## Introduction

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.[5] Catalytic activity on metal oxides is directly related to the ability of metal atoms to cycle between oxidation states.[6] 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.[7] Recently, a review on the status of DFT+*U* has been published.[8] 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.[7] 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.[7] 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.[1] For many metal oxides, accurate material properties require DFT+*U* methods with *U* corrections on the *d*-states of the metals.[9] A second approximation to these mixed metal oxide systems is to also include a *U* value to the dopant *d*-states.[10] 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.[11] 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 FeO_{x}.[12] As approaches are not standardized, comparisons between DFT examinations of M-doped CeO_{2} reactivity are challenging.

A different approach from the DFT+*U* method is the use of a hybrid exchange-correlation functional.[13] 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.[14] 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 CeO_{2} 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.