Assessing the reliability of calculated catalytic ammonia synthesis rates Andrew J. Medford et al. Science 345, 197 (2014); DOI: 10.1126/science.1253486

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20. H. Hori, T. Yonetani, J. Biol. Chem. 260, 349–355 (1985). 21. S. L. Edwards, H. X. Nguyen, R. C. Hamlin, J. Kraut, Biochemistry 26, 1503–1511 (1987). 22. A. E. Pond, G. S. Bruce, A. M. English, M. Sono, J. H. Dawson, Inorg. Chim. Acta 275–276, 250–255 (1998). 23. J. E. Erman, T. Yonetani, Biochim. Biophys. Acta 393, 350–357 (1975). 24. J. E. Huyett et al., J. Am. Chem. Soc. 117, 9033–9041 (1995). 25. Y. T. Meharenna, T. Doukov, H. Li, S. M. Soltis, T. L. Poulos, Biochemistry 49, 2984–2986 (2010). 26. A. J. Sitter, C. M. Reczek, J. Terner, J. Biol. Chem. 260, 7515–7522 (1985). 27. H. B. Dunford, Peroxidases and Catalases: Biochemistry, Biophysics, Biotechnology, and Physiology (Wiley, Chichester, UK, ed. 2, 2010). 28. E. Derat, S. Shaik, J. Phys. Chem. B 110, 10526–10533 (2006). 29. S. Shaik, D. Kumar, S. P. de Visser, A. Altun, W. Thiel, Chem. Rev. 105, 2279–2328 (2005). 30. I. Efimov et al., J. Am. Chem. Soc. 133, 15376–15383 (2011).

31. T. L. Poulos, J. Kraut, J. Biol. Chem. 255, 8199–8205 (1980). 32. T. H. Yosca et al., Science 342, 825–829 (2013). 33. J. E. Erman, L. B. Vitello, M. A. Miller, J. Kraut, J. Am. Chem. Soc. 114, 6592–6593 (1992). 34. P. Vidossich et al., J. Phys. Chem. B 114, 5161–5169 (2010). ACKN OWLED GMEN TS

We thank D. Collison for assistance with EPR, P. Ortiz de Montellano for discussions on the mechanisms, S. Fisher for assistance with the joint x-ray/neutron refinement, and the University of Leicester BioMedical Workshop for custom building of microspectrophotometry equipment, and J. Devos and the D-lab for experimental support. Atomic coordinates have been deposited in the Protein Data Bank under accession codes 4CVI for ferric CcP and 4CVJ for compound I. This work was supported by The Leverhulme Trust (grant F/00 212/Q to E.L.R./P.C.E.M.), Biotechnology and Biological Sciences Research Council (grant BB/C001184/1 to E.L.R./P.C.E.M., and a studentship to E.J.M.), The Wellcome Trust (grant WT094104MA to P.C.E.M./E.L.R.), an Institut Laue-Langevin studentship (to C.M.C.) and beam time at LADI-III and BIODIFF

CATALYSIS

Assessing the reliability of calculated catalytic ammonia synthesis rates Andrew J. Medford,1,2 Jess Wellendorff,1,2 Aleksandra Vojvodic,1 Felix Studt,1 Frank Abild-Pedersen,1 Karsten W. Jacobsen,3 Thomas Bligaard,1 Jens K. Nørskov1,2* We introduce a general method for estimating the uncertainty in calculated materials properties based on density functional theory calculations. We illustrate the approach for a calculation of the catalytic rate of ammonia synthesis over a range of transition-metal catalysts. The correlation between errors in density functional theory calculations is shown to play an important role in reducing the predicted error on calculated rates. Uncertainties depend strongly on reaction conditions and catalyst material, and the relative rates between different catalysts are considerably better described than the absolute rates. We introduce an approach for incorporating uncertainty when searching for improved catalysts by evaluating the probability that a given catalyst is better than a known standard.

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ith the surge in density functional theory (DFT) calculations of chemical and materials properties, the question of the reliability of calculated results becomes increasingly urgent (1), particularly when calculations are used to make predictions of new materials with interesting functionality (2–5). Evaluating the reliability of DFT calculations has relied mainly on comparisons to experiments or to data sets of higher-level calculations to provide a measure of the expected accuracy of directly calculated properties such as bond strengths, bond lengths, or activation energies of elementary processes. The question is how such intrinsic uncertainties in calculated microscopic properties transform into error bars on calculated complex properties, defined here as properties that depend on several microscopic properties in a 1

SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA. 2SUNCAT Center for Interface Science and Catalysis, Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA. 3Center for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark. *Corresponding author. E-mail: [email protected]

SCIENCE sciencemag.org

complex way (6). Examples of such properties include mechanical strength (7), phase stability (8), and catalytic reaction rates (5). We estimate the reliability of DFT energies by choosing an ensemble of exchange-correlation functionals to represent the known computational errors for a set of adsorption energies (9–11). This ensemble of energies is used to calculate the rates of the ammonia synthesis reaction via microkinetic modeling. We choose this process because it is well described both experimentally and theoretically (12–16) and has enough complexity to bring out important aspects of error propagation through multiple layers of simulation. This approach directly captures correlations between systematic errors in the underlying energetics, revealing that uncertainties on the calculated rates exhibit a nontrivial dependence on the reaction conditions as well as the material and that trends in catalytic activity are considerably better described than the absolute rates. To calculate energies and estimated errors, we apply the Bayesian error estimation functional with van der Waals correlation (BEEF-vdW), a recent exchange-correlation density function-

(EU FP7 NMI3-II grant 283883), Bruker UK (Sponsorship of A.J.F. and M.G.C.), and beam time at LADI-III and BIODIFF. P.C.E.M., M.P.B., and E.L.R. designed the research; C.M.C., A.G., M.P.B., A.O., S.C.M.T, T.E.S., C.L.M., E.J.M., and P.C.E.M. performed crystallographic experiments; C.M.C., A.G., M.P.B., S.C.M.T., A.O., T.E.S., E.L.R., and P.C.E.M. analyzed crystallographic data; C.M.C., A.J.F., M.G.C., J.B., and P.C.E.M. performed spectroscopic experiments; C.M.C., A.G., A.J.F., M.G.C., E.L.R., and P.C.E.M. analyzed spectra; and E.L.R., M.P.B., and P.C.E.M. wrote the paper, with contributions from all authors. SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/345/6193/193/suppl/DC1 Materials and Methods Figs. S1 to S6 Tables S1 and S2 References (35–48) 4 April 2014; accepted 29 May 2014 10.1126/science.1254398

al tailored for surface chemistry with built-in error estimation capabilities (11). The functional was fitted to describe several different properties, including common adsorbate-surface bond strengths, and an ensemble of density functionals around BEEF-vdW was generated. This Bayesian error estimation (BEE) ensemble was designed to reproduce known energetic errors by mapping them to uncertainties on the exchange-correlation model parameters. Figure 1 illustrates this. Uncertainties on new calculations may then be estimated by mapping back again: Random sampling of a probability distribution for fluctuations of the model parameters leads to a large ensemble of different predictions of the same quantity. The statistical variance of those predictions defines theffiffiffiffiffiffiffiffiffiffiffiffiffiffi error estimate on the BEEF-vdW result, sBEE ¼ p Varð→ p Þ, where the ensemble predictions are stacked in vector → p . Further details are provided in (11, 17). This approach to quantitative error estimation in DFT can be viewed as a structured analysis of the sensitivity of DFT results to the choice of exchange-correlation approximation. An appropriately designed ensemble also captures correlated variations between DFT total energies and offers a consistent approach to keeping track of possible sources of error when data from multiple calculations are folded in composite postDFT frameworks, such as microkinetic models used to analyze catalytic reactions. For the ammonia synthesis reaction, microkinetic models provide the link between the calculated microscopic properties and the reaction rate or turnover frequency (TOF). Here, we use a relatively simple kinetic model based on N2 dissociation as the rate-limiting step, following the mechanism described by Honkala et al. (16). This model has previously been shown to capture the experimentally observed trends in catalytic activity for different catalysts (18); details of the model can be found in (17, 18). We first consider in some detail the rates of ammonia synthesis over stepped Fe and Ru surfaces, which are the industry-standard catalysts (14, 19, 20). The calculated ammonia synthesis rate over iron per active (step) site (the TOF) is shown in Fig. 2A as an Arrhenius plot at industrial conditions. The red shaded area indicates that the estimated 11 JULY 2014 • VOL 345 ISSUE 6193

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error in TOF is generally less than one order of magnitude, although there is a nontrivial dependence on reaction conditions that results in larger uncertainties at extreme temperatures. We also investigated the ammonia synthesis rate over a stepped Ru surface for which there are detailed experiments for comparison (Fig. 2B). In this case, the magnitude of the prediction uncertainty is somewhat greater, around two orders of magnitude, and again varies with reaction conditions. The calculated apparent activation barrier (17) and N2 dissociation barrier are compared with experimental results in the inset of Fig. 2B; the experimental value is within the uncertainty of the calculation, corroborating the surface and kinetic models employed. One interesting feature of error propagation through the kinetic model is the cancellation of errors arising from correlation between model inputs. To assess this effect, we compared the uncertainty inferred by the BEE ensembles (red) to “uncorrelated” ensembles (gray) in Fig. 2 [see (17) for details]. The uncorrelated ensembles substantially overestimate the error at low temperatures because the TOF varies monotonically with the product of the rate constant (k) and the free site coverage (q* ). The rate constant varies exponentially with the activation energy and, in regimes of intermediate adsorbate coverage, q* will be very sensitive to variations in the energy of the most abundant surface intermediate. A positive covariance between the transition-state energy and the energies of all intermediates (see table S2) implies that an exchange-correlation functional that predicts a strong binding of the surface-bound species (lower q* ) will also yield a very stable transition state (higher k), hence the cancellation of possible errors. This reasoning is general to other surfaces, reactions, and kinetic models, although the magnitude of the effect will depend on the details of the system. Next, we compare catalytic rates and error estimates for several different metal catalysts calculated using the chosen kinetic model (see Fig. 3A). As previously indicated, the uncertainty on the TOF depends considerably on the catalyst. For the unreactive metals, the uncertainty is large, whereas it is considerably smaller for the most active catalysts, Ru and, in particular, Fe. Furthermore, Fig. 3B that  demonstrates  TOF and relative error bars on relative TOFs TOF@Fe nitrogen adsorption energies (EN – EN@Fe) are substantially smaller than on the absolute materials properties. This decrease in uncertainty is the result of the correlation of errors between various transition-metal surfaces (see table S3). Exchangecorrelation functionals that bind a given intermediate or transition state stronger on one metal are likely to bind it stronger on another, and the BEEF-vdW exchange-correlation ensemble captures this effect quantitatively. Figure 3B demonstrates that we can make statistically significant predictions about the relative ammonia synthesis rates on different metals. The predictions are in agreement with experimental data showing that Ru and Fe are the most active catalysts for ammonia synthesis at these conditions (14, 21). 198

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Fig. 1. BEEF-vdW functional and ensemble. BEEF-vdW prediction errors for 17 experimental adsorption energies (blue horizontal lines) and distribution of adsorption energy perturbations (shaded histogram) that result from applying perturbed exchange-correlation functionals from the BEEF-vdW ensemble to those systems. The inset illustrates the exchange part of the perturbed functionals (red lines) in terms of exchange enhancement factor, Fx, as a function of the reduced electron density gradient, s. The BEEF-vdW Fx is shown in blue. Fluctuations within the ensemble of functionals are scaled such that the variance of the resulting energy perturbations, s2, reproduces the average square prediction error realized by BEEF-vdW. The original ensemble from (11) reproduces errors on several properties.The present work concerns surface chemistry, so the fluctuations are rescaled to reflect the expected prediction error for surface adsorption energies only [see (17)].

A

B

Fig. 2. Simulated Arrhenius plots. Arrhenius plot for ammonia synthesis on Fe(211) (A) and Ru(0001) (B) step site as calculated with BEEF-vdW and the described microkinetic model at industrial conditions (100 bar, 50% approach to equilibrium).The shaded red area corresponds to error estimates, and the gray area corresponds to an uncorrelated model [see (17)]. In the inset of (B), the N2 dissociation barrier is plotted against the apparent Arrhenius activation barrier for the Ru(0001) step site under conditions comparable to (15). The principal component ellipses [see (17)] of the correlated (red) and uncorrelated (gray) ensembles are shown, and the predictions are compared to experimental measurements taken from (15, 19) (black error bars).

The nitrogen adsorption energy at the active site is a good descriptor of ammonia synthesis activity due to a set of linear scaling relations

between the adsorption energy of nitrogen and the adsorption energies of the other intermediates, H, NH, NH2, and the N–N transition state sciencemag.org SCIENCE

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Fig. 3. Ammonia synthesis rate trends. Absolute TOFs on metallic step sites of selected metals plotted as a function of nitrogen adsorption energy (A) and TOFs divided by the TOF of Fe as a function of nitrogen adsorption energy relative to Fe (B). The rate activity map obtained from linear scaling relations between the adsorption energy of nitrogen and other intermediates is shown in (C), where the red line/area corresponds to the uncertainty on the absolute/relative rate, respectively. Linear regression between the dissociative adsorption energy of N2 (2EN) and the N–N transition-state energy is shown in the inset, along with principal component ellipses of the metal ensembles [see (17)].The probability that a rate volcano has an optimum at a given nitrogen adsorption energy is presented in (D). The temperature is 673 K, total pressure is 100 bar, and approach to equilibrium is fixed at 10%.

A

B

C

D

(18, 22–24). Given these relations, the nitrogen adsorption energy determines all other energies in the problem (to within the accuracy of the scaling relations), and the rate can be calculated as a function of nitrogen adsorption energy. In Fig. 3C, we apply such an approach to each density functional in the BEEF-vdW ensemble to evaluate the consequence of an imperfect representation of exchange-correlation effects on a descriptor-based model of surface chemistry. The linear regression parameters between the nitrogen adsorption energy and the N–N transition state exhibit much smaller standard deviations than the underlying energies used to construct them (Fig. 3C, inset). The lower uncertainty on scaling parameters results from cancellation of error due to correlated energetics (tables S2 and S3) and is general to the other intermediates in the system, as shown in fig. S2. The resulting ensemble of “rate maps” (Fig. 3C) shows that the “volcano”-type relation between rate and nitrogen adsorption energy is well determined by the SCIENCE sciencemag.org

Fig. 4. Rate probability map. The probability that the ammonia synthesis TOF of a given catalyst is higher than the TOF of iron step sites (calculated with the self-consistent BEEF-vdW energetics) as a function of nitrogen adsorption energy relative to that of iron. Reaction conditions are given in Fig. 3. The line corresponds to linear regression between the adsorption energy of nitrogen and other intermediates as described in Fig. 3C and fig. S2.

calculations. Comparing Fig. 3, B and C, it is also evident that the calculated rates for individual metals agree with the descriptor-based model to within the BEE uncertainty. The descriptor-based analysis provides a simple method for computational design of new catalysts (3, 5). Once a set of descriptors is identified and a mapping between the descriptors and the catalytic activity has been constructed, the search for new catalysts is reduced to finding new materials with descriptor values near the optimum. An essential question is how well the position of the maximum in such activity maps is determined (25). Figure 3D shows the relative probability that the maximum of the ammonia synthesis rate occurs at a given nitrogen adsorption energy. The position of the maximum is –0.58 eV, with a standard deviation of 0.08 eV. This uncertainty in the predicted optimum is considerably smaller than the >2 eV range spanned by the nitrogen adsorption energies of the metals included in this study. The optimum descriptor

value is therefore precisely determined relative to the energetic differences between transitionmetal catalysts. In computational materials design, the magnitude of the uncertainty associated with a given predicted property is of less importance than the confidence with which one can evaluate a hypothesis. For ammonia synthesis catalysts, for instance, we could look for materials that exhibit higher TOFs than the industry-standard iron catalyst. Our analysis allows an evaluation of the probability that a given catalyst has a TOF greater than that of iron. The results in Fig. 4 provide support to the notion that despite uncertainties of several orders of magnitude in absolute values of predicted TOFs, it is possible to draw strong scientific conclusions about which new materials are promising for ammonia synthesis. The probabilities shown in Fig. 4 reflect only the expected magnitude of uncertainty on energetics from exchange-correlation effects and neglect all other possible sources of error, such as the surface model, reaction mechanism, and descriptor-based dimensionality reduction. Nonetheless, the present analysis indicates an upper bound to the reliability of theoretical predictions for ammonia synthesis TOFs, and the probabilistic approach provides a convenient framework for including additional sources of uncertainty. REFERENCES AND NOTES

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K. Burke, J. Chem. Phys. 136, 150901 (2012). G. Ceder et al., Nature 392, 694–696 (1998). S. Curtarolo et al., Nat. Mater. 12, 191–201 (2013). L. Yu, A. Zunger, Phys. Rev. Lett. 108, 068701 (2012). J. K. Nørskov, T. Bligaard, J. Rossmeisl, C. H. Christensen, Nat. Chem. 1, 37–46 (2009). 6. B. B. Machta, R. Chachra, M. K. Transtrum, J. P. Sethna, Science 342, 604–607 (2013).

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Data reported herein are available in the supplementary materials. Support from the U.S. Department of Energy Office of Basic Energy

Ultrafast low-energy electron diffraction in transmission resolves polymer/ graphene superstructure dynamics Max Gulde,1 Simon Schweda,1 Gero Storeck,1 Manisankar Maiti,1 Hak Ki Yu,2 Alec M. Wodtke,2,3 Sascha Schäfer,1 Claus Ropers1* Two-dimensional systems such as surfaces and molecular monolayers exhibit a multitude of intriguing phases and complex transitions. Ultrafast structural probing of such systems offers direct time-domain information on internal interactions and couplings to a substrate or bulk support. We have developed ultrafast low-energy electron diffraction and investigate in transmission the structural relaxation in a polymer/graphene bilayer system excited out of equilibrium. The laser-pump/electron-probe scheme resolves the ultrafast melting of a polymer superstructure consisting of folded-chain crystals registered to a free-standing graphene substrate. We extract the time scales of energy transfer across the bilayer interface, the loss of superstructure order, and the appearance of an amorphous phase with short-range correlations. The high surface sensitivity makes this experimental approach suitable for numerous problems in ultrafast surface science.

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1 4th Physical Institute, University of Göttingen, 37077 Göttingen, Germany. 2Max Planck Institute for Biophysical Chemistry, 37077 Göttingen, Germany. 3Institute for Physical Chemistry, University of Göttingen, 37077 Göttingen, Germany.

*Corresponding author. E-mail: [email protected]

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SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/345/6193/197/suppl/DC1 Materials and Methods Figs. S1 and S2 Tables S1 to S3 References (26–28)

ACKN OWLED GMEN TS

IMAGING TECHNIQUES

he investigation of atomic-scale dynamics with high spatiotemporal resolution yields insights into ultrafast structural reorganizations associated with energy transfer or phase transitions. Substantial progress was made in establishing methods for the timeresolved structural analysis of bulk media, including ultrafast implementations of x-ray crystallography (1–3), high-energy electron diffraction (4–6), and microscopy (7–9), as well as time-resolved x-ray and electron spectroscopy (10, 11). In contrast, structural dynamics at surfaces, interfaces, and ultrathin films remain largely elusive, as the surface signal in both x-ray and high-energy electron diffraction is typically masked by large bulk contributions. This limits our ability to study quasi–two-dimensional (2D) systems exhibiting characteristic phase transitions and topologically controlled ordering (12, 13), as well as the dy-

Science to the SUNCAT Center for Interface Science and Catalysis is gratefully acknowledged. A.J.M. is grateful for support by the U.S. Department of Defense through the National Defense Science and Engineering Graduate Fellowship Program.

namics of surface reconstructions and complex adsorbate superstructures (14, 15). Ultrafast electron scattering in grazing incidence enhances the surface signal (14, 16) but faces particular challenges in quantitative diffraction analysis. Optimal surface sensitivity would be attained with an ultrafast implementation of low-energy electron diffraction (LEED). At electron energies of tens to a few hundreds of electron volts, scattering cross sections are strongly increased, which allows for probing depths of only a few monolayers and has made LEED a widely used tool for surface structure determination. However, at such low energies, it has proven exceedingly difficult to implement pulsed electron sources that fulfill the requirements of an ultrafast diffraction experiment (17–19), that is, short pulse duration and low beam emittance. Laser-triggered electron emission from nanoscale photocathodes (20, 21) is expected to resolve some of these issues (22–24), providing well-collimated low-energy electron pulses and a temporal resolution that is comparable to state-of-the-art ultrafast x-ray or high-energy electron diffraction. Motivated by these prospects, we have undertaken the development of a new diffraction apparatus.

17 March 2014; accepted 2 June 2014 10.1126/science.1253486

We have developed transmission ultrafast LEED (T-ULEED) based on a nanometric needle photocathode and demonstrate its capability to resolve atomic-scale structural dynamics of surfaces and monolayer films with a temporal resolution of a few picoseconds. Specifically, we studied the ultrafast laser-driven dynamics of a polymer superstructure on freestanding monolayer graphene. In the laser-pump/electron-probe scheme (Fig. 1A), the sample is excited out of equilibrium by amplified femtosecond laser pulses (800 nm wavelength, 80-fs pulse duration, repetition rate 10 kHz, focal diameter about 100 mm). To minimize hot electron emission from graphene (25), the pump pulse is temporally stretched to 3 ps by dispersion, which, however, is still sufficiently short to resolve the processes described below. The pump-induced structural dynamics are probed by ultrashort electron pulses emitted from a sharp tungsten tip (50-nm radius of curvature), triggered by the second harmonic of the laser. These electron pulses (up to 100 electrons per pulse) are collimated and focused onto the sample at variable electron energies using an electrostatic lens assembly in a geometry that we have recently studied numerically (22). Scattered electrons are subsequently recorded in a transmission geometry by a phosphor screen microchannel plate detector (MCP, Hamamatsu F2226-24P). With our laser-triggered low-energy electron source, diffraction patterns can also be recorded in backreflection and for a range of electron energies, as demonstrated in the supplementary materials (26) (figs. S6 to S8). The electron pulse duration and the spatial and temporal overlap (delay time, Dt = 0) of the laser-pump and electron-probe pulses are determined via a transient-electric-field effect near a bare transmission electron microscopy (TEM) copper grid. Upon excitation of a single copper grid bar with high peak intensity (fluence up to 30 mJ/cm2, unstretched pump pulses), a dense electron cloud is emitted (25, 27), which may lead to a spatial deflection of the passing electron pulse (Fig. 2A). Projection images of the grid before and after Dt = 0 are shown in Fig. 2B, acquired by defocusing the pulsed electron beam. The central distortion in the lower image indicates the extension of the pump-induced electron cloud. Using a collimated electron beam passing a single mesh sciencemag.org SCIENCE

Catalysis. Assessing the reliability of calculated catalytic ammonia synthesis rates.

We introduce a general method for estimating the uncertainty in calculated materials properties based on density functional theory calculations. We il...
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