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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
(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.
Materials and Methods
Figs. S1 to S6
Tables S1 and S2
4 April 2014; accepted 29 May 2014
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
With the surge in density functional the- ory (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 pro-
vide a measure of the expected accuracy of direct-
ly calculated properties such as bond strengths,
bond lengths, or activation energies of elemen-
tary processes. The question is how such intrinsic
uncertainties in calculated microscopic proper-
ties transform into error bars on calculated com-
plex properties, defined here as properties that
depend on several microscopic properties in a
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-vd W), a
recent exchange-correlation density function-
al tailored for surface chemistry with built-in
error estimation capabilities (11). The functional
was fitted to describe several different proper-
ties, including common adsorbate-surface bond
strengths, and an ensemble of density functionals
around BEEF-vd W 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. Uncer-
tainties 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
the error estimate on the BEEF-vd W result, sBEE ¼
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 post-DFT 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
1SUNCAT 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: firstname.lastname@example.org