Uncertainty and figure selection for DFT based cluster expansions for oxygen adsorption on Au and Pt (111) surfaces
Spencer D. Miller and John R. Kitchin, “Uncertainty and figure selection for DFT based cluster expansions for oxygen adsorption on Au and Pt (111) surfaces”, Molecular Simulation 35(10-11), 920-927 (2009). https://doi.org/10.1080/08927020902833137
This summary was written by a language model from the paper and has not been checked line by line. Rely on the published paper for details.
Question
DFT is too expensive to enumerate every arrangement of adsorbates on a surface, so a small set of calculated configurations leaves open whether lower-energy structures exist. Two-dimensional cluster expansions can extrapolate a small DFT database to many more configurations, but the result depends on which clusters (“figures”) are kept in the truncated basis. This paper asks how to choose that figure set for oxygen on Au(111) and Pt(111), how to quantify the predictive uncertainty of the resulting expansion, and what the expanded configuration space reveals about the stable surface structures (Miller 2009).
Approach
The DFT data are the 15 non-zero-coverage oxygen configurations per metal reported earlier by the same authors (Miller 2009), computed with VASP, PAW potentials and PW91 on five-layer (111) slabs (bottom three layers fixed) with lattice constants of 4.18 Å (Au) and 4.00 Å (Pt). Adsorption energies are referenced to gas-phase O2 and were converged to about 0.02 eV/O on Pt and 0.03 eV/O on Au. Oxygen occupies fcc sites, mapped to an Ising lattice with spins of +1 (occupied) and -1 (vacant). The pool of candidate figures contains 18 clusters with zero to four vertices. Every complete figure set (each figure accompanied by all its sub-figures) that contains the five simplest figures (empty, point, first- and second-neighbor pairs) was fit by least squares and scored with a cross-validation (CV) score: the root-mean-square error over all leave-one-out, leave-two-out and leave-three-out predictions. Fits whose correlation matrix had a condition number above 1000 were discarded as ill conditioned, and a figure set needed at least 6000 valid prediction errors to be considered, which also excluded very large sets. Heats of formation relative to phase separation into clean and 1 ML regions (without vibrational contributions) were used to build convex hulls (Miller 2009).
Key findings
The best figure set for Pt had a CV score of 17 meV. Across the many leave-k-out fits its effective cluster interactions were roughly normally distributed around a definite positive or negative mean, and the mean magnitude shrank as figure size grew, supporting the idea that the truncated expansion captures the infinite series. Comparing prediction-error distributions for figure sets with different CV scores showed that low scores give narrow error distributions centered on zero while higher scores give wider, larger errors, so the CV score works both for ranking figure sets and as an estimate of the expansion’s uncertainty (Miller 2009).
With the best figure sets, the authors evaluated every symmetry-distinct configuration in cells of up to 15 fcc sites, 8917 configurations in total. The lowest-energy configurations at each coverage are not quite linear in coverage, especially at high coverage, unlike the roughly linear DFT trend. The configurational correlation between Au and Pt adsorption energies found in the DFT data persists across the whole expanded set, including high-energy configurations, which the authors take as evidence of a common underlying cause such as the shared d-band behavior, though with some asymmetry (Miller 2009).
The uncertainty in heats of formation, from DFT convergence (20 meV for Pt, 30 meV for Au) plus roughly 20 meV rms fit error on both surfaces, makes it hard to say which configurations lie exactly on the convex hull. To get around this, the authors defined a similarity measure as the normalized dot product of two configurations’ correlation vectors (1 for identical, approaching 0 for dissimilar) and compared every configuration with seven common reference structures. Configurations near the hull fall into a small number of structural themes, each a variant of a reference structure, so the dominant motif in each coverage range can be identified with confidence even when the exact ground state cannot. The exception is a small region near 0.5 ML on Au with no clear resemblance to the references, and its size lies within the expansion’s uncertainty (Miller 2009).
Significance
The paper gives a practical recipe for choosing figures in surface cluster expansions: exhaustive search over complete figure sets, conditioning checks and multi-fold cross-validation. It also treats the CV score as an uncertainty estimate rather than only a fitting criterion. Its argument that structural motifs, not single ground states, are the robust result of a noisy expansion is a useful way to read phase diagrams built on limited DFT data. The work extends the earlier DFT study of Au and Pt (Miller 2009) to a far larger configuration space.