Comprehensive Density Functional and Kinetic Monte Carlo Study of CO2 Hydrogenation on a Well-Defined Ni/CeO2 Model Catalyst: Role of Eley–Rideal Reactions

A detailed multiscale study of the mechanism of CO2 hydrogenation on a well-defined Ni/CeO2 model catalyst is reported that couples periodic density functional theory (DFT) calculations with kinetic Monte Carlo (kMC) simulations. The study includes an analysis of the role of Eley–Rideal elementary steps for the water formation step, which are usually neglected on the overall picture of the mechanism, catalytic activity, and selectivity. The DFT calculations for the chosen model consisting of a Ni4 cluster supported on CeO2 (111) show large enough adsorption energies along with low energy barriers that suggest this catalyst to be a good option for high selective CO2 methanation. The kMC simulations results show a synergic effect between the two 3-fold hollow sites of the supported Ni4 cluster with some elementary reactions dominant in one site, while other reactions prefer the another, nearly equivalent site. This effect is even more evident for the simulations explicitly including Eley–Rideal steps. The kMC simulations reveal that CO is formed via the dissociative pathway of the reverse water–gas shift reaction, while methane is formed via a CO2 → CO → HCO → CH → CH2 → CH3 → CH4 mechanism. Overall, our results show the importance of including the Eley–Rideal reactions and point to small Ni clusters supported on the CeO2 (111) surface as potential good catalysts for high selective CO2 methanation under mild conditions, while very active and selective toward CO formation at higher temperatures.


S1. Formation energies calculation.
The reference set for the formation energy values is {slab, H2(g), H2O(g) and CO(g)} where slab stands for the energy of the Ni4/CeO2 surface while H2(g), H2O(g) and CO(g) are the DFT energies of the gas-phase molecules.
Therefore, the formation energy on the i adsorbate is calculated as: Where  $%&' is the DFT energy of the pristine slab,  !#$%&' is the DFT energy of the adsorbate i on the slab,  ( is the number of atoms j in species i, and  ( is the reference energy of the atom j, defined in our reference set as: where  !(#) is the DFT energy for the i gas-phase species.Following the above definition, the formation energy (FE) of the different species at the different sites for the Ni4/CeO2 system is calculated and summarized in Table S1.To correctly define the energetics of the system of interest, at each surface configuration, cluster expansion Hamiltonians are used which account for the interactions between the different adsorbed species.With this definition the energy of a specific configuration is calculated as a sum of the different clusters that can represent one-body to multi-body terms.Hence, a cluster could be a single absorbed species or a group of neighboring species interacting among them.For instance, the cluster representing a two-body term interaction (normally called lateral interactions) between two species A-A is calculated as: where  4 " and  4#4 " stands for the formation energy of the A species alone and the formation energy of the two neighboring species, respectively.
Finally, the energetics of a specific i lattice configuration is expressed as the sum of cluster energies: where () is the total energy of the system (i.e., the energy of the  lattice configuration),  + is the total number of clusters included in the model,  5 is the cluster energy of cluster  and  5 () is the number of times that a pattern for k-cluster appears.Note that the formation energy of a single adsorbed species is equivalent to the cluster energy of the adsorbed species.Table S2 summarize the different two-body terms (or lateral interactions) used in the cluster expansion.

Table S1 .
Formation Energies Including the ZPE Term for All the Different Species at the Different Considered Sites on the Ni4/CeO2 system.The * and ** Symbols Stand for Monodentate (One Site) or Bidentate (Two Sites) Adsorbed Species, Respectively.

Table S2 .
Pairwise Lateral Interactions (LI), Including the ZPE term, Between the Most Relevant Species Involved in the CO2 Hydrogenation Reaction on the Ni4/CeO2 system.These Values Correspond to the Twobody Terms Used in the Cluster Expansion.The "-" Symbol is Used to Distinguish the Two Different Species in the kMC Simulation.The * and ** Symbols Stand for Monodentate (One Site) or Bidentate (Two Sites)

Table S3 .
Reaction Energies (∆ !,# ) and Forward and Reverse Energy Barriers (∆ !,%Including the ZPE Term, for the Different Elementary Reactions Considered for the CO2 Hydrogenation Reaction for the Ni4/CeO2 System with Corresponding Values for the Reaction at the Ni(111) Surface 1 Included for Comparison.For Reactions in which Two Possible Hydrogen Attacks are Considered the f and n Subscript Stands for the H Atom Being at the Site that is Far or Near the Attacking Species, Respectively.For Instance, f Stands for Situations in which H and the Other Species are at hO/NiCe or hCe/NiO, Respectively, and n Stands for Situations in which H and the Other Species are at hCe/NiCe or hO/NiO, Respectively.The * and ** Symbols Stand for Monodentate (One Site) or Bidentate (Two Sites) Adsorbed Species, Respectively.

Table S4 .
Species Coverage at the Different Considered Sites for the Simulations with the Eley-Rideal Reactions (W/ER) and without Them (Wo/ER) at the Five Different Temperatures Considered and at P(H2) = 0.528 bar and P(CO2) = 0.132 bar.Note that the Present Values are Calculated as the Mean Value of 5 Different kMC Simulations for Each Working Condition.