A comparison of chain-of-states based methods for finding minimum energy pathways (MEPs) is presented. In each method, a set of images along an initial pathway between two local minima is relaxed to find a MEP. We compare the nudged elastic band (NEB), doubly nudged elastic band, string, and simplified string methods, each with a set of commonly used optimizers. Our results show that the NEB and string methods are essentially equivalent and the most efficient methods for finding MEPs when coupled with a suitable optimizer. The most efficient optimizer was found to be a form of the limited-memory Broyden-Fletcher-Goldfarb-Shanno method in which the approximate inverse Hessian is constructed globally for all images along the path. The use of a climbing-image allows for finding the saddle point while representing the MEP with as few images as possible. If a highly accurate MEP is desired, it is found to be more efficient to descend from the saddle to the minima than to use a chain-of-states method with many images. Our results are based on a pairwise Morse potential to model rearrangements of a heptamer island on Pt(111), and plane-wave based density functional theory to model a rollover diffusion mechanism of a Pd tetramer on MgO(100) and dissociative adsorption and diffusion of oxygen on Au(111).

Designing active and stable electrocatalysts with economic efficiency for acidic oxygen evolution reaction is essential for developing proton exchange membrane water electrolyzers. Herein, we report on a cobalt oxide incorporated with iridium single atoms (Ir-Co₃O₄), prepared by a mechanochemical approach. Operando X-ray absorption spectroscopy reveals that Ir atoms are partially oxidized to active Ir^>4+ during the reaction, meanwhile Ir and Co atoms with their bridged electrophilic O ligands acting as active sites, are jointly responsible for the enhanced performance. Theoretical calculations further disclose the isolated Ir atoms can effectively boost the electronic conductivity and optimize the energy barrier. As a result, Ir-Co₃O₄ exhibits significantly higher mass activity and turnover frequency than those of benchmark IrO₂ in acidic conditions. Moreover, the catalyst preparation can be easily scaled up to gram-level per batch. The present approach highlights the concept of constructing single noble metal atoms incorporated cost-effective metal oxides catalysts for practical applications.