Density functional theory (DFT) is widely used in catalysis studies. Though numerous successful applications to metal catalysts have been achieved, the reliability of standard generalized gradient approximation (GGA) DFT approaches for metal oxide systems is more suspect.[1-4] Metal oxides with localized, strongly correlated electrons are more difficult for DFT to accurately represent. Extensions to standard GGA-DFT are often used to model these systems. Catalytic activity on metal oxides is directly related to the ability of metal atoms to cycle between oxidation states. It is essential to develop computational methods that accurately capture energetics associated with metal oxidation state changes during metal oxide catalyzed reaction cycles.
Ceria-based catalysts are particularly difficult for conventional DFT due to the self-interaction error that emerges when modeling localized f-states in partially reduced surfaces. Recently, a review on the status of DFT+U has been published. This self-interaction error causes GGA-DFT methods to improperly delocalize f-electrons in reduced ceria systems. The most widely used method to correct this self-interaction error is the addition of the Hubbard U term to the cerium f-states. The Hubbard U-term adds a semiempirical energetic penalty to partial occupation of the orbitals to which is it applied (i.e., f-states of Ce). The empirical U-term may be chosen to match known material properties, or calculated self-consistently using the linear-response approach. The empirical approach is most typically used, however, a U-value determined to match one property (i.e., band gap) may not necessarily give proper surface reduction energetics. Self-consistent U determination provides an approach to return the full ab initio nature to DFT by determining an appropriate U value for a given physical model. However, system energies cannot be trivially compared with different U-values, leaving an empirical choice of a constant U to apply within a catalytic cycle when the self-consistent U-values vary along the reaction path.
The addition of other transition metals to the ceria lattice, often occupying cerium lattice sites, can offer improved catalytic properties. Numerous studies have used DFT approaches to these “doped” ceria surfaces. Often, an onsite U potential is added only to the f-states of cerium on these doped systems. For many metal oxides, accurate material properties require DFT+U methods with U corrections on the d-states of the metals. A second approximation to these mixed metal oxide systems is to also include a U value to the dopant d-states. Correspondingly, a reasonable U value for the d-states of these metals must also be determined. A U value applied to the p-states of oxygen can also lead to a better estimate of the band gaps and reduction energies of ceria. An additional complexity is that a different U potential can be applied to the same atom in different geometric arrangements (surface vs. subsurface, near dopant vs. far from dopant). Sholl and coworkers recently used a position dependent version of DFT+U (DFT+U(R)) to correctly reproduce the properties of FeOx. As approaches are not standardized, comparisons between DFT examinations of M-doped CeO2 reactivity are challenging.
A different approach from the DFT+U method is the use of a hybrid exchange-correlation functional. Hybrid functionals add exact exchange and limit the pure GGA inaccuracies in canceling the self-repulsion, therefore, avoiding the empiricism of adding U corrections. The amount of exact exchange to use must still be chosen, retaining empiricism in hybrid functional implementation. The hybrid functional HSE06, along with other hybrid functionals, has been shown to accurately model oxygen vacancies in pure ceria. Recently, the HSE06 functional was used to examine oxygen vacancy formation and charge transfer for divalent and trivalent dopants in ceria.[15, 16]
In this perspective, we clarify the challenge to represent M-doped CeO2 surface catalytic properties with DFT+U and hybrid functionals. Our objective is to illustrate the difficulties and summarize current approaches, toward further motivating improved approaches practical for surface reactivity studies.
Challenges in Representing M-Doped CeO2 Redox Catalysis
M/CeO2 systems are useful in CO oxidation, CH4 oxidation,[2, 17, 18] and sulfur adsorptive desulfurization.[19, 20] In all of these processes, the surface cycles between oxidized and reduced conditions. CH4 and CO oxidation cycles begin with the gas phase reactant and the oxidized surface, then form an absorbed intermediate (R*-M/CeO2, example in Fig. 1) that partially reduces the surface. An oxidized product desorbs, leaving a reduced surface (M/CeO2-δ). The surface is then reoxidized by an oxidant and the cycle repeats.
To model these cycles, we need accurate energetics for transformation among the oxidized, partially reduced and reduced surfaces. For oxidized and reduced surfaces, integer formal reduction of metal atoms often occurs. However, for adsorbed intermediates (Fig. 1) and partially reduced surfaces (CO, CH3, or H absorbed to an oxygen atom, for example), it is unclear if a localized, reduction of a metal atom should occur. As characterization of adsorbed intermediates is limited and typically nonexistent if the intermediate is unstable, it is unclear if sufficient experimental data exists to verify the accuracy of DFT representation of these intermediate species. For M-doped CeO2 catalysts, it is not always known whether the dopant or the cerium atom should reduce upon intermediate formation or surface reduction.
Both Pd- and Mn-doped CeO2 shows promise as catalysts, however, the role of the dopant in the catalytic process is not fully understood. In Pd-doped CeO2, the Pd atom reduces and the cerium oxidation state is unchanged upon adsorption of methane or oxygen vacancy formation. For Mn-doped CeO2, the U value chosen for the d-states of the Mn atom changes whether the Mn atom(s) reduce or the cerium atom(s) reduce upon oxygen removal. We have recently evaluated whether the dopant serves as the reduction center or alters Ce reducibility for all transition metal doped CeO2 (111) surfaces. As the reductant adsorption and oxygen vacancy formation energetics often correlate, the energetics of formation oxygen vacancies in the surface of M-doped CeO2 is a good descriptor for the catalytic properties.
In the following sections, we briefly review the DFT+U method and discuss its application to pure and M-doped CeO2 systems with a U value on the cerium f-states. We then discuss the addition of a U term on the oxygen p-states and dopant d-states. Lastly, we examine the use of hybrid functionals for these M-doped CeO2 systems.
Strongly correlated electron materials, including middle-to-late transition metal oxides, are difficult for conventional DFT to represent. These metal oxides contain electrons in partially filled d or f shells, which are localized onto each metal atom. Conventional GGA-DFT uses spatially averaged approximations for exchange and correlation, which does not properly cancel out the electron self-interactions in localized states. This yields values for electron–electron repulsion that are artificially increased. These large values of electron–electron repulsion cause the electrons to delocalize, and often changes the metal oxides from an insulator to a conductor.
To correct these self-interaction errors, the electronic structure may be separated into localized and delocalized states. The two different states can be treated with different electronic structure methods. A common method to do this is DFT+U. The most common DFT+U implementation uses a parameterized Hamiltonian in the framework of DFT instead of an explicit Hartree–Fock calculation. The energy from DFT+U is calculated as:
where is the energy from conventional DFT, is the average Coulombic interaction, is the exchange interaction, and n is the number of electrons in a localized orbital with a given spin and magnetic quantum number. I represents the atom type, l is the angular momentum (s, p, d, or f), m is the magnetic quantum number, and σ is the spin. and are tunable parameters that may be chosen to match measurable experimental properties of the material, such as band gaps and magnetic moments. Alternatively, constrained DFT calculations, where the number of electrons on a specific site is fixed and the remaining electrons are allowed to relax, can be used to determine a U value. As the Hamiltonian only uses the difference between and [eq. (1)], we will take U to represent the difference, Ueff.
Several studies have examined what Ce-f U value give reasonable electronic behavior for reduced CeO2. Recommended U values vary between 0.2 and 7 eV depending on the experimental properties being matched and the methods used to fit U. Dissociative methane adsorption energies and surface oxygen vacancy formation energies both greatly depend on the U value chosen for Ce, as shown in Figure 2. The exchange-correlation functional [local density approximation (LDA) or which GGA] chosen can influence the U value. Using this method, Lutfalla et al. determined a U value [Perdew-Burke-Ernzerhof (PBE) functional] of 0.2 eV on the f-states of Ce to closely match the experimental CeO2 to Ce2O3 reduction energy of 138 kJ mol−1. A U value of 6.3 eV (LDA) matches the experimental band gaps of the 2p-4f and the 2p-5d. This U value also reasonably predicts the lattice constant and bulk modulus of ceria. A U of less than 5 eV (PW91) on Ce results in improper delocalization of electrons across cerium atoms of the reduced CeO2 (100) surface. A U value of 5 eV results in the reduction of two Ce atoms from Ce4+ to Ce3+ near the oxygen vacancy as well as the appearance of a single spin peak in the density of states (DOS) between the CeO2 valence band and the Fermi level (Fig. 3a). Using both the LDA and GGA functional to match the band gaps, a U value of 6 and 5 eV, respectively, was recommended. To correctly replicate the insulating properties of reduced ceria (CeO2-x), U ≥ 6 eV (LDA) and U ≥ 5 eV (GGA) are required, as can be seen in Figure 3. Figures 3b and 3c illustrates that the addition of the U terms localizes the electrons onto two cerium atoms. Overall, most studies agree that U values of around 4.5–6 eV are adequate to represent the measurable properties of ceria. Values for U lower than 4.5 eV tend to improperly delocalize electrons in the reduced surface.
Our group has examined how the U value on cerium affects the dissociative methane adsorption energy and oxygen vacancy formation energy for Zr- and Pd-doped ceria (Fig. 2). In Zr-doped CeO2, like pure CeO2, the Ce-f U value influences both methane adsorption and oxygen vacancy formation energies. However, for Pd-doped CeO2 surfaces, the Ce U value does not have a significant influence on the two energies. Surface cerium atoms reduce when Zr-doped CeO2 is reduced by either oxygen vacancy formation or dissociated methane adsorption. Therefore, the Ce-f U value impacts reduction energetics. For Pd-doped CeO2, the Pd atom reduces and the Ce-f U value, therefore, does not have a significant influence. Examining all transition metal dopants, the dopants on the left side of the periodic table are similar to Zr in which the dopant alters the reducibility of the cerium atoms. Late transition metal dopants, like Pd, become the reduction centers. The U value on Ce f-states does not have a great influence on reduction energetics or surface electronic structure for dopants on the right side of the periodic table. The use of DFT+U (with U only on cerium) provides qualitative and quantitative reduction energetics for early and late transition metal dopants. The application of a U value to the d-states of late transition metal dopants, however, can alter the reduction energetics. For dopants in the middle of the periodic table, the application of a U term to transition metal d-states will qualitatively and quantitatively alter the electronic structure of the doped surface.
Including a U Value on the d-States of Transition Metal Dopants in CeO2
A U correction can be added to the p-states of oxygen or to the d-states of a metal dopant atom to better match experimental measurable properties. The additional U value on oxygen can lead to a better estimation and description of the lattice parameters, band gaps, formation energies, reduction energies, and electronic properties for ceria. An oxygen hole (one 2p orbital with only one electron instead of two) results when a lower valency metal atom is doped into an oxide with a higher valency (trivalent dopant in ceria, for example). With a U value only on the Ce atoms, the oxygen hole is delocalized across multiple oxygen atoms for La-doped CeO2. To correct this, a U value is added to the p-states of oxygen, thereby localizing the oxygen hole onto one oxygen atom.
A U value must also be added to the d-states of many transition metal oxides to correct the self-interaction error that results in overestimation of oxidation energies as well as band gaps and magnetic moments.[9, 26] However, it has not been clearly established if this additional U correction should be added to the d-states of metal atoms doped into ceria to obtain correct energies and electronic behaviors, and if so how to choose the U value. One way to choose the U value can be from fitting a U value in bulk oxides to match experimental lattice constants and band gaps, and then using this U value for the metal dopant. This was done for Pr- and Mn-doped CeO2 and the U value chosen for M d-states was 4.5 eV. Varying the value of U changes the MnO bond lengths, as well the electronic structure. Figure 4a illustrates how the DOS changes for Mn-doped CeO2 in the oxygen vacant surface (Mn2+) as the U value on Mn changes. As the Mn-d U value increases, the valence band slowly approaches the Fermi level and shifts the antibonding Mn-oxygen states into the valence band. Ultimately, Cen et al. also settle on a U value of 4.5 eV on the d-states of Mn. Our group also examined how the U value on Mn affects Mn-doped CeO2 electronic properties. The most stable Mn oxidation state in reduced Mn-doped CeO2 (111) depends on the U value on Mn (Fig. 4b). At low U values, the most stable oxidation state for Mn in oxygen vacant Mn-doped CeO2 is Mn3+, and at U values greater than 3.5 eV, the most stable structure has Mn reducing to Mn2+. A U value of 4 eV, as determined by Wang et al. for pure MnOx, also has Mn2+ as the oxidation state in reduced Mn-doped CeO2, in agreement with X-ray absorption near edge structure (XANES) studies of Mn/CeO2 oxides. The addition of the U value to the d-states of metal dopants helps to get the correct surface reducibility, and is especially necessary for midperiodic table transition metal dopants.
In conventional DFT+U, the U value is fixed for a given atom and electronic state. The self-interaction error, however, can vary for an atom in various geometric arrangements (surface vs. subsurface, near dopant vs. far from dopant). In each of these different arrangements, a different U value might be necessary to reproduce an accurate electronic structure. One method that shows promise in being able to accomplish this is DFT+U(R). In this method, the U value is determined by examining the linear-response value of U with respect to position, (R), of every atom. This method provides promise for removing, to some extent, the empiricism of the DFT+U method. However, the linear-response value of U may also vary along the catalytic reaction coordinate, and comparison of energies at varying U values is not trivial, returning some degree of empiricism in choosing a constant value. To our knowledge, the U(R) method has not been applied to investigate catalytic properties of transition metal doped ceria surfaces.
A hybrid exchange-correlation functional may be used to alleviate the self-interaction error and improper electron delocalization. A hybrid functional combines a specified amount of nonlocal exact exchange with a GGA functional. This specified amount of exact exchange must be chosen, and therefore, this method also contains empirical dependencies; around 25% exact exchange is typically used. The nonlocal exchange added to conventional DFT more accurately captures the reduced CeO2 electronic structure, however, it also greatly increases the computational intensity (for periodic systems), making large systems or large numbers of species along a catalytic reaction pathway computationally prohibitive. Hybrid functionals can be used on smaller systems to help confirm or calibrate U values for use in DFT+U calculations.
The relative stabilities of surface to subsurface oxygen vacancies in CeO2 (111) are accurately captured by using the hybrid functional Heyd-Scuseria-Ernzerhof (HSE06), as well as PBE+U and LDA+U functionals. The vacancy formation energies between these three functionals do not match, though the HSE06 vacancy formation energy matches well with the PBE0 functional and an embedded cluster model. The adsorption energies of water on the CeO2 (111) surface using the HSE06 functional and the PBE+U functional are within 0.2 eV, and the differences in energies of various adsorption structures are within 0.1 eV. Other hybrid functionals can be used such as PBE0 or B3LYP. Of the many hybrid functionals that have been examined, all produced cell parameters that were within 2.5% of the experimental values, with the PBE0 functional providing the most accurate results. The major differences between functionals are the predicted band gaps, where the amount of exact Hartree–Fock exchange included greatly changed the band gaps as well as the reduction energies (B1-WC functional performed best).
Doped CeO2 has also been examined with hybrid functionals, which could remove the “double empiricism” of choosing Ce-f and M-d U values in DFT+U. For doping of divalent (2+ oxidation state) cations (Pd and Ni) in ceria, DFT+U (U = 5.0 eV on Ce f-states) and HSE06 yield the same qualitative descriptions of oxygen vacancy formation; however, the two functionals yield quantitatively different results. The first oxygen vacancy formation energy with HSE06 is lower than that predicted by DFT+U, and to match oxygen vacancy formation energies a U value of 10 eV would have to be used, which causes large deviation in other properties, such as shifting the Ce3+ states into the valence band. These methods still give quantitatively different results, even when a correction to the DFT O2 bond energy (0.7 eV per O atom) is added. Correcting the overestimation of the DFT O2 bond energy actually increases the difference in energy between DFT+U and HSE06 for oxygen vacancy formation. The hybrid functional HSE06 also correctly shows the formation of an oxygen hole with doping of trivalent atoms in the surface. In both of Nolan's studies, the author did not examine how the addition of a U term on the dopant would match with the hybrid functional HSE06. We examined how the addition of a U term on the d-states of Mn (coupled with a U term of Ce f-states) compares with the results from an HSE06 optimization. The qualitative preferences for Mn oxidation state match when an appropriate U value on Mn is used (U = 4.0 eV on d-states of Mn and U = 5.0 eV on Ce f-states). The absolute quantitative oxygen vacancy formation energies do not match.
Though DFT+U and hybrid functional methods can be chosen to match experimental properties of ceria and M-doped ceria systems, the transferability to surface catalytic activity is unclear. For catalytic studies, accurate reaction energies and activation barriers for transforming among reactant, adsorbed species, and products is needed. These transformations include reduction and reoxidation of the M-doped CeO2 surface, challenging accurate energetics determination with tractable DFT methods. Despite these limitations, DFT+U methods have provided qualitative insight into catalytic mechanisms and trends with dopant variations for methane and carbon monoxide oxidation and desulfurization.
To help reduce and mitigate some of these challenges faced when using DFT to study catalytic properties of M-doped ceria systems, further standardization of current “best practices,” and greater availability of experimental model systems for benchmarking would help clarify the reliability of the methods being used. Differences between studies in model construction (slab thickness, the use of mirrored slabs, the U value applied, variance in exchange-correlation functionals) complicate comparison among DFT+U studies. All DFT+U studies should be careful to test convergence or sensitivity to model parameters, and should show a full awareness of the models used by others such that we may collectively clarify sensitivity to these differences and narrow the variety. Experimental benchmark studies of single crystal, M-doped ceria surfaces, especially probing reducibility and surface reactivity, would allow for further benchmarking to establish the reliability and limitations in DFT+U. The choice of U values or hybrid functionals in DFT may be better tuned to match experimental properties and testing of their transferability could be established. More advanced approaches, such as the DFT+U(R) method, can then be tested versus benchmarks and reliability in examining surface energetics can be tested. The greater use and testing of hybrid functionals would also be a clear avenue of future work if these can be made computationally tractable within a plane-wave basis set approach. Comparison between computation and experiment M-CeO2 systems will continue to be challenged by the structural complexities of these systems. The further development of DFT(+U)-based reactive force fields will allow for more direct comparisons that include structural inhomogeneity in doping and nanostructure, and for comparison with experimental properties defined on longer length or time scales accessible to electronic structure methods.
This material is based upon work supported as part of the Center for Atomic Level Catalyst Design, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001058.
Matthew Krcha obtained his Bachelors of Science from Clarkson University in 2010. He started at the Pennsylvania State University in the fall of 2010 and is working toward his doctoral degree in Chemical Engineering. His current research examines biomass gasification effluent cleanup using ceria-based catalysts. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Michael J. Janik is an Associate Professor of Chemical Engineering at Pennsylvania State University and holds the John J. and Jean M. Brennan Clean Energy Chair. His research interests are in the use of computational methods to understand and design materials for alternative energy conversion systems. Dr. Janik received his B. S. in Chemical Engineering from Yale University. Following three years as a Process Engineer for Procter and Gamble, Janik completed his doctoral studies at the University of Virginia. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]