Machine Learning Prediction of H Adsorption Energies on Ag Alloys

Subscriber access provided by TULANE UNIVERSITY

Computational Chemistry

Machine Learning Prediction of H Adsorption Energies on Ag Alloys Robert A. Hoyt, Matthew M. Montemore, Ioanna Fampiou, Wei Chen, Georgios Tritsaris, and Efthimios Kaxiras J. Chem. Inf. Model., Just Accepted Manuscript • DOI: 10.1021/acs.jcim.8b00657 • Publication Date (Web): 21 Mar 2019 Downloaded from http://pubs.acs.org on March 24, 2019

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 26 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

Journal of Chemical Information and Modeling

Machine Learning Prediction of H Adsorption Energies on Ag Alloys Robert A. Hoyt,†,k Matthew M. Montemore,‡,¶,k Ioanna Fampiou,‡ Wei Chen,†,‡ Georgios Tritsaris,§,¶ and Efthimios Kaxiras∗,†,¶ Department of Physics, Harvard University, Cambridge, MA 02138, USA, Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA, John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA, and Institute for Applied Computational Science, Harvard University, Cambridge, MA 02138, USA E-mail: [email protected]

Abstract Adsorption energies on surfaces are excellent descriptors of their chemical properties, including their catalytic performance.

High-throughput adsorption

energy predictions can therefore help accelerate first-principles catalyst design. To this end, we present over 5,000 DFT calculations of H adsorption energies on dilute Ag alloys, and describe a general machine learning approach to rapidly predict H adsorption energies for new Ag alloy structures. We find that random forests provide accurate predictions, and that the best features are combinations of traditional chemical and structural descriptors. Further analysis of our model errors ∗

To whom correspondence should be addressed Department of Physics, Harvard University, Cambridge, MA 02138, USA ‡ Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA ¶ John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA § Institute for Applied Computational Science, Harvard University, Cambridge, MA 02138, USA k Authors contributed equally. †

1

ACS Paragon Plus Environment

Journal of Chemical Information and Modeling 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

Page 2 of 26

and the underlying forest kernel reveals unexpected finite-size electronic structure effects: embedded dopant atoms can display counterintuitive behavior such as nonmonotonic trends as a function of composition and high sensitivity to dopants far from the adsorbing H atom. We explain these behaviors with simple tight-binding Hamiltonians and d-orbital densities of states. We also use variations among forest leaves to predict the uncertainty of predictions, which allows us to mitigate the effects of larger errors.

Introduction Many important chemical processes involve hydrogenation, including biodiesel production, petrochemical conversion, fine chemical synthesis, and food processing. 1–3 For example, the selective hydrogenation of acetylene to ethylene is a necessary step to purify feedstocks for ethylene polymerization. Therefore, understanding and designing catalysts for hydrogenation reactions has been the subject of intensive research.

Adsorption

energies of key intermediates have long proved useful in rationalizing trends in catalytic performance. 4 Hydrogen itself is an important intermediate in hydrogenation reactions, and its adsorption energy can correlate with that of other intermediates, such as hydrocarbons. 5 Thus, to first approximation, the hydrogen adsorption energy can be used to screen surfaces for their catalytic performance towards hydrogenation reactions. 4,6–9 After initial screening, more detailed studies of other important intermediates for a particular reaction can be performed prior to experimental synthesis, characterization, and testing. However, accurate adsorption energy calculations generally require computationally intensive quantum chemistry methods, significantly limiting the diversity and number of candidate surfaces that can be screened.

Several approaches have been tested

for predicting adsorption energies with reduced computational effort. Some of these approaches are physically motivated, such as estimating the electronic structure from model tight-binding Hamiltonians. 10 Other approaches are data driven, 11,12 culminating in the application of modern machine-learning techniques. 13–17 To apply more powerful 2


Page 3 of 26 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


machine-learning methods and ensure the generality of the approach, larger data sets are needed. In this work we focus on a system where screening requires a very large number of calculations: H adsorption energies on stepped Ag(211) alloy surfaces. Ag surfaces generally have the weakest H adsorption energies among the common transition-metal catalysts, 18 leading to high selectivity in hydrogenation reactions 1 at the expense of catalytic throughput. To improve the reactivity of noble metal surfaces, recent work has demonstrated that even a single metal atom substitution can significantly improve activity 19,20 while retaining the host metal’s selectivity. 21 Dilute Ag alloys could therefore improve Ag’s catalytic prospects, but there is a combinatorially large number of possible alloy compositions and atomic arrangements at the surface. To enable high-throughput screening for dilute Ag alloys, we pursue machine learning models that can quickly and accurately predict H adsorption energies across a wide range of Ag alloy compositions and arrangements. We use the Ag(211) model surface both to test how well models can handle some structural complexity, and because step sites are often more catalytically active than terrace sites. 22,23 Most previous efforts at efficient prediction of adsorption energies focus on terraces, which are simpler surfaces with higher symmetry. However, it is not clear whether these models generalize to step sites. Step sites are often of greater interest for practical catalysts, which are usually nanostructured. 24 Additionally, by studying adsorption on a step site, we gain insight into how substitution at the step edge differs from substitution next to the step or in the subsurface. In this work we specifically focus on predicting H adsorption energies at the hcp hollow site adjacent to the step edge, including the bridge and top sites adjacent to this hcp hollow. We begin by mining the data for insights into catalytic behavior and constructing a very simple model as a baseline. Next, we prepare a smaller set of simple features and use them to compare the relative performance of different combinations of machine learning models and feature scaling strategies. Applying the best-performing model to the dataset and inspecting alloys with significant prediction errors provides insight into catalytic

3



properties. Finally, we apply the most effective model to a more sophisticated feature set to better understand the limits of general-purpose models. This process culminates with a high accuracy model (median absolute error of 0.34 kcal/mol), an understanding of the key properties of successful models, and the prediction of unexpected physical phenomena in dilute Ag alloy catalysts.

Data and Models

Figure 1: (a) Perspective view of all relaxed H atom positions (red dots) relative to unrelaxed Ag(211) positions (silver spheres). Lines indicate the rows of atoms above and below the step edge. (b) Triangular scatter plot of all 5457 adsorption energies; d2 /d1 and d3 /d1 are ratios of the second and third-shortest H–metal bond lengths to the shortest H–metal bond length in each relaxed structure, such that vertices correspond to H adsorption at top, bridge, and hcp sites as labeled. Each point is colored by its adsorption energy (in eV) relative to the pure Ag(211) surface. Positive (negative) values mean weaker (stronger) adsorption.

Bimetallic alloys were formed beginning with a slab structure that represents the Ag(211) host surface shown in Figures 1 and 2, with a four-atom-wide step edge and four layers in the (111) direction.

Up to eight host Ag atoms were replaced by a

single transition metal element from the following list: Ti, Zr, Hf, V, Ta, Cr, Mo, W, Mn, Re, Fe, Ru, Co, Rh, Ir, Ni, Pd, Pt, Cu, Au, Zn, Cd. For adsorption energy calculations, a single H atom was initially placed at the hcp hollow site adjacent to the step edge, its minimum-energy geometry on the pure Ag(211) lattice. Figure 1(a) shows 4


Page 4 of 26

Page 5 of 26 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


(a)

(b)

Figure 2: Views of the Ag(211) lattice from (a) the side and (b) from above. Dark spheres indicate sites in the bottom layer whose positions are fixed and occupied by Ag atoms. Gray and white spheres indicate candidate sites for one and two substitutions, and white spheres indicate the smaller set of candidate sites for three or more substitutions. Dashed lines illustrate the location of the step edge, and the red sphere indicates the initial position of the H atom before relaxation. the distribution of relaxed positions relative to the step edge, and Figure 1(b) shows the distribution of relaxed adsorption sites. The bottom layer (dark spheres in Figure 2) of the slab representing the surface is always composed of Ag to simulate the bulk, and the remaining 32 atomic sites within the slab (gray and white spheres in Figure 2) are candidates for replacement. Alloy structures were generated in three groups. First, for a single substitution, we exhaustively calculated adsorption energies for all symmetrically inequivalent arrangements for each element. Second, for two substitutions, we sampled approximately half of the symmetrically inequivalent arrangements at random for each element. Third, for three through eight substitutions, we sampled approximately half of the symmetrically inequivalent arrangements where all substitutions are made among the six closest sites to the H atom, as well as the two sites immediately below the step edge (white spheres in Figure 2). This restriction to eight total candidate sites mitigates the combinatorial increase in the number of inequivalent arrangements with the number of substitutions. The H adsorption energy for each alloy surface was obtained using density functional theory (DFT) calculations with the VASP code 25,26 (see SI for details).

Geometry

optimization for each alloy structure began with the metal and H atoms placed in the corresponding minimum-energy positions of the pure Ag(211) configuration shown in

5



Figure 2. The corresponding relaxed positions for the H atom are most often near the center of the hcp hollow, but a significant number of cases relaxed to off-center, to a bridge site, or to a top site (see Figure 1(b)). All regression models are implemented in scikit-learn, 27 an open-source package supporting a wide variety of general-purpose machine learning methods. We focus on linear and quadratic polynomials, multilayer neural networks, kernel ridge regression, random forests, 28 and extra forests. 29 When applicable, we also consider ridge and LASSO regularization to adjust the balance between model accuracy and generality. We construct and test models using four-fold splits, optimizing models exclusively using the training fold (75% of the data) and reserving the testing fold (25% of the data) to evaluate prediction errors. For each model we also consider several preconditioning methods: standardization (zero mean, unit variance), principal component analysis (PCA), and quantile transformation. These transformations are determined using only the training fold.

Results and Discussion Data Mining We begin by examining the database to understand how various positions of the substitutional foreign atoms affect H adsorption. For the structures with a single foreign atom, we calculated the standard deviation of adsorption energies for substitutions within the first, second, and third nearest-neighbor shells around the H atom, as well as more distant locations. Standard deviations decline quickly with distance: 0.30 eV among first nearest-neighbors and 0.11 eV among second nearest-neighbors, but only 0.04 eV among third nearest-neighbors or more distant atoms (see inset of Figure 3). Therefore, a single substitution significantly affects adsorption energies only if it lies within the first and second nearest-neighbor shells, a total of four symmetrically inequivalent sites. The introduction of foreign atoms (henceforth called “dopants” by analogy to semiconductor doping) results in qualitatively different effects depending on whether they 6


Page 6 of 26

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


are in the first or the second nearest-neighbor shell of the H atom. Most dopants have higher d-orbital energies and larger atomic radii than Ag and therefore tend to enhance H adsorption as first nearest-neighbors as shown in Figure 3, and noted in previous studies. 5,18,30 However, these dopants have the opposite effect when located within the second nearest-neighbor shell and weaken hydrogen adsorption. For elements with lower d-orbital energies, specifically Cd and Zn, these effects are reversed. Therefore, elements that bind strongly to H also passivate nearby Ag atoms towards H adsorption. The strong variation of adsorption energies across the d block and lattice site indicate that both the chemical identity and physical locations of dopant atoms are important features. hcp below

Standard Deviation (eV)

Page 7 of 26

0.3

0.2

0.1

0.0

1

2

3

4 (NN)

Figure 3: H Adsorption energies for selected elements at each of the four symmetrically inequivalent positions within the first nearest-neighbor (1NN) and second nearestneighbor (2NN) shells. The inset shows the standard deviations of H adsorption energies for a single dopant in the first, second, and third nearest-neighbor shells, as well as at more distant sites (≥ 4). We also examined the formation energies of the alloy surfaces, relative to the clean Ag(211) surface and the lowest-energy bulk phase of each dopant element. Our results indicate that alloy surfaces for most elements are thermodynamically unstable, with positive formation energies independent of the number of dopants (see Figure S1). These unstable elements prefer to lie as isolated atoms in the second or third layers. Of the dopants we studied, Zn, Cd, Au, Pd, and Pt form thermodynamically stable alloys, with preference for sites in the surface layer. Under mild reaction conditions, metastable alloys may remain kinetically trapped in their configuration. Therefore, we also searched for arrangements with metastable formation energies between 0 and 1 eV. We find that Ti, 7



Zr, Hf, Rh, Ni, and Cu can exist in metastable configurations for both the bare surface and in the presence of a single H atom. Adsorption of a single H atom is insufficient to qualitatively change formation energies: no formation energy changes from positive to negative or from being larger than 1 eV to being smaller that 1 eV, although higher H coverage or other reaction intermediates could produce more dramatic changes. Despite any instability, we consider all calculations in our modeling to demonstrate the ability to make high-throughput predictions across the full range of possible alloy surfaces.

Initial Features and Model To apply a machine learning model, we must first extract features from each surface in our database for use as the inputs to the model. Each atom in the Ag(211) structure is a distinct site that can be occupied by either Ag or one of the 22 transition metal dopants. Therefore, it is natural to treat the system as a static lattice. Considering the 31 lattice sites within 8 ˚ A of the H atom, we can then encode the entire structure using a binary matrix B ∈ {0, 1}31×23 , such that BiJ = 1 if lattice site i is occupied by an atom of element J. To account for the reflection symmetry present in our system, here and in the following we canonically choose the reflection yielding the highest value when this binary coding is interpreted as a binary number. With this large number of features (31 × 23 = 713) we apply LASSO-regularized linear regression. Regularization applies a penalty to large coefficients and therefore results in simpler models that are less likely to be overfitting. The root-mean-square error (RMSE) of this model on test data is 147 meV (see Figure 4). Despite not supplying any descriptors of the elements that account for their chemical properties, the linear model captures a significant amount of the variance in H adsorption, with an R2 statistic of 0.82. However, there are numerous outliers with test errors exceeding 1 eV. Therefore, very simple models can provide a surprisingly reasonable estimate for most H adsorption energies, but improvements to the features and model are needed for precise estimates. Prediction outliers of this simple model also provide insight into which data points may be unphysical. The geometries corresponding to most major outliers reveal two major 8


Page 8 of 26

Page 9 of 26 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


Figure 4: (a) Parity plot comparing simple linear model predictions to DFT calculations of H adsorption energies. The dashed line indicates 1:1 parity, and data points corresponding to surface reconstruction and H atom migration are color-coded red and blue, respectively. Four additional outliers due to reconstruction, with DFT adsorption energies of −2.83, −1.98, 1.07, and 2.03 eV are not shown. (b) Log-scale stacked error histogram with the same color coding as in plot (a). (c) Parity plot and (d) error histogram for the optimal extra forests model on the non-outlier data points, which is 97.5% of the total data. Solid bars in the histogram show the error distribution of the forest model, while the dashed outline shows the corresponding error distribution for the original model of 713 one-hot bits on the same data.

9



sources of error. The first, and most severe, is related to reconstruction, where the surface geometry changes significantly from the input structure.

In particular, “differential

reconstruction,” where the surface reconstructs differently for the bare slab compared to the H/slab systems, often results in high test errors. A second class of outliers consists of cases where the H atom relaxes by moving to a different region of the slab, beyond the top and bridge sites adjacent to the initial configuration (see SI for details). These outlier classes are color-coded in Figure 4. As indicated in the log-scale histogram in Figure 4(b), these two classes of points comprise a small fraction (2.5%) of the total data but dominate the total error. This is consistent with the lower test errors we obtained in the absence of relaxation, by predicting “adsorption energies” for the alloy surfaces in their initial bare slab and H/slab geometries.

Simple Features and Model Comparisons To determine which models perform best for this task, we compared LASSO and ridge-regularized linear and polynomial regression, ridge-regularized multilayer neural networks and radial basis function (RBF) kernel regression, random forests, 28 and extra forests. 29 Although more sophisticated variants of these models have been considered previously, 31–34 these general-purpose models are widely supported and thus especially well-suited for rapid catalyst screening. In addition, to reduce overfitting and computational effort, we also introduce a more parsimonious feature set for these comparisons, including both “arrangement” and “chemical” features that describe the locations and chemical identity of dopants, respectively. For structural features, we use a one-hot vector of length 31 indicating which sites are substituted: bi = 1 if lattice site i is occupied by a dopant. We then include the number of first, second, and third-nearest neighbors relative to the H atom, as well as the number of dopants farther away than third-nearest neighbor. This leads to 35 total arrangement features. These features are partially redundant, but linear combinations of features can yield more sparse and accurate models. 28 For chemical features, we use commonly available atomic properties to account for 10


Page 10 of 26

Page 11 of 26 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


the chemical identity of the dopant. Based on insights from previous studies, 10,30,35 we include each dopant’s tight-binding d-orbital radius and energy 36 and its column in the d block, equivalent to the number of valence d electrons. We also include the dopant’s row in the d block, Pauling electronegativity, covalent radius, mass density, enthalpy of fusion, ionization energy, bulk modulus, Poisson ratio, and conductivity, as was also done in previous work. 13,14 Combining these with the structural features yields a total of 47 features. Our tests suggest that the accuracy of the nonlinear models is insensitive to the precise feature set, as long as there are enough features that the model can effectively distinguish between different elements. The average test RMSE values and corresponding standard errors, across many randomized four-fold splits, for each model using these features is shown in Table 1 for common feature scaling strategies. Standardization shifts each feature to have zero mean and unit variance, and PCA projects these standardized features onto the corresponding principal components. These are linear transformations. In contrast, quantile scaling is a nonlinear transformation that maps each feature value to the range (0, 1) according to its percentile rank. Linear and quadratic models are generally out-performed by the other four models with lower bias for most scaling choices. Random and extra forest models consistently offer the best performance on test sets with mean RMSE values of approximately 99–100 meV. Table 1: Mean RMSE Values for Combinations of Models and Feature Scaling Strategies Regularization Model Standardized PCA LASSO linear 212 ± 6 212 ± 7 LASSO quadratic 133 ± 6 135 ± 3 Ridge neural network 106 ± 3 105 ± 3 Ridge kernel ridge 108 ± 5 109 ± 4 n/a random forest 100 ± 5 124 ± 8 n/a extra forest 100 ± 4 134 ± 4 All values are in units of meV, and the lowest RMSE value for bold font.

Quantile Unscaled 202 ± 3 212 ± 2 118 ± 8 134 ± 4 106 ± 3 fail 108 ± 3 114 ± 3 99 ± 4 99 ± 5 100 ± 4 100 ± 5 each model is shown in

Forests are particularly successful due to their if-else decisions, which handle categorical features in a natural way. In particular, forests are ideally suited for the one-hot vectors we use to encode the structure. The benefits of if-else decisions also 11



explain why PCA worsens forest performance, where the rotation in features tends to mix the one-hot bits with continuous features. We find that random and extra forests offer nearly identical performance, but extra forests are less sensitive to changes in their hyperparameters and are faster to train. Therefore we focus on extra forests in the subsequent discussion. Extra forests perform well across the entire range of adsorption energies, with an improved testing R2 score of 0.92 and a median absolute test error of only 18 meV. We find that extra forests, along with all other models, still have high test errors associated with differential reconstruction and H migration. In the following we exclude these points to focus on adsorption at the hcp site of interest and find outliers that are not due to significant relaxation. Differential reconstruction yields calculated adsorption energies that do not represent the chemical reactivity of the surface since two qualitatively different surfaces are included in the subtraction. For example, some asymmetric dopant arrangements induce the formation of (111) step facets instead of the (100) facets originally present in the (211) surface. Similarly, for significant H atom relaxation, test errors are high because the model is designed and trained to predict adsorption energies at the site of interest. By limiting the model to a single site (the hcp hollow and its adjacent bridge and top sites), we essentially assume that the catalytic properties of each site on the surface is independent, which is a common assumption in computational catalysis. Retraining extra forests on the remaining 97.5% of the data set yields a significantly lower RMSE of 56 meV.

Physical Insights We can gain insight into the model and into the underlying behavior of the system by examining cases where the model performs poorly. Test errors are highest for elements toward the left-hand side of the d block, where the two elements V and Mo have the highest RMSE values of 112 meV and 109 meV, respectively. These large errors are caused by similar configurations having significantly different adsorption energies. We quantified similarity by computing the forest kernel k(x, x0 ) which gives the correlation 12


Page 12 of 26

Page 13 of 26 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


between any two data points described by x and x0 according to the trained forest model. Forest predictions are simply k-weighted averages over all the training data (see SI for details), so training data points xi with the largest k(xi , x) values contribute the most to a prediction at x. As an example, we use the forest kernel to better understand a large outlier, the V4 arrangement in Figure 5 whose adsorption energy is significantly overestimated. Inspection of the three most similar arrangements and their corresponding H adsorption energies, shown in Figure 5, reveals the source of the error: these most similar structures all have significantly stronger H adsorption. All four arrangements have the same nearest neighbor shell around H, and similar second-nearest neighbor shells. Analyzing the density of states reveals that stronger adsorption is associated with higher d-orbital densities of states just below the Fermi level; these increases are highlighted by the green regions in Figure 5. These additional states lead to stronger H adsorption energies, consistent with previous studies of O on Pt alloys. 37 For example, the last arrangement (2)

in Figure 5 (V5 ) yields the highest increase in the density of states near the Fermi level and has the strongest H adsorption energy among these four structures.

23%

17%

9%

Figure 5: The projected densities of states on d-orbitals of V atoms for selected V/Ag(211) arrangements. Lighter atoms are in the surface layer and darker atoms are in the (1) (2) subsurface layers. V4 is a configuration with high test error, and V3 , V5 , and V5 are the three most similar arrangements according to the forest kernel k (see text). Each structure is labeled with the corresponding H adsorption energy and k value. Green shaded regions indicate excess densities of states relative to the green reference curve for V4 . Similarly, further analysis of the errors reveals cases where trends in adsorption 13



Page 14 of 26

energies are counterintuitive. In particular, two-dopant structures where both dopants lie in the first nearest-neighbor shell of the H atom have an RMSE that is twice as high as the data set as a whole, and 34% higher than all cases with at least one dopant atom in this shell. Figure 6 compares adsorption energies for dimers and isolated dopant atoms along the step edge for all dopant elements considered here. One might expect that dimers of dopant atoms would yield roughly double the effect of a single dopant atom–in most cases, strengthening adsorption. Adsorption energies for dimers of Ti, Zr, and Hf at the step edge are indeed much stronger than the corresponding isolated atom. The same is true for most elements on the right-hand side of the d block such as Co, Pd, Cu, and Zn. However, adsorption energies for dimers are significantly weaker, compared to an isolated atom, for most elements from the center of the d block, especially the third and fourth columns. The trend in H adsorption energies for dimers and isolated atoms therefore changes dramatically across the d block. In contrast, previous work has generally indicated that adsorption energies change monotonically with the composition of the nearest neighbor shell. 23,38,39 ‒

‒

isolated

dimer

‒

2

3

4

5

6

7

8

9

10

Figure 6: Comparison of H adsorption energies for isolated atoms (black bars) and dimers (green bars) at the step edge for each dopant element. Elements are grouped by column across the d block. Close inspection of the densities of states rationalizes this counterintuitive behavior. For the dimers, dopant-dopant hybridization is stronger than Ag-dopant hybridization, resulting in clear d–d bonding and antibonding peaks in most cases (see Figure S2). Previous work has shown clearly that high densities of states at or just below the Fermi energy leads to strong adsorption. 40 Dimers of elements on the left side of the d block have d–d antibonding peaks near the Fermi energy, allowing strong adsorption. For dimers of 14


Page 15 of 26 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


elements in the middle of the d block, rising d-orbital energies push the antibonding peak above the Fermi energy where it no longer contributes significantly to adsorption, while the d–d bonding peak is still much lower than the Fermi energy. In these cases, the dopant-dopant hybridization results in weaker H adsorption. For elements on the right side of the d block, the d–d bonding peak lies near the Fermi energy, again allowing strong H adsorption. The V clusters shown in Figure 5, and dimers shown in Figure 6, both illustrate how adsorption energies can be very sensitive to the precise number and location of dopant atoms. Inspecting the d-orbital densities of states and considering tight-binding Hamiltonians reveals that these changes are driven by finite size effects. Ag atoms have short d-orbital radii and low d-orbital energies, while most dopant elements have significantly higher d-orbital energies.

This difference in d-orbital energies leads to

small d-orbital hybridization between Ag atoms and most dopants. 41,42 The resulting d-orbital density of states contributed by dopant elements is therefore similar to that of an isolated dopant nanocluster whose atomic positions are restricted to the available sites in the Ag(211) surface. Similar to isolated nanoclusters, 43 the electronic states of these “embedded nanoclusters” exhibit strong d-orbital finite-size effects. The d-orbital density of states near the Fermi level is almost entirely contributed by the dopant elements since the d-orbital energy of Ag is several eV lower. These d-orbital states near the Fermi level largely dictate H adsorption energies across the d block, explaining how changes far from the H atom can yield very different H adsorption energies. These finite size effects also explain how the presence or absence of a third nearest-neighbor substitution can strongly affect H adsorption energies in multi-dopant configurations, by changing the whole cluster’s d orbital energies, while an isolated third nearest-neighbor substitution has almost no effect.

Final Model and Feature Selection In light of the observations described above, we considered a wide variety of surface features computed from unrelaxed input geometries to improve forest predictions. We 15



summarize these features here and leave additional details to the SI. Because of the sensitivity of the adsorption energy to particular arrangements of dopants, we added one-hot bits indicating the presence of dimer and trimer arrangements that may be important. We also added additional chemical features, including the formation energy of a single dopant element in the bulk-like third layer of the Ag(211) surface and H adsorption energies for isolated dopant atoms and dimers at the step edge. Since the combination of d-orbital properties and specific arrangements is important, we also included column-wise sums of the Coulomb matrix 44 using the column in the d block to which the dopant belongs as its effective charge. Finally, to account for detailed changes in orbital hybridization, we included the eigenvalues of a simplified tight-binding (TB) Hamiltonian of the Ag(211) surface and the first five moments of the tight-binding eigenvalues, weighted by their corresponding eigenvectors’ projections onto the three metal atoms closest to H. In total these contribute an additional 187 features to the original 47 features. We also tested an alternative encoding of the structure, where a radial distribution function centered at the H atom was calculated for each structure, and a weighted difference between these functions was used to define a kernel. This resulted in marginally smaller errors (by a few meV) than the one-hot vector for kernel regression, and this scheme is very difficult to implement into a forest, which is a more accurate model for this dataset. For this reason, we continued to use the one-hot vector to encode the structure. Predicting H adsorption energies using all 234 features leads to overfitting. Accordingly, we use forward selection to simplify the model as an additional form of regularization. This process was performed independently with the same four-fold crossvalidation used above, optimizing the models with training data and reserving test data for a final error estimate. Forward selection heuristically grows a selected subset F of the most important features. At each step a new forest is trained on the union of F and each one of the remaining 234 − |F | features, and then the feature yielding the lowest out-ofbag RMSE is added to F . This process was performed 40 times to obtain |F | = 40 of the most important features. Following this procedure, the training data in each fold was

16


Page 16 of 26

Page 17 of 26 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


used to optimize how many of these top 40 features to include in the final model. Forward selection performs significantly better than feature importances computed directly from the forest since the latter are more sensitive to correlations among features. The final test error was 52 meV across all four folds. To rank features by overall importance, we note that forward selection effectively ranks each feature by selecting the most beneficial feature first, followed by the second, and so on. Therefore we rank features overall according to the sum of their ranks across all four folds. Table 2 presents the overall rank of the 12 features selected by all four folds for the optimal model. Many of these features encode our previous insights into surface electronic structure and trends in H adsorption energies. For example, the number of nearest-neighbor dopants is a critical parameter and is the top overall feature. Similarly, column sums of the Coulomb matrix combine information about the spatial arrangement of dopant atoms near the H atom with their chemical identities. Likewise, the weighted second moment of tightbinding eigenvalues also depends sensitively on the atomic environment near the H atom. This second moment can also be interpreted as a measurement of the d-band width when projected onto the H atom. Compared to the previous set of 47 features, these new features modestly reduce the test RMSE with significantly fewer features: each fold selected between 20 and 33 features. Table 2: Top Features from Forward Selection Rank 1 2 3–4 5–6 7 8 9 10 11 12

Feature Number of 1st NN dopants Second moment of eigenvalues weighted by the NN step atom Coulomb matrix column sums for the NN step atom and the atom under the hcp hollow site occupied by the H atom, respectively Shortest unrelaxed dopant–dopant and H–dopant bond length, respectively Formation energy of a single subsurface substitution Dopant’s column in d block Number of 3rd NN dopants One-hot bit for the closest atom below the step edge Standard deviation of dopant positions One-hot bit for the atom under the farthest hcp hollow site adjacent to step edge from the H atom

The additional decision-making flexibility also leads to an improvement in model

17



accuracy. Figure 4 shows the corresponding parity plot and error histogram in comparison to those of the original model. Compared to the initial linear model with 713 one-hot bits, in the optimal model, test errors on the 97.5% of non-outlier points are significantly lower across the full range of H adsorption energies. In addition, fewer features can be included with only a small penalty to the RMSE. For example, retaining only half of each fold’s optimal number of features (10–16) results in a test RMSE value of 54 meV, just below the RMSE of 56 meV of the extra forest model with the same data and the previous set of 47 simple features. This corresponds to an approximately three-fold reduction in the number of features while preserving the accuracy of the original forest; hence, for future studies this feature set could reduce the amount of data needed for training. Even fewer features can be used with similar results as well; retaining a third of each fold’s optimal number of features (7–11) results in a test RMSE value of 58 meV. So far we have discussed model accuracy, but quantifying model uncertainty is also important since prediction accuracy can vary across the data set. Previous work has considered sophisticated metrics of forest confidence intervals, 45 but a simple and readily available metric is the “forest variance”: this is defined as the variance of predictions made by trees in the forest. Forest predictions are means of its trees’ predictions, so forest variances can be obtained easily alongside predictions. We define to be the prediction error of a particular point and vf to be the forest variance; the forest z-score √ is defined as zf = / vf . The z-scores here are calculated as before using four-fold splits. These z-scores are close to normally distributed and model errors are typically smaller than forest variances by a factor of 0.68. After correcting for this by scaling, the resulting histogram of z-scores for the 97.5% of non-outlier data points is extremely well fit by a normalized Gaussian with unit standard deviation. This demonstrates that the √ value = 0.68 vf is a reliable estimate of the forest uncertainty for each data point across the full range of H adsorption energies. This analysis provides an indication of the reliability of a particular prediction, which allows us to put less trust in predictions with low reliability and mitigate the effects of prediction errors.

18


Page 18 of 26

Page 19 of 26 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


Conclusion The process of constructing a machine learning model for our dataset of more than 5,000 H adsorption energies on stepped Ag alloy surfaces has given insight into both model construction and the physics underlying H adsorption. The specific conclusions we draw from constructing the model are: (i) A very simple linear model, consisting exclusively of one-hot bits, can give qualitatively useful predictions.

However, nonlinear models significantly improve accuracy, and

improved feature sets reduce test errors with many fewer features. (ii) We found random and extra forests to be the most accurate models, although other nonlinear models can give fairly similar accuracy. The optimal model is quite accurate in the vast majority of cases, with a median absolute test error of 14 meV or 0.32 kcal/mol. (iii) The model identified structures with the same first nearest-neighbor shell and a similar second nearest-neighbor shell as being “similar” (or correlated) to each other, as we would intuitively expect, and the errors are high in cases where these similar structures have significantly different adsorption energies. (iv) These structures violate a widespread assumption in machine learning: that similar input values yield similar output values. (v) The model provides reliable confidence intervals for its predictions, crucial in screening studies to mitigate the possibility of missing promising candidates due to larger-thanexpected errors. Our investigation has also provided important insights into chemical trends and found unexpected behavior in H adsorption on dilute metal alloys. Atoms that strengthen adsorption when in the first nearest neighbor shell of the H atom tend to weaken adsorption when they are in the second nearest neighbor shell. In some cases, such as the V-Ag alloys discussed above, H adsorption energies can be surprisingly sensitive to small changes in the composition of the second and even third nearest neighbor shells. We also found counterintuitive, non-monotonic behavior as a function of the composition of the nearest neighbor shell by inspecting the forest kernel. Specifically, there are several cases where two substitutions at the step edge lead to weaker H adsorption than a single 19



substitution. These counterintuitive results are driven by nanocluster-like electronic structure effects arising from weak dopant-Ag hybridization. The model tends to have higher errors in cases that are counterintuitive, suggesting that the model gains a similar “intuition” as human observers. Overall, we have found that the process of constructing a model and studying its outliers can bring physical insight beyond that given by the model itself.

Acknowledgement We acknowledge useful discussions with Tess Smidt, Kai Kohlhoff and Patrick Riley. This work was performed as part of Integrated Mesoscale Architectures for Sustainable Catalysis, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0012573. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725.

Supporting Information Available Calculation and feature descriptions, adsorption energy database, and feature extraction code.

This material is available free of charge via the Internet at http://pubs.acs.

org/.

References (1) Zaera, F. The Surface Chemistry of Metal-Based Hydrogenation Catalysis. ACS Catal. 2017, 7, 4947–4967. (2) Chen, B.; Dingerdissen, U.; Krauter, J.; Lansink Rotgerink, H.; Möbus, K.; Ostgard, D.; Panster, P.; Riermeier, T.; Seebald, S.; Tacke, T.; Trauthwein, H. New Developments in Hydrogenation Catalysis Particularly in Synthesis of Fine and Intermediate Chemicals. Appl. Catal. A Gen. 2005, 280, 17–46.

20


Page 20 of 26

Page 21 of 26 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


(3) Hu, C.; Creaser, D.; Siahrostami, S.; Grönbeck, H.; Ojagh, H.; Skoglundh, M. Catalytic Hydrogenation of C=C and C=O in Unsaturated Fatty Acid Methyl Esters. Catal. Sci. Technol. 2014, 4, 2427–2444. (4) Beeck, O. Y. Hydrogenation Catalysts. Proc. R. Soc. a 1950, 177, 118–128. (5) Montemore, M. M.; Medlin, J. W. A Unified Picture of Adsorption on Transition Metals through Different Atoms. J. Am. Chem. Soc. 2014, 136, 9272–9275. (6) Studt, F.; Abild-Pedersen, F.; Bligaard, T.; Sørensen, R. Z.; Christensen, C. H.; Nørskov, J. K. Identification of Non-Precious Metal Alloy Catalysts for Selective Hydrogenation of Acetylene. Science 2008, 320, 1320–1322. (7) Chen, W.; Santos, E. J.; Zhu, W.; Kaxiras, E.; Zhang, Z. Tuning the Electronic and Chemical Properties of Monolayer MoS2 Adsorbed on Transition Metal Substrates. Nano Lett. 2013, 13, 509–514. (8) Luo, L.; Duan, Z.; Li, H.; Kim, J.; Henkelman, G.; Crooks, R. M. Tunability of the Adsorbate Binding on Bimetallic Alloy Nanoparticles for the Optimization of Catalytic Hydrogenation. J. Am. Chem. Soc. 2017, 139, 5538–5546. (9) Zhou, Y.; Chen, W.; Cui, P.; Zeng, J.; Lin, Z.; Kaxiras, E.; Zhang, Z. Enhancing the Hydrogen Activation Reactivity of Nonprecious Metal Substrates via Confined Catalysis Underneath Graphene. Nano Lett. 2016, 16, 6058–6063. ˙ glu, N.; Kitchin, J. R. Identification of Sulfur-Tolerant Bimetallic Surfaces Using (10) Ino˘ DFT Parametrized Models and Atomistic Thermodynamics. ACS Catal. 2011, 1, 399–407. (11) Andriotis, A. N.; Mpourmpakis, G.; Broderick, S.; Rajan, K.; Datta, S.; Sunkara, M.; Menon, M. Informatics Guided Discovery of Surface StructureChemistry Relationships in Catalytic Nanoparticles. J. Chem. Phys. 2014, 140, 094705.

21



(12) Ras, E.-J.; Louwerse, M. J.; Mittelmeijer-Hazeleger, M. C.; Rothenberg, G. Predicting Adsorption on Metals: Simple Yet Effective Descriptors for Surface Catalysis. Phys. Chem. Chem. Phys. 2013, 15, 4436–43. (13) Li, Z.; Ma, X.; Xin, H. Feature Engineering of Machine-Learning Chemisorption Models for Catalyst Design. Catal. Today 2017, 280, 232–238. (14) Ma, X.; Li, Z.; Achenie, L. E. K.; Xin, H. Machine-Learning-Augmented Chemisorption Model for CO2 Electroreduction Catalyst Screening. J. Phys. Chem. Lett. 2015, 3528–3533. (15) Jinnouchi, R.; Asahi, R. Predicting Catalytic Activity of Nanoparticles by a DFTAided Machine-Learning Algorithm. J. Phys. Chem. Lett. 2017, 8, 4279–4283. (16) Ulissi, Z. W.; Tang, M. T.; Xiao, J.; Liu, X.; Torelli, D. A.; Karamad, M.; Cummins, K.; Hahn, C.; Lewis, N. S.; Jaramillo, T. F.; Chan, K.; Nørskov, J. K. Machine-Learning Methods Enable Exhaustive Searches for Active Bimetallic Facets and Reveal Active Site Motifs for CO2 Reduction. ACS Catal. 2017, 7, 6600–6608. (17) Pankajakshan, P.; Sanyal, S.; De Noord, O. E.; Bhattacharya, I.; Bhattacharyya, A.; Waghmare, U. Machine Learning and Statistical Analysis for Materials Science: Stability and Transferability of Fingerprint Descriptors and Chemical Insights. Chem. Mater. 2017, 29, 4190–4201. (18) Montemore, M. M.; Medlin, J. W. Predicting and Comparing C–M and O–M Bond Strengths for Adsorption on Transition Metal Surfaces. J. Phys. Chem. C 2014, 118, 2666–2672. (19) Kyriakou, G.; Boucher, M. B.; Jewell, A. D.; Lewis, E. A.; Lawton, T. J.; Baber, A. E.; Tierney, H. L.; Flytzani-Stephanopoulos, M.; Sykes, E. C. H. Isolated Metal Atom Geometries as a Strategy for Selective Heterogeneous Hydrogenations. Science 2012, 335, 1209–1212.

22


Page 22 of 26

Page 23 of 26 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


(20) Wang, Z.-T.; Xu, Y.; El-Soda, M.; Lucci, F. R.; Madix, R. J.; Friend, C. M.; Sykes, E. C. H. Surface Structure Dependence of the Dry Dehydrogenation of Alcohols on Cu(111) and Cu(110). J. Phys. Chem. C 2017, 121, 12800–12806. (21) Darby, M. T.; Sykes, E. C. H.; Michaelides, A.; Stamatakis, M. Carbon Monoxide Poisoning Resistance and Structural Stability of Single Atom Alloys. Top. Catal. 2018, 0, 0. (22) Honkala, K.; Hellman, A.; Remediakis, I. N.; Logadottir, A.; Carlsson, A.; Dahl, S.; Christensen, C. H.; Nørskov, J. K. Ammonia Synthesis from First-Principles Calculations. Science 2005, 307, 555–558. (23) Montemore, M. M.; Madix, R. J.; Kaxiras, E. How Does Nanoporous Gold Dissociate Molecular Oxygen? J. Phys. Chem. C 2016, 120, 16636–16640. (24) Tritsaris, G. A.; Greeley, J.; Rossmeisl, J.; Nørskov, J. K. Atomic-Scale Modeling of Particle Size Effects for the Oxygen Reduction Reaction on Pt. Catal. Lett. 2011, 141, 909–913. (25) Kresse, G.; Hafner, J. Ab Initio Molecular Dynamics for Liquid Metals. Phys. Rev. B 1993, 47, 558–561. (26) Kresse, G.; Furthm¨ uller, 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. (27) Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res. 2011, 12, 2825–2830. (28) Breiman, L. Random Forests. Mach. Learn. 2001, 45, 5–32. (29) Geurts, P.; Ernst, D.; Wehenkel, L. Extremely Randomized Trees. Mach. Learn. 2006, 63, 3–42.

23



(30) Hammer, B.; Nørskov, J. Theoretical Surface Science and Catalysis – Calculations and Concepts. Adv. Catal. 2000, 45, 71–129. (31) Zou, H.; Hastie, T. Regularization and Variable Selection via the Elastic Net. Journal of the Royal Statistical Society. Series B (Statistical Methodology) 2005, 67, 301–320. (32) LeCun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. Gradient-Based Learning Applied to Document Recognition. Proc. IEEE 1998, 86, 2278–2324. (33) Duvenaud, D. K.; Maclaurin, D.; Iparraguirre, J.; Bombarell, F.; Hirzel, T.; AspuruGuzik, A. Convolutional Networks on Graphs for Learning Molecular Fingerprints. NIPS 2015. 2015; pp 2224–2232. (34) Tipping, M. E. Sparse Bayesian Learning and the Relevance Vector Machine. J. Mach. Learn. Res. 2001, 1, 211–244. (35) Muscat, J. P.; Newns, D. M. Theoretical Model of Chemisorption on Metals: I. Formalism. Surf. Sci. 1979, 87, 643–655. (36) Harrison, W. A. Electronic Structure and the Properties of Solids: the Physics of the Chemical Bond; Dover Publications, Inc.: New York, New York, USA, 1989; p Appendix E. (37) Hyman, M. P.; Medlin, J. W. Effects of Electronic Structure Modifications on the Adsorption of Oxygen Reduction Reaction Intermediates on Model Pt(111)-Alloy Surfaces. J. Phys. Chem. C 2007, 111, 17052–17060. (38) Greeley, J.; Jaramillo, T. F.; Bonde, J.; Chorkendorff, I.; Nørskov, J. K. Computational High-Throughput Screening of Electrocatalytic Materials for Hydrogen Evolution. Nat. Mater. 2006, 5, 909. (39) Yu, W.-Y.; Zhang, L.; Mullen, G. M.; Henkelman, G.; Mullins, C. B. Oxygen Activation and Reaction on Pd–Au Bimetallic Surfaces. J. Phys. Chem. C 2015, 119, 11754–11762. 24


Page 24 of 26

Page 25 of 26 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


(40) Nørskov, J. K.; Abild-Pedersen, F.; Studt, F.; Bligaard, T. Density Functional Theory in Surface Chemistry and Catalysis. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 937–943. (41) Thirumalai, H.; Kitchin, J. R. Investigating the Reactivity of Single Atom Alloys Using Density Functional Theory. Top. Catal. 2018, 61, 462–474. (42) Greiner, M. T.; Jones, T. E.; Beeg, S.; Zwiener, L.; Scherzer, M.; Girgsdies, F.; Piccinin, S.; Armbr¨ uster, M.; Knop-Gericke, A.; Schlögl, R. Free-Atom-like d States in Single-Atom Alloy Catalysts. Nat. Chem. 1008–1015. (43) Crampton, A. S.; Rötzer, M. D.; Ridge, C. J.; Schweinberger, F. F.; Heiz, U.; Yoon, B.; Landman, U. Structure Sensitivity in the Nonscalable Regime Explored via Catalysed Ethylene Hydrogenation on Supported Platinum nanoclusters. Nat. Commun. 2016, 7 . (44) Rupp, M.; Tkatchenko, A.; M¨ uller, K.-R.; von Lilienfeld, O. A. Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning. Phys. Rev. Lett. 2012, 108, 058301. (45) Wager, S.; Hastie, T.; Efron, B. Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife. J. Mach. Learn. Res. 2014, 15, 1625– 1651.

25



For Table of Contents Use Only Title: “Machine Learning Prediction of H Adsorption Energies on Ag Alloys” Authors: • Robert A. Hoyt • Matthew M. Montemore • Ioanna Fampiou • Wei Chen • Georgios Tritsaris • Efthimios Kaxiras Table of Contents graphic:

26


Page 26 of 26

Machine Learning Prediction of H Adsorption Energies on Ag Alloys

Recommend Documents