Examining the Linearity of Transition State Scaling Relations - The

Apr 17, 2015 - We have investigated how to relate the concepts from linear adsorption energy scaling to transition state energies. We expect that stro...
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Examining the Linearity of Transition State Scaling Relations Philipp N. Plessow†,‡ and Frank Abild-Pedersen*,† †

SUNCAT Center for Interface Science and Catalysis, Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States ‡ SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States S Supporting Information *

ABSTRACT: The dissociation of strong bonds in molecules shows large variations in the geometric structure of the transition state depending on the reactivity of the surface. It is therefore remarkable that the transition state energy can be accurately described through linear relations such as the Brønsted-Evans−Polanyi relations. Linear scaling relations for adsorbates with fixed structure can be understood in terms of bond order conservation but such arguments should not apply to transition states where the geometric structure varies. We have investigated how to relate the concepts from linear adsorption energy scaling to transition state energies. We expect that strong deviations from linearity only occur for very early or very late transition states. According to the Sabatier principle, the rate-limiting step of the best catalysts is not expected to be in either of these regions. Our results therefore support the use of linear transition state scaling relations for the optimization of catalysts.

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The scaling parameters, γAHx and ξAHx, can be thought of as coefficients to a best linear fit, but more generally, ξAHx describes the weak interaction limit, where EA = 0 and the slope, γAHx, how the energies of different adsorbates depend on the surface and measures their relative reactivity. The coefficient γ is therefore a measure of the bond-order of the adsorbate−surface bond. For the AHx species it was found to equal the number of available bonds of the central atom A, n − x, normalized to the maximum number of bonds, n: γ = (n − x)/n. In general, energy descriptors Ei and slopes γi can be of any dimension, depending on the complexity of the adsorbate.12−17 In a similar fashion, different types of linear relations have been developed to provide information about the kinetics of chemical reactions as a function of simpler variables, the BrønstedEvans-Polanyi (BEP) relation being the most common. In BEP relations, one uses the reaction energy, ΔER, as a single descriptor for the activation energy such that1,18

INTRODUCTION

The introduction of linear scaling relations for adsorption, as well as transition state energies, has proven extremely useful in the quest to identify, understand, and predict reactivity trends of solid catalysts.1−7 Their ability to relate complex properties, such as the energies of adsorbates or transition states to simple descriptors provides all the ingredients required to calculate trends in rates, equilibria, and turnover frequencies of complex reaction networks.8−10 The existence of these relations for reaction intermediates can be understood in terms of simple bond order arguments and, hence, their accuracy depends on the variations in the adsorbate−surface bond strength. Linear adsorbate scaling can therefore not be easily generalized to transition states that have large variations in their structure and bonding. However, theoretical modeling over the past decade has shown clearly that linear approximations describe transition state energies with surprisingly high accuracy,1 and in this paper, we will explore in detail why that is. Our investigation of transition states will build on the linear scaling relations for adsorbates with fixed structures and we shall briefly review these first. For adsorbed hydrates AHx (x = 0, ..., n − 1), with A = {C, N, S, O}, in which all bonds in the adsorbate except the adsorbate−surface bond are kept fixed, the adsorption energies are linearly dependent.11 Here n denotes the maximum number of bonds the central atom A can form with its surroundings. Using the adsorption energy of A as the descriptor, the energies of adsorbed AHx can now be written: E AHx = γAH × EA + ξAHx x

© 2015 American Chemical Society

EaBEP = α × ΔE R + β

(2)

Transition state geometries can vary significantly: As an example, we have found variations in the N−N bond distance of about 0.4 Å for N2 splitting ranging from coinage to very reactive metal surfaces. Such variations are expected to be important and it is therefore somewhat surprising that the transition state energies can be successfully described by one linear scaling relation. Received: March 2, 2015 Revised: April 15, 2015 Published: April 17, 2015

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DOI: 10.1021/acs.jpcc.5b02055 J. Phys. Chem. C 2015, 119, 10448−10453

Article

The Journal of Physical Chemistry C Clearly, ΔER can be estimated using adsorbate scaling relations for reactants and products; hence, the BEP relations can be viewed as a class of truncated relations with a restricted functional form. The parameter α is often related to a late or early transition state and therefore to the adsorbate−surface bond which, when related to the above, is expected to work only for fixed structures of the transition state. In this work we will study how the geometry of the transition state influences the surface−adsorbate bond, and we will show how one based on knowledge from linear scaling relations can account explicitly for these variations and develop a structuredependent model for transition states. Armed with the functional form for the transition state energies, we can address the still open questions about linearity and the use of linear transition state relations in computational chemistry.

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COMPUTATIONAL DETAILS Since we are interested in describing the correct trends of surface reactivity rather than obtaining the correct result for specific elements and their most stable surfaces and spin states, all calculations have been carried out spin-restricted on unrelaxed surfaces. The fcc(211) surfaces of 23 pure transition metals have been studied. Calculations have been carried out with the RPBE density functional19 and ultrasoft pseudopotentials20 with the DACAPO code21 using a plane-wave cutoff of 340 eV and a k-point mesh of (4,4,1). Slabs with four surface layers were separated by about 10 Å, and the dipole correction22 has been used to reduce the artificial interaction between slabs.

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Figure 1. Energy profiles for N2 dissociation for various fcc(211) surfaces and corresponding transition states are shown in eV and relative to E(dNN = 1.4 Å) = 0 in the upper panel. The lower panel shows scaling parameters obtained for different N−N distances along the reaction path. For γ, the error bars enclose a variation of γ that changes the MAE of the fitted scaling relations by less than 10%.

RESULTS AND DISCUSSION

Linear adsorbate scaling scaling relations describe the energy of a given adsorbate well as a function of some descriptor and we will base our approach to transition states entirely on these relations. If one applies adsorbate scaling relations to the structures along a reaction path, a linear scaling relation [γ(r),ξ(r)] should hold for each fixed value of the reaction coordinate r. The transition state is the energy-maximum of this reaction path. In an idealized case where the transition state structure is exactly the same for different surfaces, the linear adsorbate scaling relation for this structure is identical to the transition state scaling relation, γTS = γ(rTS) and ξTS = ξ(rTS). In the following, we will study cases where this does not hold and we will try to account for the variations in transition state structure. We will limit ourselves to the case where changing the surface reactivity does not lead to a new reaction path. This allows us to approximate the structure variations as occurring only along the onedimensional reaction path r. Since there exist linear adsorbate scaling relations for all the structures on this reaction path, we should be able to derive new, general transition state scaling relations. We will now investigate the validity of the assumptions in the above considerations for the dissociation of N2 on a large number of fcc(211) metal surfaces, where the N−N bond is broken across the step as shown on the insets in Figure 1. Since the reaction coordinate in this example can be described simply by the N−N distance, it is a very intuitive example for a transition state where structure variation occurs only along the same reaction path. To reduce the dimensionality and thereby allowing us to easier visualize and illustrate some general points we have chosen the N binding energy, Ead(N) at the hcp site on the upper terrace as the only energy descriptor in our analysis. Nevertheless, we want to point out that our conclusions for

transition state scaling relations hold for any surface-catalyzed reaction that occurs along the same reaction path for the surfaces of interest. Furthermore, the descriptor is not limited to a single dimension. If one generalizes eq 1 to any multidimensional descriptor ε, the linear scaling relations for different values of the reaction coordinate becomes E(r , ε) = γ(r ) ·ε + ξ(r )

(3)

The upper part of Figure 1 shows the energy of constrained minimizations E(dNN) using fixed N−N distances dNN for five different transition metals. In order to compare the shape of the energy profiles and the geometry of the transition state, the curves are shifted to E(dNN = 1.4 Å) = 0. The obtained energy curves are smooth and the transition state occurs at different values of dNN. With the energy profiles E(dNN) for different surfaces and corresponding adsorption energies Ead(N) as descriptors, linear scaling relations can be fitted for each dNN using eq 1. The resulting slopes γ and offsets ξ are shown in the lower panel of Figure 1 as a function of dNN and the underlying data points and the linear regression are depicted in Figure 2. The intermediate structures are well described by linear scaling relations with an average mean absolute error (MAE) of less than 0.1 eV. As N2 is stretched, ξ and γ increase reflecting a higher energy of the stretched molecule in the gas phase and a stronger interaction with the surface, respectively. It is also obvious, that these two parameters, and their interpretation (strain and reactivity) have to be correlated. Since for N2 the descriptor is one-dimensional, one can visualize E(r, ε) in a 3D plot (Figure 2) as a function of the reaction coordinate dNN and the energy descriptor Ead(N). The fitted linear scaling relations for different metal surfaces at fixed 10449

DOI: 10.1021/acs.jpcc.5b02055 J. Phys. Chem. C 2015, 119, 10448−10453

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The Journal of Physical Chemistry C

Figure 2. Energy profile for the splitting of N2. Black dots are the calculated energies for fixed N−N geometries on a number of fcc(211) surfaces and the black lines are the best linear fits to the points. The red line shows the energy profile for the gas phase stretch of spin restricted N2 and the green plane defines a cut along Ead(N) = 0 showing the continuous variation in the offset ξ as a function of dNN. The blue curve shows the energy profile of the transition state for N2 splitting and its projection shows how rTS behaves as a function of Ead(N).

N−N bond distances are shown in black in Figure 2. The potential energy curves from Figure 1 correspond to E(dNN,Ead(N)) at fixed Ead(N), a cut of the surface parallel to the dNN axis. For each fixed energy descriptor, ε, there is a certain structure such that rTS = rTS(ε), as indicated in Figure 2. Inserting this into eq 3, we obtain E TS(ε) = γ(ε) ·ε + ξ(ε)

Figure 3. Lower figure shows the contributions to ETS, ξ and the absolute/negative value of (γ × Ead) (the reactant-like transition state has been chosen as reference). The upper figure shows the first and second derivative of ETS with respect to ε.

(4)

The canceling of the last terms in eq 6 follows from eq 5 and provides an interpretation of the scaling parameters γ and ξ. Since the descriptor is chosen relative to atoms in the gas phase, ξ(r) corresponds to a reaction mediated by a surface that is noninteracting to weakly repulsive. In this case, ξ(r) is an approximation to the activation strain of a reaction often discussed in energy-decomposition schemes.23−25 Equation 6 means that to first order in εi, the resulting approximate strain of the molecule, δξ, is canceled by a changed reactivity Δγ. This can be understood as bond-order conservation, where loss of bonding internally in the adsorbate is compensated by bonding to the surface. The lower panel of Figure 3 shows how the two quantities ξ and (γ × εad) in eq 6 behave as a function of the chosen descriptor for N2-dissociation. Even though both curves have a nonsimple behavior as a function of the descriptor they are seen to be correlated, such that the requirement in eq 6 is fulfilled. Equation 6 gives us the first derivatives of ETS but says nothing about the curvature other than implying that ETS has to bend to connect the initial state like and final state like slope γ. Since our formulation of transition state energy scaling is based on linear energy scaling at each geometry E(r, ε), curvature of ETS(ε) can only arise from changes in geometry. To get an expression for the curvature of ETS, we estimate the change in transition state structure with the Newton step (δr = −h−1g). Here, g and h are the gradient and the Hessian eigenvalue along the reaction coordinate at the given value of ε and r. Starting from a transition state at a given descriptor ε and corresponding transition state geometry rTS(ε), an infinitesimal change of the descriptor δε at fixed geometry will lead to a finite gradient:

Using the (in general, unknown) function rTS(ε), we have thus transformed a set of geometry dependent scaling relations into a nonlinear transition state scaling relation in energy. We will now continue with the analysis of some general properties of eq 4 which we again illustrate for N2-dissociation. For N2 dissociation, ETS(ε) is shown in the lower panel of Figure 3. On very reactive surfaces, the reaction eventually becomes downhill in energy with no transition state and for very noble surfaces the reaction becomes entirely uphill in energy. Therefore, a transition state exists only within a certain region of the descriptor space, ε. At the boundaries of this region, the transition state (a first-order saddle point) and reactant or product (a minimum) merge into a single inflection point that is characterized by a zero Hessian eigenvalue along the reaction coordinate. To calculate the slope of the transition scaling relations, we differentiate eq 3: d E(rTS(ε), ε) = γi(rTS(ε)) + dεi

⎛ ∂E(r , ε) ⎞ ⎜ ⎟ ⎝ ∂r ⎠r = r (ε) TS =0

×

∂rTS(ε) ∂εi

(5)

Here we have used that the transition state is a stationary point so that its gradient vanishes. Since the same must hold if we differentiate eq 4, we obtain dE TS(ε) ∂ξ(ε) ∂γ(ε) + = γi(ε) + ε· ∂ε ∂ε dεi ii =0

∂E(rTS(ε), ε + δε) d γ (r ) = ·δε ∂r dr

(6) 10450

(7)

DOI: 10.1021/acs.jpcc.5b02055 J. Phys. Chem. C 2015, 119, 10448−10453

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The Journal of Physical Chemistry C Within the quadratic approximation of the energy in r, the change in transition state geometry and energy is therefore given by δrTS ≈ −h−1

d γ (r ) ·δε dr

⎛ dγ(r )) ⎞2 δE TS ≈ −0.5h−1⎜δε· ⎟ ⎝ dr ⎠

(8)

(9)

We now obtain the curvature of ETS by differentiating eq 9 with respect to the descriptor: d2E TS(ε) dεi2

⎡ d γ (r ) ⎤ 2 ≈ −h−1⎢ i ⎥ ⎣ dr ⎦

(10)

This expression agrees with the numerical second derivative shown in the upper panel of Figure 3 to numerical accuracy and is indistinguishable when plotted in the same graph. Equation 10 tells us that the curvature along one descriptor is always positive since h < 0 and diverges when approaching the inflection points, where h = 0, which is exactly what one sees in Figure 3. The change in the transition state coordinate, δrTS, resulting from a change of the descriptor δεi will be largest if the potential energy surface is flat, when |h| is small. Generally, one would therefore expect ΔETS to exhibit least curvature in the middle region. The energy change depends quadratically on how γ(r) changes with the reaction coordinate. For N2 and similar bond dissociations, the transition state with final state like structure scales almost like the dissociated molecule, which means that (dγi(r))/(dr) is small. This leads to a smaller curvature of ΔETS toward the upper inflection point than one might expect. The transition state with initial state like structure, however, merges into a minimum that has N2 significantly stretched resulting in a higher (dγi(r))/(dr). Consequently, in such a case, significant curvature is only expected for transition states with strong initial state character, as one observes in the upper panel in Figure 3. According to the Sabatier principle,26 the best catalyst is typically neither a very reactive nor a very unreactive surface. Therefore, we expect the transition states of the optimal catalysts to lie in the region where ΔETS has little curvature and linear scaling works well. This is illustrated in a simple activity volcano for ammonia formation (Figure 4), where we assume competition between rate-limiting N2-dissociation and formation of ammonia from adsorbed N atoms. In order to see how general our observations on N2 are, we have modified γ(r) and ξ(r) independently. The results show that even strong perturbations that change the character of the reactivity of N2 significantly do not lead to dramatic deviations from linear scaling (see Supporting Information for details). Also, deviations from linearity are the smallest when γ(r) and ξ(r) are correlated as in N2 so that increased strain is typically accompanied by increased reactivity. We expect this condition to be fulfilled for most surface reactions of interest, because it is generally a necessity for efficient catalysis. If the adsorbate, while undergoing its surface-mediated reaction, did not become more reactive toward the surface in the transition state, then the surface would be no catalyst for this reaction. So far we have discussed general properties of transition state scaling relations primarily for the case of N2, because it can be described with only one descriptor and is therefore easier to visualize. We have reached the same conclusions when applying

Figure 4. Plot of (negative) free activation barrier for N2 dissociation and free energy of ammonia relative to adsorbed atomic nitrogen at p = 200 bar and T = 700 K.

our general scheme to CO and NO dissociation which require two descriptors, one for C/N and one for O.27 These results will now be discussed along with N2 to quantify how the prediction of transition state energies based on few reference calculations is affected by the geometry dependence of transition states. To this end, we will compare the classical linear transition state fitting approach with a model we have developed for predicting transition state energies and structures based on few data points on the potential energy surface. For this model it is always assumed that reactant- and product-like scaling information is available since for the evaluation of rate constants it is required, too. The challenge is therefore to use as little data in the transition state region as possible. There are two ways to improve the efficiency of fitting the linear scaling coefficients of the energy curve E(r). First, one can try to reduce the number of geometries, r, required to allow for a reasonable representation of γ(r) and ξ(r). Second, one can reduce the number of surfaces required to obtain a scaling relation for a given r. An approach to the first is to fit the adsorption energy of the structures, Ead(r), rather than the reaction energy. As shown in Figure 5, subtracting the gasphase energy, Ead(r) = E(r) − Egas(r), gives a well-behaved monotonous curve. Ead(r) can be interpolated with fewer points than E(r). Interpolation using only five points gives quantitative agreement. If only one point in the inner region (besides reactant and product) is used, the resulting curve is still qualitatively correct and the deviation is only about 0.05 eV. It is important to note that Egas(r) can be computed using spinrestricted DFT, since the only requirement is a resulting wellbehaved adsorption energy. Using one more approach to reduce the required number of surfaces (see Supporting Information), the proposed model requires the following input: (i) Scaling relations for reactant- and product-like structures that bracket the transition state space • Example N2-dissociation: This requires the calculation of adsorption energies of N and N2 on a sufficiently large number of surfaces. One can then obtain scaling relations for initial Ead(N2) and final state Ead(2N) using the descriptor Ead(N). 10451

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Table 1. Table Shows the Average of the MAE in eV That Can Be Obtained Using n of N fcc(211) Surfacesa linear TS-fit N2 CO NO

r scaling

BEP

N

n=3

n=4

n=N

n=2

n=3

n=N

14 19 11

0.15 0.23 0.31

0.14 0.19 0.26

0.11 0.12 0.17

0.14 0.21 0.27

0.12 0.20 0.26

0.12 0.20

a

The adsorption energy at the upper terrace hcp sites is used as the descriptor. The linear transition state fit requires explicit calculation of n transition states. For r-dependent scaling, initial and final state scaling parameters, as well as a single energy on one or two surface for a fixed geometry in the transition state region are required. Bond distances for initial and final state scaling are r = 1.25 and 2.60 Å, respectively and r = 1.85 Å for the near transition state.

Figure 5. Energy profile of the surface mediated dissociation of the N−N bond over an Fe fcc(211) step. The black points are calculated and the decomposition into gas phase and adsorption energy is shown. Indirect interpolation of E(r) based on interpolation of the adsorption energy Ead(r) is shown using 9, 3, and 1 points in addition to initial and final point.

(ii) At least one point in the transition state region for at least as many surfaces as there are descriptors (typically one or two) • Example N2-dissociation: A single constrained optimization with dNN = 1.85 Å at the appropriate site for one surface is sufficient. (iii) Gas-phase energy curve for the reaction, with sufficient points (≈10) • Example N2-dissociation: The gas-phase energy of N2 as a function of dNN= {1.2, ..., 2.6} Å is required. The above data provides the adsorption energy Ead(r) along the reaction path at the initial, final and one intermediate point for one or two surfaces. As described before and depicted in Figure 5, this is sufficient to interpolate Ead(r) for these surfaces. Together with the interpolated gas-phase energy, one can now obtain a continuous interpolation of the energy curve E(r) = Ead(r) + Egas(r). Scaling parameters γ(r) and ξ(r) as a function of r are now fitted such that (1) E(r) is reproduced for the reference surfaces and (2) the scaling parameters vary monotonously between their initial and final value. Numerically, this fit is carried out on a grid, both in r- and in descriptor space. Interpolation gives a continuous set of descriptors {γ(r), ξ(r)}, which in turn give a continuous energy curve E(r) for any given descriptor ε. Transition state energies for a given descriptor ε are easily determined numerically as local maxima in the one-dimensional reaction coordinate space ETS(ε) = maxr[E(ε,r)]. We will now evaluate the performance of different models to predict transition state energies by selecting n representative surfaces out of all N computed surfaces for the dissociation of N2, CO, and NO. The predictions are then evaluated by the mean absolute error (MAE) of the N transition state predictions. To discuss the quality of a model, the mean of these (Nn ) individual MAEs from the different possible combinations of surfaces is used. This allows us to estimate how reliable a model on average is in predicting the N transition state energies of our test set employing only n reference surfaces. Overall, the accuracy of a classical linear fit and the geometry-dependent prediction are rather similar (Table 1). If only three or less surfaces are provided, fitting can lead to large errors relative to individual points. The predictions of the linear fit depend more strongly on the specific choice of reference surfaces for N2 and NO and are similar to r-dependent scaling for CO. This can be seen from the distribution of the MAEs shown in Figure 6.

Figure 6. Distribution of mean absolute errors (MAE) of the prediction of transition state energies for N2, CO, and NO (left to right) using linear scaling (top) and r-dependent scaling (bottom). The columns represent the percentage of individual predictions within 0.01 eV blocks using more (blue) or less (green) reference surfaces: 3(4) for linear scaling and 2(3) for r-dependent scaling. The vertical lines show the corresponding mean value of the MAEs as in Table 1

If more explicitly computed transition states are provided, the linear fit eventually performs better than our simpler model, which relies only on one or two calculation at any point in the vicinity of the transition state.

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SUMMARY AND CONCLUSION We have established a connection between linear adsorbateand transition state scaling. The adsorbate scaling parameters vary smoothly through the transition state region from reactant to product, corroborating their interpretation as a measure of bond-order. This allows the prediction of transition state structure and energy based on the geometry-dependence of the scaling parameters, γ(r) and ξ(r). A predictive model has been described that mainly reproduces classical linear transition state scaling. Analytically, we have found general constraints on the functional form of scaling relations, ETS(ε). To first order in the descriptor, ε, effects on the transition state energy resulting from change of the transition state geometry cancel. This can be interpreted as bond-order conservation where loss of intramolecular bonding is compensated by adsorbate−surface bonding. In general, the curvature of ETS(ε) is positive and is expected to be small in the intermediate ε-range that is neither very close to being completely up- or downhill. This region is where, according to the Sabatier principle, we expect the best catalysts.26 For N2 dissociation and related reactions, we predict 10452

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the highest deviation from linearity for very low activation barriers, which do generally not factor into the overall rate of catalysis. This justifies the use of linear scaling relations for transition state energies in the interpretation of the kinetics of chemical reactions.

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ASSOCIATED CONTENT

S Supporting Information *

DFT-computed data and a more detailed derivation of the theory and methods in this text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from the Office of Basic Energy Sciences of the U.S. Department of Energy to the SUNCAT Center for Interface Science and Catalysis at SLAC/Stanford is gratefully acknowledged.

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DOI: 10.1021/acs.jpcc.5b02055 J. Phys. Chem. C 2015, 119, 10448−10453