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).
A generalized solid-state nudged elastic band (G-SSNEB) method is presented for determining reaction pathways of solid-solid transformations involving both atomic and unit-cell degrees of freedom. We combine atomic and cell degrees of freedom into a unified description of the crystal structure so that calculated reaction paths are insensitive to the choice of periodic cell. For the rock-salt to wurtzite transition in CdSe, we demonstrate that the method is robust for mechanisms dominated either by atomic motion or by unit-cell deformation; notably, the lowest-energy transition mechanism found by our G-SSNEB changes with cell size from a concerted transformation of the cell coordinates in small cells to a nucleation event in large cells. The method is efficient and can be applied to systems in which the force and stress tensor are calculated using density functional theory.
Kinetic pathways of Li-ion diffusion in olivine phosphates are calculated from density functional theory (DFT). Previously reported theoretical diffusion rates for Li ions and vacancies in defect-free crystalline FePO4 and LiFePO4 are six orders of magnitude faster than experimentally measured values. This discrepancy can be resolved by considering the different components of Li kinetics, including diffusion in the bulk, on the surface, in the presence of defects, and in varying local environments. Using DFT+U, we quantify each of these effects and determine that, while bulk diffusion is affected by strain and Li concentration, these are not significant enough to explain the slow diffusion observed in experiment. However, surface diffusion is observed to have have high barriers, which could contribute to slow kinetics in nanostructured cathodes. Anti-site defects also provide a possible explanation for slow diffusion, but only for vacancy diffusion in LiFePO4, which has a barrier of 0.71 eV, compared to 0.29 eV in defect-free channels. In FePO4, a concerted Li-ion diffusion mechanism around the anti-site defect is found to have a low barrier of 0.35 eV, allowing for facile cross-channel diffusion at room temperature. The difference between Li-ion and vacancy diffusion is understood in terms of a favorable coordination between Li ions and localized electrons on Fe centers at the transition states for Li-ion hopping in FePO4. Greater distances between vacancies and holes at the transition states for vacancy diffusion lead to higher diffusion barriers.
The combined experimental and theoretical study of intrinsic hydrogen diffusion on bridge-bonded oxygen (BBO) rows of TiO 2(110) is presented. Sequences of isothermal scanning tunneling microscopy images demonstrate a complex behavior of hydrogen formed by water dissociation on BBO vacancies. Different diffusion rates are observed for the two hydrogens in the original geminate OH pair suggesting the presence of a long-lived polaronic state. For the case of separated hydroxyls, both theory and experiment yield comparable temperature-dependent diffusion rates. Density functional theory calculations show that there are two comparable low energy diffusion pathways for hydrogen motion along the BBO from one BBO to its neighbor, one by a direct hop and the other by an intermediate minimum at a terrace O. The values of kinetic parameters (prefactors and diffusion barriers) determined experimentally and theoretically are significantly different and indicate the presence of a more complex diffusion mechanism. We speculate that the hydrogen diffusion proceeds via a two-step mechanism: the initial diffusion of localized charge, followed by the diffusion of hydrogen. Both experiment and theory show the presence of repulsive OH-OH interactions.
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