is able to predict the formation energy of independent NO−O−vac configurations accurately across all coverages, while TBST developed in this work is biased toward predicting NO to bind more strongly than the DFT calculated values. However, the relative trends for surface coverages and reaction rates with varying conditions observed using the two models are consistent and lead to the same qualitative conclusion. A shorter TBST model can therefore be used for high throughput screening of adsorbates, catalysts, and surface facets to identify the systems for detailed investigation. Our work also provides insight into NO−O coadsorption and kinetics of NO oxidation under dual adsorbate conditions, and motivates extensions to other systems.

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