Cluster Expansion
A cluster expansion is a lattice model that writes the energy of a configuration as a sum over clusters of lattice sites (singlets, pairs, triplets and so on) of products of occupation or spin variables, each weighted by a fitted coefficient. The sites can be atoms in an alloy or adsorption sites on a surface, and the coefficients are fit to DFT energies of a set of enumerated configurations. Once fit, the expansion evaluates new configurations very cheaply, which makes it well suited to Monte Carlo simulations of ordering, segregation and coverage-dependent adsorption. Its main limitations follow from its construction: it is tied to the lattice it was trained on, it describes relaxation only implicitly through the relaxed energies it was fit to, and it cannot represent off-lattice dynamics or reconstructions. See also Coverage-dependent adsorption.
Figure selection and uncertainty for O on Au(111) and Pt(111) (2009)
Miller and Kitchin fit two-dimensional cluster expansions to 15 DFT oxygen configurations per metal on Au(111) and Pt(111), choosing among all complete figure sets drawn from 18 candidate figures by a cross-validation score built from every leave-one-, two- and three-out fit, with ill-conditioned fits (condition number above 1000) excluded. The best Pt set had a CV score of 17 meV, and its interaction coefficients shrank in magnitude with figure size. Applying the expansions to all 8917 configurations in cells of up to 15 sites, they found that roughly 20 meV of fit uncertainty plus DFT convergence error prevents pinning down the exact convex-hull structures, but a correlation-vector similarity measure showed that near-hull structures cluster into a few structural motifs (Miller 2009). Summary: miller2009a. See also Cross-validation.
Ground state FCC and BCC Cu-Pd structures (2015)
Boes et al. used ATAT cluster expansions on FCC and BCC lattices to find the ground state Cu1-xPdx configurations, whose convex hulls agreed with the known phase diagram and predicted the B2 structure at x = 0.5 to be more stable than FCC at that composition (Boes et al. 2015). These ordered structures served as references for DFT core level shift calculations that distinguished ordered B2 from disordered FCC alloys. Summary: boes2015.
Cluster expansions compared with a neural network potential for O/Pd(111) (2017)
Boes and Kitchin contrasted cluster expansions with a Behler–Parrinello neural network potential for oxygen on Pd(111). The network is not restricted to a lattice or to ground-state configurations, so it can also drive NEB and MD calculations, although in their Python implementation it was more expensive than a cluster expansion and they ran grand-canonical MC without relaxation. Their MC gave a mean fcc oxygen coverage of 0.50 ± 0.05 ML at μ(O2) = 0.75 eV and 600 K, 0.13 ML above a published cluster-expansion result for the same conditions, which they attribute mostly to differences in DFT formation energies. They note that in principle a network can represent the surface reconstructions that cluster expansions cannot (Boes 2017). Summary: boes2017a. See also Neural network potentials.
A neural network alternative for alloy segregation (2017)
Boes and Kitchin noted that DFT-based cluster expansions had been used to model segregation at alloy surfaces but described them as difficult to extend and the codes as not readily available. For AuPd(111) they instead trained a neural network potential, with training structures chosen by EMT enumeration of energy-unique slab configurations, and ran Monte Carlo on a fixed lattice without relaxation. They also point out that earlier ab initio AuPd segregation studies fit an effective Ising model to fully relaxed dilute-limit energies, which hides the separate contribution of relaxation; their own DFT tests found relaxation energies small and nearly uniform near the bulk composition (Boes 2017). Summary: boes2017; see also Surface segregation.
Finite-temperature Cu-Pd phase boundaries (2017)
Geng, Boes and Kitchin built separate ATAT cluster expansions for Cu-Pd on the fcc (69 configurations, 40 clusters) and bcc (65 configurations, 35 clusters) lattices, with leave-one-out cross-validation scores near 5 meV/atom. To keep the lattice model meaningful, they excluded DFT structures whose relaxation strain tensor norm exceeded 0.05, and after the cross-validation fit they pruned four-body interactions by hand until the score or ground-state predictions degraded. Monte Carlo phase-boundary tracing with these expansions reproduced the shape of the Cu-rich L12/B2 region but placed the order-disorder transition near 720 K rather than the experimental 823 K (Geng et al. 2017). Summary: geng2017.