Neural network predictions of oxygen interactions on a dynamic Pd surface
Jacob R. Boes and John R. Kitchin, “Neural network predictions of oxygen interactions on a dynamic Pd surface”, Molecular Simulation 43(5-6), 346–354 (2017). https://doi.org/10.1080/08927022.2016.1274984
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Question
Oxygen adsorbed on a metal surface sits on a potential energy surface with many dimensions, both from adsorbate-adsorbate interactions and from thermally displaced metal atoms. DFT samples this surface accurately but too expensively for molecular dynamics (MD) or Monte Carlo (MC). Cluster expansions are cheap but are tied to a fixed lattice and to relaxed ground-state configurations, and Cartesian neural networks are tied to a fixed number of atoms. The paper asks whether a high-dimensional Behler–Parrinello neural network (NN) can reproduce coverage-dependent oxygen interactions with a dynamic Pd(111) surface, and develops a methodology for building one (Boes 2017).
Approach
The authors trained a Behler–Parrinello NN with the AMP code on 11,925 DFT calculations (VASP, PBE, PAW, 800 eV cutoff for oxygen-containing structures) of four-layer Pd(111) slabs with 0 to 1 ML oxygen, plus a small set of bulk Pd calculations. Initial structures came from enumerating oxygen occupations of fcc, hcp, bridge and top sites on a 3 × 3 × 4 slab (512 configurations per site type), with effective medium theory used to discard energetically degenerate ones. Every relaxation step was used as training data, and later data were generated by running MD and coverage calculations with earlier versions of the NN. The best network used a 6.5 Å cutoff, 20 G2 and G4 symmetry functions per element, and two tanh hidden layers (reported as 2 and 12 nodes). Ten percent of the data was held out for validation. The NN was then applied to bulk and clean-slab properties, layer-slipping barriers, oxygen adsorption and formation energies, grand-canonical MC on a 10 × 10 × 4 slab, nudged elastic band (NEB) diffusion paths, and 600 K MD of one to three oxygen atoms, with DFT single points used for validation (Boes 2017).
Key findings
The validation RMSE was 0.009 eV/atom, about the same as the training error. For pure Pd the NN matched DFT closely: lattice constant 3.939 Å for both, bulk modulus 173 versus 175 GPa, and four-layer surface formation energy 0.563 versus 0.561 eV/site. Because only four-layer slabs were in the training set, the NN surface energy stops changing beyond four layers. Layer-slipping NEBs showed that the NN under-predicts the energy of hcp stacking (by 0.04 and 0.08 eV/site for one and two slipped layers), which was attributed to sparse training data in that region.
For 107 energy-unique oxygen configurations on 3 × 3 Pd(111), NN adsorption energies were within about 0.05 eV/O of DFT for most coverages, with fcc configurations over-predicted by about 0.1 eV/O (worst at low coverage, which was under-represented in the data) and hcp configurations under-predicted by about 0.05 eV/O. DFT showed strong surface reconstruction at 2/3 ML, where rows of oxygen pull a row of Pd out of the surface; the NN was not trained to describe these reconstructions. Unrelaxed grand-canonical MC with the NN at μ(O2) = 0.75 eV and 600 K gave a mean fcc coverage of 0.50 ± 0.05 ML, 0.13 ML higher than a published cluster expansion under the same conditions. The authors attribute this to lower formation energies in their DFT data (about 0.1 eV/site lower at mid coverage and over 0.2 eV lower at 1 ML), probably from the O2 reference energy, and note that neglecting relaxation likely biases the NN coverage low.
For single-oxygen diffusion the NN gave a bridge-site barrier of 0.40 eV versus 0.56 eV from DFT, both consistent with an experimental range of 0.4 to 0.5 eV, and an fcc-hcp energy difference of 0.07 eV versus 0.18 eV from DFT. Two-oxygen NEBs showed the second-nearest-neighbor arrangement about 0.14 eV less stable than the fourth, and a sharp energy rise when oxygens occupy adjacent fcc and hcp sites. MD at 600 K was stable and within about 0.10 eV/site of DFT for coverages up to 0.22 ML (runs up to 10,000 steps), with oxygen pairs pulling a Pd atom partly out of the surface. At 0.33 ML and above, oxygen eventually entered the subsurface, a region absent from training, and the surface deconstructed within 2000 steps.
Significance
The work shows that a Behler–Parrinello NN can carry DFT accuracy for O/Pd(111) to larger cells and to dynamic simulations (NEB, MD, grand-canonical MC) that neither fixed-size Cartesian networks nor lattice cluster expansions handle, while making clear that failures occur in unsampled regions (hcp stacking, reconstructions, subsurface oxygen). The authors suggest iterative training and cross-checking with a second, differently built NN to find such regions, and they release the full DFT database and trained parameters in the supporting information for reuse (Boes 2017).