Modeling palladium surfaces with density functional theory, neural networks and molecular dynamics
Tianyu Gao and John R. Kitchin, “Modeling palladium surfaces with density functional theory, neural networks and molecular dynamics”, Catalysis Today 312, 132–140 (2018). https://doi.org/10.1016/j.cattod.2018.03.045
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Question
DFT-based molecular dynamics and Monte Carlo are limited to hundreds of picoseconds and a few thousand atoms, so faster potentials with near-DFT accuracy are needed to study surface processes relevant to catalysis. Following earlier group work on Au, AuPd segregation and O on Pd, this paper asks how well a Behler–Parrinello neural network (NN) potential for a single element, palladium, can reproduce bulk, defect and surface properties, including activated processes such as vacancy and adatom diffusion, and what it costs relative to DFT. Adatom diffusion is treated as a stepping stone toward modeling activated catalytic processes at surfaces (Gao 2018).
Approach
Reference data came from VASP with the PBE functional, PAW potentials, a 350 eV cutoff and 14 × 14 × 14 k-point grids for primitive cells (about 1 meV convergence). Slabs had five layers with the bottom two fixed and 6 Å of vacuum on each side, and barriers were found with the nudged elastic band (NEB) method. The potential was trained with the AMP code using a 6.5 Å cutoff, AMP’s default eight Gaussian symmetry functions (four radial, four angular) and two hidden layers of 18 nodes, with energy and force goals of 1 meV and 0.01 eV/Å. The training set was grown iteratively: two NNs of different architecture were trained, one was used to run MD or other simulations, and configurations where the two networks disagreed were computed with DFT and added. The final set contained 3035 bulk and slab configurations, including relaxation steps, NEB paths and MD snapshots, with 10% held out for validation. MD with the NN as the ASE calculator was used for adatom diffusion on a five-layer 2 × 2 Pd(111) slab (up to 8 runs of 10 to 15 ps per temperature) (Gao 2018).
Key findings
The network reached residual errors of 5.51 meV/atom on the training set and 1.92 meV/atom on the validation set, with errors roughly normal (standard deviation about 0.006 eV/atom) and no sign of overfitting; the conclusions quote a 5.7 meV/atom RMSE overall. Equations of state for fcc, hcp, bcc, simple cubic and diamond Pd were nearly indistinguishable from DFT, and cohesive energies, equilibrium volumes and bulk moduli agreed closely (fcc: 3.756 versus 3.755 eV cohesive energy, 175.8 versus 175.5 GPa). Elastic constants, a harder test because they depend on curvature, were within about 20 GPa of DFT (C11 277.8 versus 254.8, C12 124.3 versus 135.8, C44 69.4 versus 58.7 GPa). Bulk vacancy formation energies were within about 0.1 eV of DFT across supercells, and both methods gave a bulk vacancy migration barrier of about 0.676 eV. Langevin MD of bulk fcc at 900 K, used to add training data, gave NN energies within 3 meV/atom of DFT on average.
For surfaces, the Pd(111) surface energy was 84.7 meV/Ų from the NN versus about 83 meV/Ų from DFT, independent of supercell size; both are well below experiment, which the authors attribute to PBE. Surface vacancy formation energies were underestimated, with errors growing from 0.047 to 0.222 eV as the supercell grew, signaling too little vacancy training data. The surface vacancy diffusion barrier was underestimated by about 0.06 eV, while a closed adatom path through top, fcc, bridge and hcp sites matched NEB closely. On stepped Pd(211), total energies of perfect, vacancy and adatom slabs agreed with DFT to within 0.01 to 0.11 eV, and diffusion paths along the step edge also agreed well. From the linear (Brownian) region of the mean square displacement and an Arrhenius fit, MD gave an adatom diffusion activation energy of 0.132 eV, close to NN-NEB bridge-site barriers of 0.136 eV (fcc-bridge) and 0.133 eV (hcp-bridge). Single-point NN evaluations took under 1 s, about four orders of magnitude faster than DFT, and the cost grew more slowly with system size (Gao 2018).
Significance
The paper gives a careful, property-by-property validation of a single-element NN potential, showing it can replace DFT for bulk, defect, surface and diffusion simulations of Pd while being far cheaper. It also states the limits plainly: the potential is interpolative, so it should not be trusted for high pressure or nanoparticles absent from the training data, and it cannot be used for adsorbates, alloys or oxides, because it has no fingerprints or networks for non-Pd atoms and cannot simply be extended to them. Validating with similar geometries in the training set, and growing the set by simulation-driven disagreement, are the central practical lessons (Gao 2018).