Quantifying Uncertainty in Activity Volcano Relationships for Oxygen

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Quantifying Uncertainty in Activity Volcano Relationships for Oxygen Reduction Reaction Siddharth Deshpande, John R. Kitchin, and Venkatasubramanian Viswanathan ACS Catal., Just Accepted Manuscript • DOI: 10.1021/acscatal.6b00509 • Publication Date (Web): 29 Jun 2016 Downloaded from http://pubs.acs.org on July 6, 2016

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Quantifying Uncertainty in Activity Volcano Relationships for Oxygen Reduction Reaction Siddharth Deshpande, John R. Kitchin, and Venkatasubramanian Viswanathan∗ Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, 15213 E-mail: [email protected]

Abstract The Oxygen Reduction Reaction (ORR) is an important electrochemical reaction and a major bottleneck for fuel cells. Due to the existence of a scaling relation between the adsorption energies of two key intermediates involved in ORR, OOH∗ and OH∗ , the electrocatalytic activity for the ORR, to a first approximation, is determined by a single descriptor. This descriptor-based approach has been used to screen for electrocatalyst materials that have an optimal binding energy of oxygen intermediates. However, given that this descriptor-based search relies on several approximations, it is crucial to determine the overall predictability of the descriptor-based model to determine the activity of a catalyst. In this work, we develop a formalism for estimating uncertainty for an electrocatalytic reaction scheme and apply this framework to determine errors involved in describing the ORR activity. We perform density functional theory calculations using BEEF (Bayesian Error Estimation Functional)-vdW exchange correlation functional to determine the adsorption energies of ORR intermediates on transition metal (111) and (100) facets. We show that the error estimates for the adsorption energies calculated ∗

To whom correspondence should be addressed

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with a reference metal surface, chosen here to be Pt(111), are much smaller than those calculated with gas-phase molecules as reference. We demonstrate that ∆GOH and ∆GOOH are the optimal descriptors for the 4e− and the 2e− ORR respectively. We show that for the 4e− ORR with ∆GOH as the descriptor, the uncertainty in activity is determined by the error associated with the adsorption energy of OH∗ (∼0.1 eV) for materials that lie on the strong binding leg and the error involved in the scaling relation between OOH∗ and OH∗ (∼0.2 eV) determines the uncertainty in activity for the weak binding leg. We propose a parameter, the expected limiting potential, UEL , which is the expected value of UL . The deviation of the expected limiting potential, UEL , from the thermodynamic limiting potential, UL , provides a qualitative estimate of the prediction error and can be used to identify trends in predictability. We believe that the concept of the expected limiting potential will be crucial in descriptor-based screening studies for multi-electron electrochemical reactions.

Keywords uncertainty, optimal descriptor, activity volcano, scaling relation, contour, fuel cells

1

Introduction

The oxygen reduction reaction (ORR) has been a subject of major interest in electrochemistry owing to its technological importance in fuel cells. 1 Oxygen electrochemistry along with the hydrogen electrochemistry, has served to lay the foundational scientific understanding of surface electrocatalysis 2 and has served as the test-bed for the development of new materials based on computational screening. 3–8 Based on a thermodynamic formalism, it has been argued that an ideal catalyst for the ORR needs to activate molecular oxygen as OOH∗ , while at the same time, bind oxygen species such as O∗ and OH∗ weakly. 3 Using Density Functional Theory (DFT) calculations,

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it has been shown that there exists a linear scaling relationship between the adsorption energies of the intermediates OOH∗ and OH∗ . 9,10 The scaling relationship leads to an universal activity volcano on metal surfaces, with the electrocatalytic activity being determined by a single activity descriptor. 10 This thermodynamic analysis remains qualitatively and even semi-quantitatively accurate even when kinetics are taken into account. 11 This analysis has been used in the computational search of new catalysts and has led to the discovery of catalysts for improved oxygen reduction activity. 12,13 Given the many approximations involved in the simple descriptor-based screening, an important issue emerges on how the errors involved in the calculated microscopic properties, for e.g., adsorption energies, propagate through the complex reaction network in determining macroscopic properties, for e.g. catalytic activity. This issue becomes even more important given the tremendous increase in the use of DFTbased screening using simple descriptors 14–17 and the increased functionality demanded in these screening studies. Given the overall time and resource that is required for synthesis, characterization and testing of new candidate materials, there is a need to address the reliability of the identified candidates through computational screening. Uncertainty quantification has emerged as an important frontier in computational engineering and physics 18 and the recently developed exchange correlation functional, Bayesian Error Estimation Functional with van der Waals correlation (BEEF-vdW), has brought error estimation capability to DFT calculations. 19 This approach uses an ensemble of exchange correlation functionals to quantify the errors in DFT and this approach can be seen as a structured way to test the sensitivity of DFT results to the choice of the exchange-correlation functional. Recently, Medford et al. used the DFT calculations using BEEF-vdW exchange correlation functional to assess the reliability of calculated reaction rates for a prototype heterogeneous catalytic reaction scheme, the ammonia synthesis. 20 In this work, we develop the formalism for error estimation for an electrocatalytic reaction scheme and apply this approach to the ORR. Using BEEF-vdW as the exchange correlation functional, we perform DFT calculations to determine the adsorption energies of oxygen

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intermediates on fcc metal (111) and (100) facets. We demonstrate that the optimal activity descriptor for the 4e− ORR that minimizes the overall prediction error is ∆GOH , the free energy of adsorbed OH∗ . Using this descriptor, we show that the uncertainty in activity for materials that lie on the strong binding leg of the activity volcano is determined by the error associated with the adsorption energy of OH∗ (∼0.1 eV) while the uncertainty in activity for materials that lie on the weak binding leg of the activity volcano is determined by the error involved in the scaling relation between OOH∗ and OH∗ (∼0.2 eV). We perform a similar analysis for the 2e− ORR and show that while ∆GOOH is the optimal descriptor, using ∆GOH as a descriptor provides sufficient accuracy for determining the 2e− oxygen reduction activity. It is advantageous to use ∆GOH as a descriptor for the 2e− ORR as then the 2e− and the 4e− ORR could be compared directly. Based on our error estimation analysis, we propose a new parameter, the expected limiting potential, UEL , defined as the expected value of the limiting potential UL for a given value of the activity descriptor. The expected limiting potential, UEL , and the limiting potential, UL only deviate from each other close to the top of the activity volcano indicating that the activity predictions from the thermodynamic activity volcano become less reliable in this region.

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Methods

Calculations were performed using the Vienna ab initio simulation package (VASP) 21 with BEEF-vdW 19 as the the exchange-correlation functional. Core electrons were described using the Projector Augmented Wave function (PAW) and k-Points were represented using Monkhorst Pack grids. 22 The Kohn Sham orbitals were expanded up to energy cutoffs of 450 eV with a Fermi smearing of 0.1 eV. All structures were converged with a force criterion < 0.05 eV/˚ A. A 6 × 6 × 1 k-point grid was used for a unit cell having 2 atoms each in the x and y direction and with 4 layers in the z-direction (2 × 2 × 4). The bottom two layers of the slab were fixed and the remaining were allowed to be relaxed. For other types of unit cells

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√ √ considered in the calculation ( 3 × 3 × 4, 2 × 3 × 4), the k-points were suitably scaled. The adsorption energies were calculated for intermediates O∗ , OH∗ , OOH∗ and H2 O for the transition metals Pt, Pd, Ag and Au and the alloy Pt3 Ni on the fcc(100) and fcc(111) facets. The free energies of these intermediates were calculated at standard conditions, and a potential, U = 0 V versus the Reversible Hydrogen Electrode (RHE), using the relation ∆G0 = ∆Ew,water + ∆ZP E − T ∆S,

(1)

where ∆Ew,water is the formation energy of an intermediate including the stabilization of water, calculated relative to H2 O and H2 , ∆ZPE is the difference in zero point energies, T is the temperature of the system and ∆S is the change in entropy. We use the zero point energies and entropic corrections as reported in earlier work. 3,23 A potential of U = 0 V versus the RHE implies the relation, ∆GH+ + ∆Ge− = 1/2∆GH2 (g) . This relation provides a way to computationally calculate the sum of the free energies of a proton and an electron and is termed as the computational hydrogen electrode. 3 The effect of potential U is included by shifting the free energy of an electron by -eU. Within the computational hydrogen electrode formulation, there is no explicit pH dependence for the reaction. It is worth pointing out that pH effects have been observed experimentally for oxygen reduction 24 and a theoretical analysis of the pH effect remains outside the scope of the computational hydrogen electrode approach and our present work. The free energy at a potential U, can thus be calculated using the relation, ∆G = ∆G0 − eU where U is the potential relative to the RHE. This procedure is used to construct the free energy diagrams (section 3.1) and to deduce an important parameter, the potential-determining step. The potential determining step is defined as the first step that becomes uphill in free energy when the potential is increased 3 and the highest potential at which all the steps in the reaction scheme are downhill in free energy is known as the limiting potential UL . In order to determine the uncertainty in the calculated adsorption energies and the asso-

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ciated free energies, we utilize the error estimation capabilities of BEEF-vdW. The methodology for estimating error within the BEEF-vdW functional is briefly discussed below and more detailed description can be found in the work by Wellendorf et al.. 19

2.1

Bayesian Error Estimation Functional (BEEF-vdW)

Bayesian Error Estimation Functional (BEEF-vdW) is a semi-empirical type of exchange correlation functional that takes into account the view of both a reductionist and an empiricist. 19 The method uses different types of data sets as the empirical data and fits the exchange enhancement factor, Fx (s) to it. These data sets can be categorized as, molecular formation energies, molecular reaction energies, molecular reaction barriers, noncovalent interactions, solid state properties, and chemisorption on solid surfaces. The expression for the Exchange correlation energy for BEEF-vdW is given by,

Exc =

M x −1 X m=0

GGA−x am Em + αc E LDA−c + (1 − αc )E P BE−c + E nl−c ,

(2)

where, Mx represents the degree of the polynomial. The coefficients am and αc are the fitting parameters which are optimized over the data sets in such a way that over fitting is avoided while maintaining dependency on the data. EGGA−x represents the GGA exchange energy, m EPBE−c and ELDA−c represent the PBE and LDA correlation energies and Enl−c represents the non-local correlation energy obtained from the functional vdW-DF2 . 25 BEEF-vdW uses an ensemble of functionals to calculate errors. 26 After calculating the optimum value for the coefficients am and αc , each coefficient is perturbed around its optimal value. An ensemble matrix is calculated (Ω−1 ) with the dimensions (Mx × Mx ). Then the eigenvalues(wΩ2 −1 ) and the eigenvectors (VΩ−1 ) of the ensemble matrix are used along with a matrix of randomly generated numbers of size k with mean zero and standard deviation one (vk ), to generate an ensemble of coefficients (ak = VΩ−1

.

wΩ−1

.

vk ). This ensemble of

coefficients can be used to generate an ensemble of energies (eq 2). The ensemble of energies

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provides a systematic way to calculate the uncertainty associated within the GGA class of functionals for a given calculation. This approach is more systematic than calculating the uncertainty using a small set of different GGA functionals. Further, this approach provides a computationally tractable way to estimate the uncertainty associated for a given calculation. An important issue arises regarding the comparison of the uncertainty estimate from the ensemble approach and fully self-consistent calculations. It has been shown that the ensemble estimate performs well in bounding the uncertainty from the self-consistent approach for several datasets including the CE17 dataset, which comprises of chemisorption energies of small molecules on late transition metal surfaces. 19 Uncertainty estimates for chemisorption forms the focus of this paper and we could expect the ensemble approach to provide a reliable estimate for the uncertainty associated with the chemisorption calculations. However, we caution that the reliability of error estimates from the ensemble approach still needs to be established for other classes of calculations (for e.g. solid state properties, molecular reaction energies). A more detailed discussion on the error estimate from the ensemble approach can be found in the Supporting Information.

2.2

Adsorption Energy Distribution

Using the BEEF-vdW functional, ensembles of adsorption energies for various intermediates involved in the ORR were calculated. Referencing the adsorption energy of any intermediate with respect to gas phase molecules (H2 (g), H2 O(g)) leads to large estimated error (see Supporting Information). However, when the reference is changed from the gas phase molecules to a reference element, for e.g., Pt(111), it was found that this leads to much smaller error estimates due to similarity in metal-adsorbate bonding characteristics (see Supporting Information). This supports the common notion that trends in DFT calculations are more accurate than the individual values. Building on this approach, we use the following methodology to get a combined error estimate for the adsorption energies of various intermediates. We illustrate the approach using the example of the intermediate OH∗ . 7


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Figure 1 shows the combined adsorption energy distribution for the intermediate OH∗ (EOH ) calculated for metals Pt, Pd, Ag and Au and the alloy Pt3 Ni for the fcc(100) and the fcc(111) facet. In all the distribution contains 20000 values, 2000 for each metal facet considered. The standard deviation for the distribution (σOH ) is 0.11 eV. Based on this analysis, we approximate the uncertainty involved in determining the adsorption energy of an intermediate by the the standard deviation of the calculated adsorption energy ensemble on the various metal facets. This estimate remains valid when other transition metal surfaces, such as Ni(111), Rh(111), Ru(0001) and Co(0001) are considered. The standard deviation for this set is ∼ 0.15 eV and the distribution is shown in the Supporting Information.

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Results and Discussion

We follow the associative mechanism for the 4e− ORR which comprises of sequential adding of a proton and electron in each step, written as: 3,27 O2 + 4H+ + 4e− → OOH∗ + 3H+ + 3e−

(6a)

OOH∗ + 3H+ + 3e− → O∗ + H2 O + 2H+ + 2e−

(6b)

O∗ + 2H+ + 2e− + H2 O → OH∗ + H+ + e− + H2 O

(6c)

OH∗ + H+ + e− + H2 O → 2H2 O

(6d)

It has been demonstrated that the intermediates OOH∗ and OH∗ are stabilized by hydrogen bonding with water while there is minimal effect of hydrogen bonding on the intermediate O∗ . 28,29 To account for this hydrogen bonding, the intermediates OOH∗ and OH∗ were mod√ √ eled using a water layer in an 3 × 3 configuration with a 1/3 monolayer (ML) coverage, shown in Figure 2a. O∗ is modeled in a 2 × 2 configuration in the fcc site with a 1/4 monolayer (ML) coverage, shown in Figure 2a. To account for the free energy of water that is in

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equilibrium with the intermediates OOH∗ and OH∗ , a

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√

3 ×

√

3 water layer is considered. 30

(See Supporting Information) For modeling adsorbates on the fcc(100) surface, it is important to point out that the fcc(100) surface undergoes a hex reconstruction when annealed. 29 However, in an electrochemical environment, the hex reconstruction gets lifted and reforms into its original square lattice configuration. 31 Thus, we use the unreconstructed fcc(100) surfaces to model the reaction mechanism on the fcc(100) facet. The thermodynamically stable intermediate structures (H2 O, OH∗ and OOH∗ ) for the fcc(100) facet are different when compared to the hexagonal fcc(111) facet due to the mismatch between the preferred “hexagonal” water overlayer and the underlying square symmetry of the (100) surface. For the fcc(100) case, the intermediates OOH∗ and OH∗ were modeled using a water layer with a 1/3 coverage on a 2 × 3 × 4 unit cell shown in Figure 2b. The intermediate O∗ is assumed to have no stabilization due to water and is modeled as a 1/4 ML coverage on the hollow site. Stable water layer structures, other than the 1/3 ML coverage considered in this study for the (111) facet could be important. However, we expect the trends to remain quantitatively similar. Figure 1 √ √ shows the ensemble of adsorption energies for the intermediate OH∗ in a 3 × 3 water layer on the fcc(111) facet and a water layer with 1/3 coverage on the fcc(100) facet. This ensemble remains robust with a standard deviation of 0.13 eV even when additional water layer structures, 1/4 ML OH∗ on fcc(111) and fcc(100), 1/2 ML OH∗ -H2 O on fcc(100), are considered. These distributions could be found in the Supporting Information. Apart from the 4 e− reduction of oxygen to form H2 O (eq 6), oxygen reduction can also proceed through the 2e− pathway in which the intermediate OOH∗ reduces to hydrogen peroxide (H2 O2 ). 32 Hydrogen peroxide (H2 O2 ) though undesirable in a fuel cell, is a strong and environmentally benign oxidizing agent. 32,33 Therefore, understanding the selectivity of the 2e− and the 4 e− ORR is beneficial. 33–35 The associate 2 e− ORR can be written as

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follows: O2 + 2H+ + 2e− → OOH∗ + H+ + e−

(7a)

OOH∗ + H+ + e− → H2 O2 .

(7b)

The adsorption energies of the intermediate OOH∗ calculated in the 4e− ORR model could be directly used to understand the 2e− oxygen reduction to hydrogen peroxide (H2 O2 ).

3.1

Free Energy Diagram

The calculated adsorption energies can be used to construct a free energy diagram, which is a plot showing the free energy of the intermediates involved in a reaction scheme plotted at a given potential U. Figure 2 shows the calculated free energy diagram for Pt(111) and Pt(100) plotted at a potential of, U of 0.76 V and UL of 0.63 V. The figure also shows the standard deviation for the free energy of each intermediate calculated with respect to the reference gas phase molecules, H2 (g) and H2 O(g). For Pt(111), BEEF-vdW functional predicts that the step going from the intermediate O∗ to OH∗ is the potential determining step. Using RPBE functional, it has been shown previously, 28 that for Pt(111) the final step 6d associated with the reduction of OH∗ to H2 O(l) is the potential determining step and this is consistent with experimental findings. 36 This inconsistency arises as the functional BEEF-vdW overstabilizes the intermediate O∗ . However, the difference in the thermodynamic limiting potential for the two steps, 6c and 6d is very small (∼0.05 V) and within the uncertainty of the calculations. In order to maintain consistency, we proceed with the limiting potential of 0.76 V identified through previous analysis. 10 For the Pt(100) case, step 6d associated with the reduction of OH∗ to H2 O(l) is the potential determining step. As seen from Figure 2, the uncertainty involved in some of the adsorption energies is as large as 0.5 eV. However, as we demonstrated earlier, trends in adsorption energies have much smaller uncertainties as compared to the individual values.

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relationship. The following relation, derived from statistics, was used to generate the stan2 2 2 dard deviation of the intercept, σ(OH ∗ −OOH∗ ) = σOH∗ + σOOH∗ − 2(µ(OH∗ OOH∗ ) - µOH∗ µOOH∗ ).

σOH∗ , µOH∗ and σOOH∗ , µOOH∗ are the standard deviations and means for EOH and EOOH calculated with respect to gas phase molecules H2 and H2 O respectively. µOH∗ OOH∗ is the mean associated with the distribution of EOH ×EOOH . σ(OH∗ −OOH∗ ) is the standard deviation for the scaling relationship intercept assuming EOH and EOOH are linearly related with slope one. Adsorption energies (EOH and EOOH ) and the respective ensemble of energies was calculated for the metals Pt, Pd, Ag and Au and the alloy Pt3 Ni on the fcc(111) and fcc(100) facets. A detailed description of the procedure used to calculate σOH∗ and σOOH∗ is presented in the supporting information (see Supporting Information). The uncertainty in individual OH∗ and OOH∗ adsorption energies, calculated referenced to gas phase molecules, is of the order of 0.3-0.4 eV where as the value for σ(OH∗ −OOH∗ ) is found to be 0.19 eV. Figure 3 shows the plot for scaling relationship (black line) along with the adsorption energies (EOH and EOOH ) for the metals Pt, Pd, Ag and Au and the alloy Pt3 Ni (red dots), plotted with their respective ensemble energies (yellow dots). Two pairs of error lines (σ(OH∗ −OOH∗ ) and 2 × σ(OH∗ −OOH∗ ) ) are also plotted as error bounds. The uncertainty is well captured in the error bounds as the 2 × σ(OH∗ −OOH∗ ) lines include 90 percent of the points. The R2 value for the scaling relation with slope 1 and intercept 3.2 is 0.80. The Supporting Information file also shows a plot of the best fit line for the generated data points and this line has a slope of 0.943, an intercept of 3.28 and a R2 value of 0.89. In this work, we use the scaling relation with slope 1 and intercept 3.2 to be consistent with our prior works. 10,32

3.3

Volcano Relations

A consequence of the scaling relation is that the two intermediates OH∗ and OOH∗ have a fixed energy difference of 3.2 eV between them. This leaves only ∆Gavail = −4 × 1.23 + 3.2 = −1.72 eV. Where ∆Gavail is the available free energy for the remaining two steps (6a and 6d). 13


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Using adsorbed OH∗ as the descriptor, for all the metal facets that bind the oxygen intermediate too strongly, the limiting potential, UL for the 4e− ORR is given by:

UL − UL (P t(111)) = ∆GOH − ∆GOH (P t(111)).

(8)

In this formalism, an ensemble of free energies leads to an ensemble of limiting potentials, UL . The uncertainty associated with the calculated OH∗ is determined using the adsorption energy distribution method (section 2.2) and this determines the uncertainty associated with the determined limiting potentials, UL . For catalysts that bind the oxygen intermediates weakly, the limiting potential is given by, 10

UL − UL (P t(111)) = 0.2 − (∆GOOH − ∆GOOH (P t(111))) ≈ 0.2 − (∆GOH − ∆GOH (P t(111))). (9) Invoking the scaling relation between OOH∗ and OH∗ , the activity can be described, to a first approximation, in terms of a single descriptor, the free energy of adsorbed OH∗ . Note that in this case the uncertainty in the limiting potentials, UL , is associated with the uncertainty in the intercept of the scaling relationship (section 3.2). Figure 4a shows the calculated activity volcano for the 4e− ORR including the error bounds. The y-axis in the figure represents the limiting potential calculated with respect to the limiting potential of Pt(111) and the x-axis represents the free energy of the intermediate OH∗ calculated with respect to the free energy of the intermediate OH∗ for Pt(111). The figure also shows the calculated free energy of the intermediate OH∗ , calculated with an explicit water layer to account for hydrogen bonding. This plot shows the calculated values for 111 facet of the metals Pt, Pd, Au and the alloy Pt3 Ni and the 100 facet of the metals Pt, Ag, Au and the alloy Pt3 Ni, calculated with respect to Pt(111) (blue dots) plotted against there respective experimental limiting potential. The choice of the metal facets and alloys are made such that they have high activity for ORR. It is to be noted that the trend for the adsorption energies 15


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including the water layer with the BEEF-vdW exchange correlation functional is similar to that reported in previous work without an explicit water layer with the RPBE functional. 10 The uncertainties in the calculated adsorption energy for all the metal facets considered are small and typically of the order of ∼ 0.1 eV. The figure includes two pair of error lines, yellow representing 1σ and green representing 2σ for the calculated limiting potential. It can be seen from the figure that almost all the calculated free energies and the corresponding experimental values (blue dots) lie within the one standard deviation error bound of the two legs of the volcano. This leads to the conclusion that the experimentally reported activity lies within the prediction error of the theoretically calculated limiting potential. For the 2e− oxygen reduction, the optimal descriptor is the free energy of adsorbed OOH∗ since it directly determines the activity for both legs of the volcano and leads to the least prediction error as shown in the Supporting Information. The free energy of adsorbed OOH∗ has been used as the descriptor for identifying new electrocatalyst materials for selective H2 O2 production. 38,39 Another way to determine the activity for 2e− ORR is through the scaling relation between the intermediates OH∗ and OOH∗ . This allows the 2e− and 4e− ORR pathways to be compared in a single activity volcano plot. We would like to highlight that due to the relatively small uncertainty associated with the scaling relation between OOH∗ and OH∗ (∼ 0.2 eV), as we will demonstrate later, OH∗ is a descriptor that provides sufficient accuracy in determining the 2e− ORR activity. The potential determining steps for the 2e− ORR can be described in an analogous way to the 4e− ORR. For all the species that bind oxygen intermediates too strongly, eq 7b is the potential determining step, given by, 32

UL = ∆GOOH − ∆GH2 O2

(10)

The formation energy of ∆GH2 O2 is 3.56 eV, calculated using thermodynamic tables to

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avoid well-known issues involved in calculating molecular reaction energies using DFT. Using the scaling relation between the intermediates OH∗ and OOH∗ , the above equation can be written as, UL = ∆GOH − 0.36. Note that the limiting potentials for the 2e− ORR are referenced relative to the limiting potential for Pt(111). The 2e− activity for the strong binding leg is given as

UL − UL (P t(111)) = ∆GOH − ∆GOH (P t(111))

(11)

For materials which bind oxygen intermediates weakly, eq 7a is the potential determining step. The 2e− ORR activity for the weak binding leg is given by 32

UL − UL (P t(111)) = 0.56 − (∆GOH − ∆GOH (P t(111))).

(12)

As the above equation is now referenced with respect to the limiting potential (UL (Pt(111))) for the 2e− ORR, there is an offset when compared to eq 9. Figure 4b shows the 2e− ORR volcano plot using free energy of adsorbed OH∗ as the descriptor. The uncertainty associated with the weak and the strong binding legs of the 2e− volcano is the uncertainty in determining the intercept of the scaling relationship between OOH∗ and OH∗ . As the 2e− ORR process involves tuning the adsorption energy of only one intermediate, it is possible to attain a limiting potential that is equal to the equilibrium potential of the reaction. The outer bounds of the errors cap out at the equilibrium potential for the reaction. The plot also incorporates the free energy of the intermediate OH∗ along with the respective uncertainty for metal facets Pt(111), Au(111), Ag(100), Au(100) and Pt(100) calculated with respect to Pt(111). Given that the descriptor for the 2e− and the 4 e− ORR volcano is the free energy of the intermediate OH∗ (∆GOH ), the 2e− and the 4e− ORR volcano plots could be compared directly. The comparison can be used to comment upon the selectivity of a given catalyst to a particular process. The inclusion of uncertainty associated with a given calculation can further help in commenting upon the reliability of 17


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volcanoes.). But, when we take the uncertainty associated with it in account, it could be seen that the prediction is more sensitive to uncertainties in the 4e− ORR case. In case of the 2e− ORR it can be seen that taking uncertainty into account the prediction for Au(100) roughly goes from one leg to another predicting almost the same catalytic activity through out. This points to the conclusion that taking uncertainty into account, the calculation for Au(100) is more reliable for the 2e− ORR than the 4e− one. While Figures 4a and 4b provide a way to understand the uncertainties associated with the limiting potentials, they do not provide a visual representation of the probability distribution associated with the limiting potential, UL . In order to determine the probability distribution for the limiting potential, we construct a contour plot representing the probability distribution of the limiting potential UL as a function of the free energy of the intermediate OH∗ (∆GOH ). Consider a random variable ∆GOH ∼ N(h∆GOH i, σOH ) where, h∆GOH i is the mean and σOH is the standard deviation corresponding to the adsorption energy of the intermediate OH∗ . A given value of the mean, h∆GOH i represents a calculated value of the free energy while the random variable, ∆GOH , accounts for the uncertainty of the calculated value. This generates a probability distribution of the limiting potential, UL as function of the mean, h∆GOH i. Now for a given value x taken from the distribution around a given h∆GOH i, the distribution of UL for x lying on the strong binding leg of the activity volcano for the 4e− ORR is given by, UL ∼ N (x, σOH ). Invoking the scaling relationship, for x lying on the weak binding leg, UL is given by, UL ∼ N (4.92 − (3.2 + x), σscal ). Hence the distribution for UL for the 4e− ORR for any x taken from the distribution around a given h∆GOH i could be written as UL ∼ min(N (x, σOH ), N (4.92 − (3.2 + x), σscal )).

(13)

Similarly for the 2e− ORR, the distribution for UL for any value x taken from the distri-

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(a)

(b) Figure 5: A contour plot of the probability distribution of the limiting potential UL as a function of the mean free energy of the intermediate OH∗ (h∆GOH i) for (a) 4e− and (b) 2e− ORR is shown. The color map represents the probability distribution of UL as a function of mean free energy h∆GOH i. The blue curve represents the expected limiting potential UEL and the solid black lines represent the theoretical activity volcano. Also plotted on Figure (b) is the green curve which represents the expected limiting potential using the free energy of the adsorbed OOH∗ as the descriptor (represented on the top x-axis).

bution around a given h∆GOH i could be written as

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UL ∼ min(N (x − 0.36, σscal ), N (4.92 − (3.2 + x), σscal )),

(14)

where, σscal is the standard deviation associated with the intercept of the scaling relation between OH∗ and OOH∗ . The values for σOH and σscal are taken to be 0.11 and 0.19 eV respectively. The above mentioned procedure is used to construct the probability distribution of UL as a function of h∆GOH i. From the probability distribution, an important quantity termed as the Expected limiting potential (UEL ) can be derived given by the expectation value of the limiting potential, UL . For a given value of h∆GOH i, the expected limiting potential represents the value that would be expected given a large number of experiments on materials with the same calculated h∆GOH i. Further, the uncertainty in the calculated limiting potential (UL ) for a material with a given mean h∆GOH i, can be estimated by the departure of the calculated UL of the chosen material from the UEL for that given h∆GOH i. The departure would give the average error in the calculated UL . Another useful metric could be the most probable limiting potential, UM L which would represent the limiting potential that has the maximum probability. Figures 5a and 5b represent the 4e− and 2e− ORR contours with the probability distribution of the limiting potential, UL plotted as a function of h∆GOH i. The black dashed lines in the plot represent the theoretical activity volcano. The blue curve in Figures 5a and 5b represents the expected limiting potential (UEL ). It can be seen that the theoretical activity volcano and UEL track each other and deviate only near the top of the activity volcano. This happens because, near the top of the volcano, the number of points in the distribution of UL lying on the strong and the weak binding legs are of the same order. As a result of which the expectation value drops below the theoretical limiting potential. This leads to an important conclusion that the uncertainty in calculation increases as one approaches the top of a given theoretical volcano. Also plotted in Figure 5b is the expected limiting potential when the free energy of the intermediate OOH∗ is chosen as the descriptor (green 21


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curve). The standard deviation for OOH∗ as the descriptor, σOOH , is ∼ 0.12 eV for the mean of the UL distribution lying on either legs of the volcano (See supporting Information for the Figure). As could be seen from Figure 5b, the distance between the green curve and the top of the theoretical volcano is lesser when compared to the blue one. This leads to the conclusion that the expected limiting potential UEL and the theoretical activity volcano (black dashed lines), would track each other completely in the limit of the uncertainty in calculation tending to zero. The most probable limiting potential, UM L for the 4e− and 2e− ORR is shown in the Supporting Information. For a normal distribution, UEL and UM L should be the same. If the distribution deviates from a Normal Distribution, UEL and UM L could deviate. Near the top of the theoretical activity volcano, the distributions for UL for the 4e− and the 2e− ORR deviate from a Normal Distribution. The deviation happens in the 4e− case as the standard deviations for both legs of the activity volcano are different (σOH = 0.11 eV and σscal = 0.19 eV). While in the 2e− case the deviation happens because of the constraint that the value of UL cannot exceed 0.68 V. As shown above, the error estimation capability of BEEF-vdW can be effectively used to get the probability distribution of the limiting potential UL as a function of the activity descriptor. The probability distribution can then be further used to derive quantities such as the expected and most probable limiting potential, UEL and UM L . We believe that the expected limiting potential, UEL , could prove to be a helpful parameter in accurately identifying new candidate catalysts for multi-electron electrochemical reduction reactions such as reduction of CO2 , N2 etc.

4

Conclusion

We show a systematic approach to calculating the uncertainty involved in DFT-predicted activities for an electrocatalytic reaction scheme and apply this to oxygen reduction reaction. It is shown that relative errors (∼0.1 eV) are smaller compared to the absolute ones(∼0.5

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ACS Catalysis

eV), supporting the common notion that trends in DFT calculations are more reliable than absolute values. We show that the optimal descriptor for the 4e− ORR is the free energy of the intermediate OH∗ (∆GOH ). Using ∆GOH as the descriptor we show that the standard deviation calculated using the adsorption energy distribution method for the intermediate OH∗ (∼ 0.1 eV) and the error in the scaling relationship intercept (∼ 0.2 eV) can be used as the error associated with the strong and the weak binding legs of the 4e− ORR volcano. We further show that while, the optimal descriptor for the 2 e− ORR should be the free energy of intermediate OOH∗ (∆GOOH ), using ∆GOH as the descriptor still provides a reasonable way to predict activity of various catalysts for the 2e− ORR. The use of the same descriptor for the 4e− and 2e− ORR is beneficial as the two processes can then be compared directly and using error estimates for a given prediction, one can comment upon the selectivity of a given catalyst as well as the reliability of the prediction. The work further demonstrates a method to visualize the probability distribution of the limiting potential UL as a function of the mean free energy (h∆GOH i). We introduce a new parameter UEL , the expected limiting potential, which can be effectively used to understand the effect of errors on the prediction of the limiting potential UL . It is shown that UL and UEL track each other and deviate only near the top of the activity volcano, demonstrating the effective use of UEL to identify regions on the activity plot with high uncertainty.

Supporting Information Available The figures for the combined adsorption energy distribution for the intermediate OOH*, 2e− activity volcano with OOH∗ as the descriptor, contour plots comparing UM L and UEL and the 4e− activity volcano with O∗ as the descriptor are included. All the calculated data along with the codes used to produce the figures presented in the work are also included. This material is available free of charge via the Internet at http://pubs.acs.org/.

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Acknowledgement Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research. This work is also supported in part by the Pennsylvania Infrastructure Technology Alliance, a partnership of Carnegie Mellon, Lehigh University and the Commonwealth of Pennsylvanias Department of Community and Economic Development (DCED). V. V. gratefully acknowledges support from the National Science Foundation CAREER award CBET-1554273.

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Quantifying Uncertainty in Activity Volcano Relationships for Oxygen

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