Extrapolating Energetics on Clusters and Single-Crystal Surfaces to

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Extrapolating Energetics on Clusters and Single Crystal Surfaces to Nanoparticles by Machine-Learning Scheme Ryosuke Jinnouchi, Hirohito Hirata, and Ryoji Asahi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b08686 • Publication Date (Web): 03 Nov 2017 Downloaded from http://pubs.acs.org on November 4, 2017

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The Journal of Physical Chemistry

Manuscript for

Extrapolating energetics on clusters and single crystal surfaces to nanoparticles by machine-learning scheme

Ryosuke Jinnouchi,*1 Hirohito Hirata,2 and Ryoji Asahi1 1

Toyota Central R&D Labs, Inc., 41-1 Yokomichi Nagakute, Aichi 480-1192, Japan

2

Toyota Motor Corporation, 1200 Mishuku, Susono, Shizuoka 410-1193, Japan

*Correspondence to: [email protected]

1

ACS Paragon Plus Environment


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Abstract A Bayesian linear regression scheme using a local structural similarity kernel as a descriptor is used to predict the energetics of atoms and molecules on nanoparticles. Examination of the binding energies of N, O, and NO with RhAu alloy single crystal surfaces and particles indicate that regression models predict the binding energies on nanoparticles having diameters longer than 1.5 nm within the an error range of 100 to 150 meV when the DFT data on single crystal surfaces are used for training. By contrast, when the DFT data on small clusters are used for training, the regression models produce greater an error range of 200 to 400 meV. Kinetic analyses using the predicted energetics of the direct decomposition of NO on RhAu nanoparticles indicate that catalytic activity increases with a decrease in the particle diameter up to 2.0 nm, whereas the activity drops when the diameter decreases to 1.5 nm. Detailed examinations of free energy diagrams and structures of active sites indicate that the drop in catalytic activity derives from the disappearance of active alloyed corner sites on the small nanoparticle as a result of Au segregations at the corners of narrow facets.

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1. Introduction Designing heterogeneous catalysts that realize high activity and selectivity for desired chemical reactions is one of the greatest challenges to achieving sustainable energy systems and material synthesis processes. Nanoparticles are widely used in industry because of their high surface-to-volume ratios. Their surfaces are composed of a variety of surface sites, and their catalytic activity and selectivity are often dominated by a few specific surface sites.1-4 To identify and optimize the dominant surface sites, intensive experimental and theoretical analyses have been conducted on well-defined model catalysts such as clusters5 and single crystal surfaces,6-15 and valuable information has been obtained on the detailed relationships between structures and compositions of surface sites in terms of activity and selectivity. Based on the obtained information, many high-performance catalysts have been developed by optimizing the composition, size, and shape of nanoparticles using state-of-the-art catalyst synthesis processes.16-20 However, major structural gaps exist between model catalysts and practically used nanoparticle catalysts, and those gaps often create difficulties in extrapolating information from model catalysts to nanoparticles, particularly in the case of highly heterogeneous alloy nanoparticles. Descriptor-based approaches are powerful methods to bridge structural gaps. In these approaches, reaction energetics dominating catalytic activity and selectivity are described by simple functions of descriptors that can be easily computed. In conventional approaches, descriptors and functions are constructed based on simple physical models and/or empirical rules. A typical model of a descriptor and function deduced from physical models is the d-band model.10, 12 In this model, 3



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moment characteristics of the projected density of states onto valence orbitals of a transition metal atom (e.g., d-band center) are used as the descriptors. These moment characteristics are related to binding energies of reaction intermediates with catalyst surfaces based on a simple tight-binding model that describes metal-adsorbate interactions. Moment characteristics can be obtained using electronic structure calculations such as density functional theory (DFT) calculations;21-22 the functions relating them to the binding energies include adjustable parameters that must be determined in order to reproduce the “training data” on the binding energies. The d-band model has been successfully used to explain observed trends in catalytic activities and to explore new active alloy catalysts.13-14,

23

In addition, other electronic descriptors exist such as work function, the

number of valence electrons, integrated crystal orbital overlap populations (COOP), and crystal orbital Hamiltonian populations (COHP), and their correlations with binding energies provide new physical insights into observed trends in metal-adsorbate interactions.24 Generalized coordination number (GCN) is a geometrical descriptor constructed based on an empirical law that states that a surface atom with low coordination numbers binds more strongly with other atoms and molecules.25-27 This descriptor was recently utilized to explore optimal surface morphologies of pure Pt electrocatalysts for oxygen reduction reaction.26 Combinations of several physics-based descriptors have also been recently suggested.28-30 Ma and Xin29 constructed orbitalwise coordination numbers calculated by two-center hopping integrals to include both the electronic and geometrical effects into the descriptors based on a tight-binding model. The model was used to predict the binding energies of O and CO on an Au shell on Ag or Cu core nanoparticles. 4


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Machine-learning schemes such as neural network28 or gradient-boosting regression30 have also been suggested in order to mix the multi-descriptors automatically, and the regression models were used to predict the binding energies of CO on close-packed single crystal alloy surfaces and on pure Pt nanoparticles. An advantage of predictions that employ physics-based descriptors is that the determined functions possess a certain level of universality beyond the small number of sets (typically 10–100) of training data. Unfortunately, however, many limitations exist within these physical or empirical rules, and a trade-off often occurs between their accuracies and universality. Recently, our group31 suggested a Bayesian linear regression scheme that employs a multi-dimensional structural descriptor32-33 measuring a local configurational similarity between two surface sites. This method relies on the fact that catalytic active sites are determined by their local atomic configurations. In other words, two sites having a similar atomic configuration should generate similar catalytic activity. Unlike regressions using physics-based descriptors, this model does not rely on any other physical models or empirical rules except for the locality assumption and, therefore, the accuracy of a prediction can be improved by simply increasing the number of training data sets for any materials. Furthermore, the descriptors can be easily calculated without performing expensive DFT calculations on nanoparticles. This is in contrast to conventional approaches that require DFT calculations on nanoparticles in order to construct descriptors and/or training data. However, two essential questions remain concerning the applicability of the proposed machine-learning scheme. In the previous study,31 the regression model that uses training data on the single crystal surfaces was confirmed to be applicable to a truncated octahedral nanoparticle 5



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having a diameter longer than 2 nm. However, whether the same model can predict energetics on smaller particles remains unclear. Another essential question concerns the availability of data on small clusters. Energetic data on small clusters can be easily obtained by DFT calculations, and therefore, data preparation processes can be considerably accelerated by using cluster data. This study examines the accuracies of regression models generated from training data on single crystal surfaces and/or small clusters. After the accuracies of their predictions are confirmed, the regression models are used to predict catalytic activity of the direct decomposition of NO on Rh(1−x)Aux nanoparticles. 2. Method 2.1 Machine-learning scheme A schematic of the suggested machine-learning algorithm is given in Fig. 1. As described in our previous study,31 this algorithm consists of two steps: learning the stored DFT data and extrapolating the obtained lessons to nanoparticles. In the learning step, DFT data on the catalytic surfaces are stored in the database. The stored data consist of geometrical information on the unrelaxed surface sites determined from bulk geometries and binding energies of reaction intermediates with the relaxed surface sites. The word “relaxed” refers to the geometries optimized by the DFT calculations. In the previous study,31 DFT calculations on single crystal surfaces were executed, and the data were stored in a database. In this study, as illustrated in Fig. 1, DFT data on the small clusters are also stored in the database. Next, the local structural similarity between I-th and J-th unrelaxed intermediates on the surfaces is 6


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evaluated as KIJ by using a similarity kernel known as a smooth overlap atomic position (SOAP).31-33 The SOAP KIJ consists of overlap integrals between three-dimensional atomic distributions within a cutoff radius Rcut from the I-th surface site and those from the J-th surface site as follows.

 FIJ K IJ =   F F  II JJ

FIJ =

   

nhyp

,

(1)

λττ λτ τ λτ τ ∫ Sττ (Rˆ )Sτ τ (Rˆ )dRˆ , ∑ ∑ δ ∑∑∑∑ τ τ τ τ

Natom

Natom

′

′

ij

′

i =1,i∈I j =1, j∈J

′′

′ ′′

′′ ′′′

ij

′′ ′′′

ij

()

) S ijττ ′ Rˆ = κ ττ ′ ∫ ρ iτ (r )R ρ τj ′ (r )d r ,

ρi (r ) = τ

Natom

∑δττ j =1

j

f cut

(

(3)

  1 2 2 ri − r j exp r r r −   + −  , i j (πη )3   η  

)

(2)

′′′

1

 1  r    + 1 , r ≤ Rcut cos π f cut (r ) =  2   Rcut   ,    0, r > Rcut

(4)

(5)

where Natom denotes the number of atoms composing the reaction intermediate; the symbol i and j refer to the atoms included in the I-th and J-th intermediates, respectively; ri denotes the position vector of the i-th atom in the unrelaxed geometry, details of which are provided in the previous study;31 and ρiτ is the distribution function of the τ-th element inside the cutoff radius Rcut from the i-th atom. As indicated by (4), ρiτ is calculated by summing the atomic positions described by the Gaussian function. To ensure that only atoms inside the cutoff radius Rcut contribute to this summation, the cutoff function defined as (5) is introduced in (4). Sijττ’ is the overlap integral between the atomic distribution functions ρiτ and Rˆ ρjτ’, where Rˆ denotes the operation rotating 7



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the distribution function ρjτ’. To make the similarity measure rotationally invariant, FIJ is constructed by integrating the product of Sijττ’ and Sijτ”τ”’ over Rˆ as shown in (2), and the SOAP KIJ is obtained by normalizing FIJ as shown in (1). Based on a suggestion of De et al.,33 the parameters λττ’ and κττ are described by the following equations.

λττ ′ =1

(6) 

κ ττ ′ = exp −

(ε

d,τ



2 − ε d,τ ′ )  , ∆2 

(7)

where εd,τ is the d-band center of the τ-th element in its pure bulk state. As expected from (1) and (2), the SOAP KIJ defined in (1)–(7) approaches unity (or zero) when the two atomic distributions

ρiτ and ρjτ’ overlap more or less strongly. The binding energy Eb,I of the I-th relaxed reaction intermediate is assumed to be described by a linear function of KIJ as follows.

Eb,I =

Nbasis

∑w K J

IJ

,

(8)

J

where the regression coefficients wJ are determined to reproduce the stored DFT training data Eb,IDFT by using a Bayesian linear regression method.31, 34 Here, Nbasis is the number of basis sets in the linear regression and equals the number of training data Ndata sets in this study. In the extrapolating step, an unrelaxed nanoparticle model is constructed from the bulk geometry, and the local similarity between the K-th and J-th intermediates on the nanoparticle and single crystal surface, respectively, is evaluated as KKJ by the SOAP. By using KKJ and the regression coefficients wJ, the binding energy Eb,K of the K-th relaxed intermediate on the nanoparticle is predicted as follows. 8


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Eb,K =

Nbasis

∑w K J

KJ

.

(9)

J

Although we call this step “extrapolation,” thanks to the reasonable locality assumption described previously, the descriptor SOAP can efficiently convert the extrapolation problems in nanoparticle sizes to interpolation ones in terms of local structural similarity. This conversion realizes accurate predictions of the energetics as described later in this study. The parameters used to calculate the SOAP, as given in (1)–(5), are tabulated in Table S1 in the section “Supporting Information.” The parameters are the same as those used in the previous studies.31-33 Because parameter optimizations were already examined in those studies, only their essences are briefly explained as follows. The most crucial parameter is the cutoff radius Rcut. From a physical view point, Rcut determines the interaction range considered by the SOAP descriptor. In general, the longer Rcut is, the more accurate is the prediction. By contrast, from the perspective of the machine-learning algorithm, Rcut determines the size of the space spanned by the SOAP descriptors. The longer Rcut is, the greater is the dimension of the SOAP descriptor, and the larger the descriptor space. Thus, an increase in Rcut improves the accuracy of the prediction but increases the necessary number of sets of training data. A typical example is given in Fig. 2 for a case involving the binding energies of an oxygen atom with single crystal Rh(1−x)Aux alloy surfaces.31 At any length of Rcut, the error σ decreases with the number of sets of training data. However, the reachable accuracy and the amount of data achieving the greatest accuracy both increase with an increase in Rcut as previously explained. In this study, to balance the accuracy and amount of data, 9



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we set Rcut to 0.6 nm. The polynomial parameter nhyp is a kind of hyper-parameter that determines the sharpness of the SOAP in (1); the larger nhyp is, the sharper the SOAP. In general, a sharper similarity measure detects structural differences more precisely and allows more correct interrogations of the binding energies in the database. However, too sharp of a SOAP worsens the performance of interpolations among the training data points, and therefore, the necessary number of sets of training data again increases. The parameter nhyp was optimized as 2–4 in previous studies31-33 and, in this study, was set to 3. The broadening width η = 0.05 nm of the atomic positions in (4) is much smaller than the metal-metal bond lengths of approximately 0.28 nm and the metal-adsorbate bond lengths of approximately 0.2 nm. This indicates that the Gaussian atomic distribution function in (4) is localized to each atomic site, and the onsite overlap integrals of the Gaussian distributions mainly contribute to the total overlap integral in (3). Eq. (7), with the parameter ∆, was originally introduced by De et al.33 to measure the similarity between different elements using the element descriptor εd,τ. If the difference εd,τ−εd,τ’ is much smaller than the parameter ∆, the SOAP judges that the elements τ and τ’ possess a similar chemical nature, and effects of element τ’ on the binding energy are replaced with those of element τ. If εd,τ. is the good descriptor of the binding energy, the necessary number of sets of training data can be considerably decreased by the introduction of (7) because the data on element τ’ becomes unnecessary. Unfortunately, however, the d-band center εd,τ of the pure bulk metal is not a sufficiently good descriptor of the binding energy. Therefore, we set ∆ to a small value of 0.03 eV. 2.2 DFT calculations on clusters 10


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Training data on binding energies of N, O, and NO with Rh(1−x)Aux alloy clusters were obtained by a method used to obtain data on single crystal Rh(1−x)Aux alloy surfaces of (111), (100), (110), (211), (221), and (532) (x = 0.00, 0.25, 0.75, and 1.00) in our previous study.31 The clusters consist of 6, 19, and 38 atoms with x = 0.17, 0.25, 0.50, 0.63, 0.75, and 0.83 as shown in Fig. 3(A), (B), and (C), and their binding energies with the atoms and molecules were obtained by structural optimizations using DFT calculations. Initial locations of Au atoms were randomly determined in the calculations of the training data. By contrast, in the calculations of the test data, they were determined by Monte Carlo simulations at 500 K using a regression model of interactions among metal atoms that themselves were determined by a machine-learning scheme similar to that explained in the previous section. Details of the regression model and Monte Carlo simulations are provided in our previous study.31 To check the accuracies of the predicted binding energies on larger particles, test data on Rh(1−x)Aux alloy particles comprised of 79 and 201 atoms, as shown in Fig. 3(D) and (E), respectively, were also obtained using the DFT calculations. The number of sets of test data on single crystal surfaces, nanoparticles, and clusters is summarized in Table S2 in the section “Supporting Information.” In the DFT calculations, each nanoparticle was located in a unit cell to form a close-packed structure, where the distance from the centers of nanoparticles was equispaced by 1.2, 1.5, 2.0, 2.0, and 2.5 nm for the clusters consisting of 6, 19, 38, 79, and 201 atoms, respectively. Similar to the previous study, all DFT calculations were executed using the Vienna Ab initio Simulation Package (VASP5.1).35-36 Plane wave basis sets with an energy cutoff of 400 eV were used to expand the one-electron Kohn-Sham orbitals, and the projector augmented 11



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wave (PAW) method37-38 was used to describe interactions between valence electrons and effective cores composed of atomic nuclei and inner electrons. Exchange-correlation interactions among valence electrons were described by a revised Perdew-Burke-Ernzerhof functional proposed by Hammer et al.39 Brillouin zone integrations were executed using a Gaussian broadening scheme40 with a broadening width of 0.1 eV and Γ-point sampling for the k-point. The structural optimizations were executed until the maximum forces acting on atoms become less than 0.05 eV⋅Å−1. Similar to our previous study,31 after structural optimizations, the maximum atomic displacements and SOAP between initial and final structures were calculated. Then, the data with a maximum displacement greater than 0.1 nm and SOAP less than 0.5 were excluded from the database. 2.3 Kinetic model of the direct decomposition of NO After confirming the accuracies of the predicted binding energies, the machine-learning method was applied to predict the catalytic activity of the direct decomposition of NO on the Rh(1−x)Aux alloy nanoparticles with diameters of 1.5 to 5.0 nm, as shown in Fig. 3(D) to (I). Details of the kinetic analysis are given in our previous study.31 In this section, only the essence of the analysis is described. The following reaction pathway consisting of dissociations and formations of diatomic molecules41-42 is examined by kinetic analysis. 2NO(g) → NO(ad) + NO(g),

(R1)

NO(ad) + NO(g) → N(ad) + O(ad) + NO(g),

(R2) 12


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N(ad) + O(ad) + NO(g) → N(ad) + O(ad) + NO(ad),

(R3)

N(ad) + O(ad) + NO(ad) → 2N(ad) + 2O(ad),

(R4)

2N(ad) + 2O(ad) → N2(g) + 2O(ad),

(R5)

N2(g) + 2O(ad) → N2(g) + O2(g),

(R6)

where (g) and (ad) denote the chemical specie in the gas and adsorbed phases, respectively,. In addition, the net catalytic activity of the direct NO decomposition is evaluated by a method that assumes the so-called nearsightedness, where a high surface area catalyst is regarded as a distribution of surfaces with different local geometries, and a perturbation to the surface is only measurable within a screening length of sub nm or on the order of one lattice constant as described by Nørskov et al.3 Based on these assumptions, the turnover frequency (TOF) of the catalytic reaction is calculated as a summation of the local TOFs over all the surface sites as follows.

TOF = ∑ TOFs ,

(10)

s

where s indicates the s-th surface atom. The local turnover frequency TOFs is calculated by Sabatier analysis.11, 43-44 In this analysis, a free energy diagram at each surface site is evaluated by using the machine-learning scheme described in Section 2.1 and Brønsted-Evans-Polanyi relationships derived from Falsig et al.45 In addition, activation free energies of all elementary reactions are calculated from the free energy diagram as ∆Gts*, where t denotes the t-th elementary step. By using ∆Gts* and based on the assumption that the local catalytic reaction is dominated by the elementary reaction with the highest activation free energy, the TOFs is evaluated as follows. 13



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TOFs =

 ∆Gs*  k BT  , exp − h  k BT 

(11)

( )

∆Gs* = max ∆Gts* . t

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(12)

3. Results and discussion 3.1 Accuracies of predictions on binding energies Figures 4(A)–(C) show the mean absolute errors, σN, σO, and σNO of the binding energies of N, O, and NO, respectively, with the Rh(1−x)Aux alloy particles of diameters d = 0.5 to 2.0 nm and single crystal surfaces (d = ∞) predicted from the training data on single crystal surfaces. All errors are shown as functions of the number of sets of training data Ndata. The mean absolute errors at the largest Ndata are shown for each diameter in Fig. 4(D), and the predicted binding energies of Eb,NML, Eb,OML, and Eb,NOML of N, O, and NO, respectively, are plotted as functions of those obtained by the DFT calculations in Figs. S1(A) to (C) in the section “Supporting Information.” As indicated by these figures, the mean absolute errors of nanoparticles with d ≥ 1.5 nm decrease to than 100–150 meV with each increase in Ndata, whereas for smaller particles, the errors are always greater than 200–400 meV. The results indicate that the energetic information on single crystal surfaces can be extrapolated up to the nanoparticle with a diameter of 1.5 nm (approximately 80 atoms) within the accuracies of 100–150 meV. The large errors for small clusters derive from their considerably different surface structures from the bulk surfaces; the structures are so different that the cluster data are located far from the training single crystal data in the descriptor space.

14


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Figures 5(A)–(D) and S2(A)–(C) show the same properties predicted from the training data on the small clusters. For all particles, the mean absolute errors decrease with the increase in the training data. However, for the nanoparticles with d ≥ 1.5 nm, the errors do not decrease to than 200–400 meV. The results indicate that to predict the reactivity of practically used nanoparticles with d ≥ 1.5 nm in many applications,16-19, 46-48 extrapolations from the single crystal surfaces give more accurate results than those from small clusters. The trend reasonably agrees with the empirically well known facts shown in previous studies6-15 that the single crystal model catalysts give valuable information for designing nanoparticles, whereas small clusters possess unique reactivity that is different from the bulk surfaces.5 Although the regression models generated from the training data on the single crystals can more accurately predict the energetics on nanoparticles with d ≥ 1.5 nm, those models generate significantly large errors for smaller particles. The errors can be partially suppressed by adding the training data on small clusters to those on single crystal surfaces, as shown in Figs. 6(A)–(D) and S3(A)–(C). The suppression effects can be more clearly shown in Figs. 7(A) and (B) in the case of the oxygen adsorbate. Although adding the cluster data does not decrease the errors on particles larger than 1.5 nm, it does decrease errors on small clusters up to the values achieved by the regression models generated from the training data on small clusters. It should also be noted that the errors can be further decreased by increasing the cutoff radius from 0.6 to 0.8 nm to detect structural differences between the small clusters and single crystal surfaces more precisely. The results shown in Figs. 6 and 7 mean that the data on clusters are compatible with those on single 15



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crystal surfaces in our Bayesian linear regression scheme. Using this data on clusters can extend the applicability of the regression models generated from the single crystal data. 3.2 Composition- and size-dependent catalytic activity The TOFs of the direct decomposition of NO on the Rh(1−x)Aux alloy nanoparticles with d = 1.5 nm were calculated by using the regression models generated from the training data on single crystal surfaces, which most accurately predicts the energetics on nanoparticles with d ≥ 1.5 nm. The results are summarized in Fig. 8 and compared with the results on larger nanoparticles obtained in our previous study. 31 For nanoparticles with 2 ≤ d ≤ 5 nm, the Au atomic ratio xmax giving the maximum TOF (TOFmax) increases with the decrease in the diameter d as shown in Figs. 8(A) and (B). The TOFmax also increases with the decrease in d as shown in Fig. 8(C). As discussed in our previous study,31 the trend in xmax derives from the surface segregations of Au illustrated in Fig. 9. By means of the surface segregation, the Au atomic ratio on the surface xsurf becomes proportional to x⋅d. Therefore, a larger x is necessary for the smaller nanoparticles to achieve the optimal surface ratio of xmaxsurf = 0.5–0.6 as shown by the small triangles in Fig. 8(B). The trend in TOFmax derives from the location of the active site. As illustrated in the activity map in Fig. 9, the alloyed corner site of Rh(1-x)Aux nanoparticles is identified as the most active site by the kinetic analysis. Because the surface density of the corner site is proportional to d−2 on the truncated octahedral nanoparticles, the TOF per surface area linearly depends on d−2 as indicated by the solid line in Fig. 8(C).

16


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The nanoparticle with d = 1.5 nm essentially satisfies the trend in xmax observed in the larger nanoparticles. However, the TOFmax on this small nanoparticle is much lower than the value expected from the d−2 trend as shown in Fig. 8(C). The rapid drop is caused by the narrow facets on this nanoparticle. To demonstrate the effects of these narrow facets, a free energy diagram of the direct NO decompositions at the corner site on this small nanoparticle is compared with that on the nanoparticle with d = 2.0 nm in Fig. 10(A). The result indicates that the free energy barrier of the second NO dissociation (from the state number 5–7) is considerably raised because the dissociated intermediates composed of atomic nitrogen and oxygen adsorbates are destabilized. This destabilization of adsorbates is caused by the narrow facets. As shown in Figs. 10(B) and (C), Au atoms are segregated at edges and corners of the Rh(1-x)Aux nanoparticles for both d = 1.5 and 2.0 nm. However, for d = 1.5 nm, the (100)-facet is purely covered by Au atoms because the facet consists of edges and corners only. Through this full coverage of Au, both the nitrogen and oxygen atoms are repelled from the (100)-facet. Another major effect is caused by the narrow (111)-facet. Although Rh atoms remain at the center of the (111)-facet, their coverage considerably decreases because of the high ratio of the edge atoms. In this situation, the oxygen atoms must interact with more Au atoms, and their binding energies considerably decrease. In past studies, the sudden drops of the catalytic activity across certain thresholds of the diameter were also reported on several reactions such as dissociations of diatomic molecules and ammonia synthesis on pure transition metal surfaces.7, 49-51 Detailed experimental and theoretical analyses indicate that these drastic changes are caused by either the disappearance or appearance of 17



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active sites known as double-step edges4 at the thresholds. The structure of active sites in our study does not consist of these double-step edges. In addition, our kinetic model still must include effects produced by the lateral interactions and adsorbate diffusions for more accurate predictions. Nevertheless, the results described in this section demonstrate that the kinetic analysis using machine-learning energetics can be a useful tool for predicting the size-sensitive catalytic activity of nanometer-scale alloy catalysts. 4. Conclusion In this study, we examined the applicability of the recently developed machine-learning scheme that predicts the reaction energetics of catalysis. Our examination revealed that training data on both single crystal surfaces and clusters are available in the machine-learning scheme, and detailed analyses showed that extrapolations from the training data on single crystal surfaces generate accurate energetics on nanoparticle catalysts with diameters longer than 1.5 nm, whereas for smaller particles, the extrapolation generates large errors. By adding cluster data to the training data, the errors on smaller particles are suppressed, indicating that the single crystal and cluster data are compatible in the machine-learning scheme. Constructed regression models were applied to the direct decomposition of NO on Rh(1−x)Aux alloy nanoparticles with diameters from 1.5 to 5.0 nm, and the applications indicated that kinetic analysis using machine-learning energetics can capture major atomic-scale structural effects on alloy nanoparticle catalytic activity. Further improvements to this scheme are described at the end of this article. Kinetic analysis relies on activation barriers estimated from the Brøsted-Evans-Polanyi relations, and this 18


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approximation can generate errors of 100–200 meV in activation free energies. In addition, the model neglects effects by lateral interactions among adsorbates and adsorbate diffusions, and their inclusion can considerably affect the results of catalytic activity. However, we must stress that our machine-learning scheme can potentially predict all necessary properties for including those effects: activation barriers of dissociations and diffusions, and the coverage-dependent binding energies that include lateral interactions. Thus, we believe that the proposed machine-learning scheme can have wide applications in the design of high-performance composition-, size- and shape-controlled nano-catalysts used for energy conversions and material synthesis. Supporting Information The supporting information is available free of charge on the ACS Publications website at DOI: SOAP parameters, predicted binding energies versus DFT binding energies and the amount of test data. References (1)

Taylor, H. S. A Theory of the Catalytic Surface. Proc. Roy. Soc. A 1925, 108, 105-111.

(2) Boudart, M. Catalysis by Supported Metals. Adv. Catal. 1969, 20, 153-166. (3) Norskov, J. K.; Bligaard, T.; Hvolbaek, B.; Abild-Pedersen, F.; Chorkendorff, I.; Christensen, C. H. The Nature of the Active Site in Heterogeneous Metal Catalysis. Chem. Soc. Rev. 2008, 37, 2163-2171. (4) Van Santen, R. A. Complementary Structure Sensitive and Insensitive Catalytic Relationships. Acc. Chem. Res. 2009, 42, 57-66. (5) Ferrando, R.; Jellinek, J.; Johnston, R. L. Nanoalloys: From Theory to Applications of Alloy Clusters and Nanoparticles. Chem.Rev. 2008, 108, 845-910. (6) Somorjai, G. A.; Joyner, R. W.; Lang, B. The Reactivity of Low Index [(111) and (100)] and Stepped Platinum Single Crystal Surfaces. Proc. Roy. Soc. A Mathematical and Physical Sciences 1972, 331, 335-346.

19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(7)

Zambelli, T.; Wintterlin, J.; Trost, J.; Ertl, G. Identification of the ''Active Sites'' of a

Surface-Catalyzed Reaction. Science 1996, 273, 1688-1690. (8) Dahl, S.; Logadottir, A.; Egeberg, R. C.; Larsen, J. H.; Chorkendorff, I.; Tornqvist, E.; Norskov, J. K. Role of Steps in N-2 Activation on Ru(0001). Phys. Rev. Lett. 1999, 83, 1814-1817. (9) Hammer, B.; Norskov, J. K. Why Gold Is the Noblest of All the Metals. Nature 1995, 376, 238-240. (10) Hammer, B.; Norskov, J. K. Theoretical Surface Science and Catalysis - Calculations and Concepts. In Adv. Catal. Vol 45: Impact of Surface Science on Catalysis, Gates, B. C.; Knozinger, H. Eds. Elsevier Academic Press Inc: San Diego, 2000; Vol. 45, pp 71-129. (11) Norskov, J. K.; Bligaard, T.; Rossmeisl, J.; Christensen, C. H. Towards the Computational Design of Solid Catalysts. Nat. Chem. 2009, 1, 37-46. (12) Hammer, B.; Morikawa, Y.; Norskov, J. K. Co Chemisorption at Metal Surfaces and Overlayers. Phys. Rev. Lett. 1996, 76, 2141-2144. (13) Stamenkovic, V. R.; Fowler, B.; Mun, B. S.; Wang, G. F.; Ross, P. N.; Lucas, C. A.; Markovic, N. M. Improved Oxygen Reduction Activity on Pt3Ni(111) via Increased Surface Site Availability. Science 2007, 315, 493-497. (14) Stamenkovic, V. R.; Mun, B. S.; Arenz, M.; Mayrhofer, K. J. J.; Lucas, C. A.; Wang, G. F.; Ross, P. N.; Markovic, N. M. Trends in Electrocatalysis on Extended and Nanoscale Pt-Bimetallic Alloy Surfaces. Nat. Mater. 2007, 6, 241-247. (15) Calle-Vallejo, F.; Pohl, M. D.; Reinisch, D.; Loffreda, D.; Sautet, P.; Bandarenka, A. S. Why Conclusions from Platinum Model Surfaces Do Not Necessarily Lead to Enhanced Nanoparticle Catalysts for the Oxygen Reduction Reaction. Chem. Sci. 2017, 8, 2283-2289. (16) Cui, C. H.; Gan, L.; Li, H. H.; Yu, S. H.; Heggen, M.; Strasser, P., Octahedral PtNi Nanoparticle Catalysts: Exceptional Oxygen Reduction Activity by Tuning the Alloy Particle Surface Composition. Nano Lett. 2012, 12, 5885-5889. (17) Chen, C.; Kang, Y.; Huo, Z.; Zhu, Z.; Huang, W.; Xin, H. L.; Snyder, J. D.; Li, D; Herron, J. A.; Mavrikakis, M., et al. Highly Crystalline Multimetallic Nanoframes with Three-Dimensional Electrocatalytic Surfaces. Science 2014, 343, 1339-1343. (18) Huang, X.; Zhao Z.; Cao, L.; Chen, Y.; Zhu, E.; Lin, Z.; Li, M.; Yan, A.; Zettl, A.; Wang, Y. M., et al. High-Performance Transition Metal-Doped Pt3Ni Octahedra for Oxygen Reduction Reaction. Science 2015, 348, 1230-1234. (19) Bu, L.; Zhang N.; Guo S.; Zhang X.; Li J.; Yao J.; Wu, T.; Lu G.; Ma J.-Y.; Su, D., et al. Biaxially Strained PtPb/Pt Core/Shell Nanoplate Boosts Oxygen Reduction Catalysis. Science 2016, 354, 1410-1414. (20) Li, M.; Zhao, Z.; Cheng, T.; Fortunelli, A.; Chen, C.-Y.; Yu, R.; Zhang, Q.; Gu, L.; Merinov, B. V.; Lin Z., et al. Ultrafine Jagged Platinum Nanowires Enable Ultrahigh Mass Activity for the Oxygen Reduction Reaction. Science 2016, 354, 1414-1419. (21) Hoenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864-B871.

20


Page 20 of 35

Page 21 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(22)

Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation

Effects. Phys. Rev. 1965, 140, A1133-A1138. (23) Stamenkovic, V.; Mun, B. S.; Mayrhofer, K. J. J.; Ross, P. N.; Markovic, N. M.; Rossmeisl, J.; Greeley, J.; Norskov, J. K. Changing the Activity of Electrocatalysts for Oxygen Reduction by Tuning the Surface Electronic Structure. Angew. Chem. Int. Edit. 2006, 45, 2897-2901. (24) Su, H.-Y.; Sun, K.; Wang, W.-Q.; Zeng, Z.; Calle-Vallejo, F.; Li, W.-X. Establishing and Understanding Adsorption–Energy Scaling Relations with Negative Slopes. J. Phys. Chem. Lett. 2016, 7, 5302-5306. (25) Calle-Vallejo, F.; Martinez, J. I.; Garcia-Lastra, J. M.; Sautet, P.; Loffreda, D. Fast Prediction of Adsorption Properties for Platinum Nanocatalysts with Generalized Coordination Numbers. Angew. Chem. Int. Edit. 2014, 53, 8316-8319. (26) Calle-Vallejo, F.; Tymoczko, J.; Colic, V.; Vu, Q. H.; Pohl, M. D.; Morgenstern, K.; Loffreda, D.; Sautet, P.; Schuhmann, W.; Bandarenka, A. S. Finding Optimal Surface Sites on Heterogeneous Catalysts by Counting Nearest Neighbors. Science 2015, 350, 185-189. (27) Calle-Vallejo, F.; Loffreda, D.; Koper, M. T. M.; Sautet, P. Introducing Structural Sensitivity into Adsorption-Energy Scaling Relations by Means of Coordination Numbers. Nat. Chem. 2015, 7, 403-410. (28) Ma, X. F.; Li, Z.; Achenie, L. E. K.; Xin, H. L. Machine-Learning-Augmented Chemisorption Model for CO2 Electroreduction Catalyst Screening. J. Phys. Chem. Lett. 2015, 6, 3528-3533. (29) Ma, X. F.; Xin, H. L. Orbitalwise Coordination Number for Predicting Adsorption Properties of Metal Nanocatalysts. Phys. Rev. Lett. 2017, 118. (30) Gasper, R.; Shi, H. B.; Ramasubramaniam, A. Adsorption of CO on Low-Energy, Low-Symmetry Pt Nanoparticles: Energy Decomposition Analysis and Prediction via Machine-Learning Models. J. Phys. Chem. C 2017, 121, 5612-5619. (31) Jinnouchi, R.; Asahi, R. Predicting Catalytic Activity of Nanoparticles by a DFT-Aided Machine-Learning Algorithm. J. Phys. Chem. Lett. 2017, 8, 4279-4283. (32) Bartok, A. P.; Kondor, R.; Csanyi, G. On Representing Chemical Environments. Phys. Rev. B 2013, 87, 16. (33) De, S.; Bartok, A. P.; Csanyi, G.; Ceriotti, M. Comparing Molecules and Solids across Structural and Alchemical Space. Phys. Chem. Chem. Phys. 2016, 18, 13754-13769. (34) Bishop, C. Pattern Recognition and Machine Learning (Information Science and Statistics); Springer, 2006. (35) Kresse, G.; Furthmuller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phy. Rev. B 1996, 54, 11169-11186. (36) Kresse, G.; Furthmuller, J. Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15-50. (37) Blochl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953-17979.

21



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(38)

Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave

Method. Phys. Rev. B 1999, 59, 1758-1775. (39) Hammer, B.; Hansen, L. B.; Norskov, J. K. Improved Adsorption Energetics within Density-Functional Theory Using Revised Perdew-Burke-Ernzerhof Functionals. Phys. Rev. B 1999, 59, 7413-7421. (40)

Fu, C. L.; Ho, K. M. 1st-Principles Calculation of the Equilibrium Ground-State Properties

of Transition-Metals - Applications to Nb and Mo. Phys. Rev. B 1983, 28, 5480-5486. (41) Falsig, H.; Bligaard, T.; Rass-Hansen, J.; Kustov, A. L.; Christensen, C. H.; Norskov, J. K. Trends in Catalytic NO Decomposition over Transition Metal Surfaces. Top. Catal. 2007, 45, 117-120. (42)

Falsig, H.; Bligaard, T.; Christensen, C. H.; Norskov, J. K. Direct NO Decomposition over

Stepped Transition-Metal Surfaces. Pure Appl. Chem. 2007, 79, 1895-1903. (43) Bligaard, T.; Norskov, J. K.; Dahl, S.; Matthiesen, J.; Christensen, C. H.; Sehested, J. The Bronsted-Evans-Polanyi Relation and the Volcano Curve in Heterogeneous Catalysis. J. Catal. 2004, 224, 206-217. (44) Norskov, J. K.; Rossmeisl, J.; Logadottir, A.; Lindqvist, L.; Kitchin, J. R.; Bligaard, T.; Jonsson, H. Origin of the Overpotential for Oxygen Reduction at a Fuel-Cell Cathode. J. Phys. Chem. B 2004, 108, 17886-17892. (45) Falsig, H.; Shen, J.; Khan, T. S.; Guo, W.; Jones, G.; Dahl, S.; Bligaard, T. On the Structure Sensitivity of Direct NO Decomposition over Low-Index Transition Metal Facets. Top. Catal. 2014, 57, 80-88. (46) Hirata, H. Recent Research Progress in Automotive Exhaust Gas Purification Catalyst. Catal. Surv. Asia 2014, 18, 128-133. (47) Kondratenko, E. V.; Takahashi, N.; Nagata, N.; Ibe, M.; Hirata, H.; Takahashi, H. Operando Uv/Vis Analysis of the Synergy Effect between Copper and Gold in Nitric Oxide Reduction over Gold and Copper on Alumina Catalysts. ChemCatChem 2015, 7, 3956-3962. (48) Liu, L. C.; Guan, X.; Li, Z. M.; Zi, X. H.; Dai, H. X.; He, H. Supported Bimetallic AuRh/Gamma-Al2O3 Nanocatalyst for the Selective Catalytic Reduction of NO by Propylene. Appl. Catal. B-Environ. 2009, 90, 1-9. (49) van Hardeveld, R.; van Montfoort, A. The Influence of Crystallite Size on the Adsorption of Molecular Nitrogen on Nickel, Palladium and Platinum. Surf. Sci. 1966, 4, 396-430. (50) Honkala, K.; Hellman, A.; Remediakis, I. N.; Logadottir, A.; Carlsson, A.; Dahl, S.; Christensen, C. H.; Norskov, J. K. Ammonia Synthesis from First-Principles Calculations. Science 2005, 307, 555-558. (51) Spencer, N. O.; Schoonmaker, R. C.; R. C. Somorjai, G. A. Iron Single Crystals as Ammonia Synthesis Catalysts: Effect of Surface Structure on Catalyst Activity. J. Catal. 1982, 74, 129-135.

Figure captions 22


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Figure 1 Schematic of the Bayesian linear regression scheme for predicting energetics of reaction intermediates of catalytic reactions on nanoparticles when using stored DFT energetic data on single crystal surfaces and small clusters. Figure 2 Mean absolute errors σO of the binding energies of O on single crystal Rh(1−x)Aux alloy surfaces as functions of the number of sets of training data Ndata (A), and the minimum mean absolute error σO,min as a function of the cutoff radius Rcut (B). Figure 3 Truncated octahedral Rh(1−x)Aux alloy nanoparticle models used in DFT and machine-learning calculations. Figure 4 Mean absolute errors of the binding energies of (A) N, (B) O, and (C) NO with the Rh(1−x)Aux alloy single crystal and particles predicted when using DFT data on single crystal surfaces as training data and as functions of the number of sets of training data that equals the number of basis sets used for the regression. The bar graph (D) shows the mean absolute errors in the binding energies predicted when using the maximum amount of training data shown in Figures (A)–(C). Figure 5 Mean absolute errors of the binding energies of N, O, and NO with the Rh(1−x)Aux alloy single crystal and particles predicted when using DFT data on small clusters as training data. The figures are similar to those in Fig. 3. Figure 6 Mean absolute errors of the binding energies of N, O, and NO with the Rh(1−x)Aux alloy single crystal and particles predicted when using DFT data on single crystal surfaces and small clusters as training data. The figures are similar to those in Fig. 3. 23



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Figure 7 Mean absolute errors of the binding energies of O with the Rh(1−x)Aux alloy single crystal and particles predicted when using DFT data on single crystal surfaces and/or small clusters as training data. The results in (A) and (B) were obtained using the cutoff radii Rcut = 0.6 and 0.8 nm, respectively. Figure 8(A) Predicted TOFs of the direct decomposition of NO on Rh(1−x)Aux alloy nanoparticles as functions of the Au atomic ratio x, (B) the Au atomic ratio (xmax) and that on the surface (xmaxsurf) yielding the maximum TOF (TOFmax) as functions of the particle diameter d, and (C) TOFmax as a function of d. Figure 9 Distributions of atoms on the surfaces and activation free energy ∆Gs* of the direct decomposition of NO on the Rh(1−x)Aux alloy nanoparticles with diameters of 1.5 and 3.0 nm. Figure 10(A) Free energy diagrams of the direct decomposition of NO at the alloyed corner sites of the Rh(1−x)Aux alloy nanoparticles with diameters of 1.5 and 2.0 nm, and (B and C) stable adsorption sites for nitrogen and oxygen atoms formed by the NO dissociations near the alloyed corner sites of the Rh(1−x)Aux alloy nanoparticles. The values shown in (B and C) indicate the binding energies of oxygen atoms with the catalytic surfaces. The unit of the binding energies is eV.

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Figure 1

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1

σO (eV)

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Rcut=0.35 nm Rcut=0.40 nm Rcut=0.50 nm Rcut=0.60 nm

0.01 1

10

100

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Ndata 0.3 (B)

σO.min min (eV)

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Figure 2

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Figure 3

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Figure 4

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Figure 6

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Extrapolating Energetics on Clusters and Single-Crystal Surfaces to

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