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By Aravind Asthagiri, Michael J. Janik
The Royal Society of ChemistryCopyright © 2014 The Royal Society of Chemistry
All rights reserved.
Computational Catalyst Screening
LARS C. GRABOW
Chemical & Biomolecular Engineering, University of Houston, Houston, Texas, 77204-4004, USA
Brute-force attacks are known in cryptography as (typically illegal) attempts to hack into encrypted data by systematically trying all possible key combinations of letters, digits and special characters until the correct access key or password has been found. Although a brute-force attack is guaranteed to be successful, its application is limited to very small problems because of the time required to generate and test all possible key combinations. For example, a standard 128-bit encryption key has 2128 possible permutations. If we simply assume that a typical central processing unit (CPU) can generate 109 bit flips per second (~1 GHz), then the total time that is required to test all possible permutations is 2128/109 = 3.4×1029 seconds or 1022 years! For obvious reasons a brute-force attack is most likely going to fail for this problem and a more targeted strategy is needed.
The above example illustrates the shortcomings of a brute-force attack, but a variation of it is still one of the most widely used strategies for the development of heterogeneous catalysts in practice. By using a combinatorial chemistry approach with completely automated, high-throughput experimentation equipment, one can synthesize and test enormous libraries of catalysts for their catalytic activity for a specific reaction. A good example is the search for advanced water–gas shift catalysts, in which Yaccato et al. have synthesized over 50 000 catalysts and tested them in more than 250 000 experiments for low, medium and high temperature water–gas shift conditions. Their effort led to a proprietary noble metal catalyst that can reduce the reactor volume by an order of magnitude without increasing the reactor cost. Although this trial-and-error approach almost always leads to an acceptable catalyst, the search space is restricted by the amount of time and resources available and many, possibly far better, candidates can be missed. The quickly evolving alternative to experimental high-throughput catalyst testing is computational catalyst screening. This approach relies on the fact that the catalyst activity for many catalytic reactions is usually determined by a small number of descriptors, which can be calculated from first-principles density functional theory (DFT) simulations and stored in a large property database. Populating this property database with DFT data is the most time-consuming step in this process, but the resulting database is applicable to any reaction and only has to be generated once. With a comprehensive database in place, it becomes a very easy task to screen thousands of database records in a short amount of time to identify catalyst candidates that possess descriptor values within the optimal range for a given reaction. Although the computational screening process can still be interpreted as a brute-force attack, the complexity of the problem has been greatly reduced. Hence, the number of materials that can be screened computationally increases drastically when compared with the experimental counterpart. The list of catalysts that fall into the desired range of descriptor values may be narrowed down further by using cost, stability, environmental friendliness, or any other applicable criteria. The remaining materials can then be synthesized and experimentally tested under realistic reaction conditions. In general, not all computationally screened candidates will be good catalysts, but good catalysts will usually be included in the candidate list.
Somorjai and Li have recently reviewed the major advances in modern surface science that only became possible through the successful symbiosis of theory and surface sensitive experimental techniques. The recent literature also contains several examples where a descriptor-based approach, both theoretically and experimentally, has led to the discovery of new catalytic materials. The following list should not be understood as an exhaustive review, but is meant to serve as inspiration to the reader and to demonstrate the wide applicability of this method. Early on, Besenbacher et al. discovered graphite resistant Ni/Au alloy catalysts for steam reforming, Jacobsen et al. found an active Co/Mo alloy for ammonia synthesis by interpolation in the periodic table, and Toulhoat and Raybaud showed that the metal–sulfur bond strength can correctly predict trends in hydro-desulfurization activity on metal–sulfide catalysts. These initial successes were followed by other prominent examples that include CO-tolerant fuel cell anodes, Cu/Ag alloys as selective ethylene epoxidation catalysts, near-surface alloys for hydrogen activation and evolution, Ru/Pt core–shell particles for preferential CO oxidation, Ni/Zn alloys for the selective hydrogenation of acetylene, Sc and Y modified Pt and Pd electrodes and mixed-metal Pt monolayer catalysts for electro-chemical oxygen reduction, and the rediscovery of Pt as the most active and selective catalyst for the production of hydrogen cyanide.
1.1.1 A Walk through a Computational Catalyst Design Process: Methanation
The most comprehensive example of a success story in computational catalyst design comes from the group of Jens Nørskov, who has pioneered the descriptor-based design approach and has applied it to numerous reactions. In several publications his group has studied the methanation reaction (CO + 2H2 -> CH4 + H2O), starting from a detailed electronic structure analysis and leading to the development of a patented technical methanation catalyst based on a Fe/Ni alloy. In the beginning of any descriptor-based design study one must first answer the question: "What is the most suitable reactivity descriptor for the reaction?" This question is typically answered by thoroughly studying the underlying reaction mechanism and identifying the rate-limiting step and most abundant surface intermediates. However, intuition can sometimes replace a detailed mechanistic study and a descriptor can be found through an educated guess. In the case of the methanation reaction, CO dissociation is the most critical step in the reaction mechanism. For weakly interacting metal catalysts, the dissociation is rate limiting, whereas for strongly interacting catalysts, the surface is poisoned by adsorbed C and O atoms. This leads to the volcano curve in Figure 1.1(a), which shows the experimentally measured methanation activity as a function of the calculated CO dissociation energy. The top of the volcano corresponds to the maximum methanation activity and indicates the optimal value of the CO dissociation energy, which is the activity descriptor in this case. The next step in the catalyst design process is to screen a database of CO dissociation energies and search for catalysts with CO dissociation energies near the optimum. This screening may be combined with a cost estimation of the resulting material and can further be linked to a stability test. Figure 1.1(b and c) show pareto plots of binary transition metal alloys for which the CO dissociation energy was estimated through a simple interpolation scheme owing to the lack of an existing database. The most active catalysts, characterized by CO dissociation energies close to the optimum, lie to the left of the graph and are connected with a solid line indicating the pareto-optimal set. The pareto-optimal set of Figure 1.1(b) contains the cheapest catalysts for a given value of CO dissociation energies and, similarly, Figure 1.1(c) can be used to screen for alloy stability. Only alloys with a negative alloy formation energy are stable and their stability increases as the alloy formation energy becomes more negative. Upon careful inspection, one notices that FeNi3 is not only contained in both sets, but it is also located at the "knee" of the activity pareto-optimal set, which indicates that neighboring solutions are worse with respect to either activity or cost. Clearly, FeNi3 is a very promising catalyst candidate for the methanation reaction. This catalyst identification step concludes the theoretical design process and experimental verification of the theoretical prediction is necessary. Experimentally obtained methanation rates of Fe/Ni alloys as a function of Ni content are displayed in Figure 1.1(d) and clearly show that the computationally predicted FeNi3 alloy is significantly more active than its components. As an outcome of this tour de force in computational catalyst design, a process based on Fe/Ni alloys has been patented for the hydrogenation of carbon oxides.
In the remainder of this chapter, the background information that leads to the identification of appropriate catalyst descriptors (e.g. d-band model, scaling relationships) is reviewed and the basic strategy for successful catalyst screening using various levels of detail (e.g. Sabatier rate vs. microkinetic model) is outlined. A step-by-step illustration of the method will be given using ammonia synthesis and CO oxidation as examples. The interested reader is encouraged to work through the examples independently at her/his own pace.
1.2 Starting from the Electronic Structure
1.2.1 Density Functional Theory
Computational catalyst screening would not be possible without the existence of a theory that enables us to calculate the chemical properties of the catalyst and the reaction of interest. Fortunately, in the mid 1960s Hohenberg, Kohn and Sham published two seminal papers formulating two theorems, which led to the development of density functional theory (DFT). The contributions of Walter Kohn to the development of this theory were later honored in 1998 with the Nobel Prize in Chemistry. DFT is nowadays widely used in many different areas of science and engineering, including computational chemistry, catalysis, materials science, physics, and geology. The two theorems can be summarized as:
1. The ground state properties of a many-electron system are uniquely determined by the electron density.
2. The total energy of a system has a minimum for the ground state electron density.
DFT provides a solution to the Schrödinger equation:
[??]Ψ = EΨ (1.1)
and is in principle an exact theory, but in practice the exact formulation of the kinetic energy term for a system of interacting electrons is unknown. In the Kohn–Sham approach, the kinetic energy is therefore approximated with the kinetic energy of a system of non-interacting electrons and a correction term, EXC, which accounts for exchange and correlation effects in the interacting system. Although approximations for the description of the exchange-correlation energy must be made, DFT has the huge advantage over wave function based methods that the electron density is a function of only three spatial coordinates, while the many-body wave function for N electrons depends on 3N coordinates. Thus, DFT significantly reduces the computational intensity of the problem and enables the treatment of systems of several hundred atoms. From the electron density, n(r), all other properties of the system are determined (Theorem 1) and the total energy E is calculated using
[MATHEMATICAL EXPRESSION OMITTED] (1.2)
The first term, TKS[n(r)], is the kinetic energy of fictitious, non-interacting electrons and is obtained from the single-electron Kohn–Sham equations
[MATHEMATICAL EXPRESSION OMITTED] (1.3)
where veff is the effective field defined by the nuclei and the current electron density. The second and third term in the total energy equation describe the electrostatic electron–electron interactions (Hartree energy) and the electron– nuclei interactions, respectively. The last term, EXC, in the total energy equation depends on the unknown exchange-correlation functional, for which several approximations exist. The simplest approximation is the local density approximation (LDA), which can be derived from the case of a homogeneous electron gas and only depends on the electron density at a single point. In this case, the exchange contribution in the LDA is exact, but the correlation still has to be approximated. The LDA works remarkably well for bulk materials where the electron density varies slowly, but has insufficient accuracy for most applications in chemistry, including atoms, molecules, clusters, and surfaces.
An obvious extension to the LDA is the generalized gradient approximation (GGA), which depends not only on the local density but on the density gradient. Because the gradient correction can be implemented into a GGA functional in many different ways, there exist a variety of different GGA flavors. The most widely used GGA functionals are the Perdew–Wang 91 (PW91) and the Perdew–Burke-Ernzerhof (PBE) functional. Both GGA functionals have good accuracy for a wider range of problems than the LDA because they contain more physical information; however, they are not necessarily always better. The PBE functional was later revised by Hammer, Hansen and Nørskov (RPBE), in order to improve the accuracy of chemisorption energies of atoms and small molecules on transition metal surfaces. These GGA functionals are very good general-purpose functionals and may be used as a starting point for computational catalyst design. However, the GGA still fails for problems such as the accurate prediction of band gaps in semiconductors, systems where van der Waals interactions are dominant, or for electronic structure calculations in materials with strongly correlated electrons, where self-interaction errors can be encountered. Several improvements to the GGA have been suggested (e.g. DFT + U, DFT-D, meta-GGA, hybrid-GGA), but many of these functionals are problem specific or contain adjustable parameters that need to be fitted for each system. This empirical nature, along with the increased computational effort, renders these functionals generally unsuitable for computational catalyst screening. Work to improve XC functionals further is ongoing in the community, and in the next few years faster computers and new functionals will have a positive impact on the quality of DFT calculations.
Although DFT calculations are at the heart of computational catalysis, it is not strictly necessary to perform your own calculations for a catalyst design project. DFT calculations for many reactions have already been published and efforts are undertaken to make these data easily available to the whole catalysis community, even on mobile devices. Yes, there is an app for that! But even a non-DFT expert should understand the basic principles that underlie the theoretical results, before using them in a research project. For those readers that have a deeper interest in DFT and want to perform their own calculations, the tutorial-style book Density Functional Theory – A Practical Introduction by David S. Sholl and Janice A. Steckel is a highly recommended starting point.
1.2.2 The d-Band Model
Computational catalyst screening relies on the prediction of correct trends across different catalysts rather than the prediction of quantitative rates and selectivities for each catalyst. Understanding the origin of the observed trends in terms of the underlying electronic structure can therefore be very helpful during the screening process. For transition metal surfaces, trends in reactivity can be very well described and understood in terms of the d-band model (Figure 1.2) developed by Hammer and Nørskov.
Many of us have likely seen a schematic drawing of the bonding structure in a hydrogen molecule in one of our previous chemistry classes. Upon bond formation, the two atomic orbitals form two new molecular orbitals and they can be distinguished as a bonding and an anti-bonding orbital. In the case of a hydrogen molecule, two electrons can distribute into these orbitals and since each orbital can accommodate up to two electrons, naturally both electrons occupy the lower-lying bonding orbital. The energy that is gained by stabilizing the electrons in this process is the bond energy.
Excerpted from Computational Catalysis by Aravind Asthagiri, Michael J. Janik. Copyright © 2014 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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