(659e) Determining the Predictability of Clusters Expansions By Quantifying Their Associated Errors

Quantifying
the lateral interactions between adsorbates on the surface of a catalyst using
a first-principles-based lattice gas Hamiltonian provides an opportunity to
obtain ab initio insights into large-scale
phenomena and presents a path forward to bridging the atomistic and meso-scale
length and time scales of a heterogenous catalytic reactions. However, it has
recently been shown that supercell volume and geometric relaxations—often
critical to capturing the true chemistry of a system—can cause anything from
poor convergence of the associated cluster expansion (CE) to outright CE
failure [1-3].
Indeed, when the positions of the surface metal atoms deviate significantly
from those of the bulk, a cluster expansion stops becoming useful unless
many-body terms are added to a lattice gas Hamiltonian, which in turn affects
its convergence. We are therefore motivated to determine how relaxations quantitatively
affect cluster expansions. To do this, we have developed a robust method of
quantifying errors in the effective cluster interactions (ECIs) of CEs with
minimal statistical assumptions about the underlying data and model. The errors
are found by direct evaluation of the ECI fluctuations occurring during the
calculation of the leave-multiple-out cross-validation (LMO CV) score [4].

ECI
errors were determined for two separate systems (Figure 1): O/Fe(100) with a fully relaxed lattice as well as O/Fe(100) with
an idealized, fixed lattice (data for the former taken from Bray et al.[5]). The
optimum CEs for these systems were found with the newly developed ab initio Mean-field Augmented Lattice
Gas Modeling (AMALGM) code, and Figure 1 shows the results for the relaxed
(Figure 1A and 1B) and fixed (Figure 1C and 1D) lattice CEs. Both systems were
evaluated using the optimum CEs for the relaxed (Figure 1A and 1C) and fixed
(Figure 1B and 1D) systems to fairly compare the two. Our results show that
relaxations do indeed result in more uncertain CEs Calibri;mso-bidi-theme-font:minor-latin">: regardless of which CE is used, the
LMO CV score of the fixed system is consistently two to three times smaller
than that for the relaxed system—and our method can quantify this uncertainty.
For instance, the error in the adsorption energy value (the ECI for Cluster ID
#1) is four times greater for the relaxed system (Figure 1A) than for the fixed
system (Figure 1C) even when using the optimum CE for the relaxed system. Many
ECIs in both systems have ECI errors that are larger than the ECI values
themselves; thus, our method reveals when it is difficult to ascertain whether
a particular ECI should be regarded as repulsive or attractive and when an ECI
value of 0 meV would potentially be more appropriate. This suggests that we may
be able to use these errors as a CE “filter”, allowing only CEs with
significant ECIs to be considered during the CE optimization procedure. A
particularly important observation based on our evaluated errors here is that the
ECIs between the two systems are often not statistically differentiable,
meaning the fixed system CE (Figure 1C) could be used in the place of the
relaxed system CE (Figure 1A) with potentially little negative consequences. While
this does not mean that CEs are unable to capture relaxations outright, it does
highlight the importance of being able to quantify the errors inherent within a
CE, and our work here provides the tools necessary to make that evaluation.

[1] J. M. Sanchez, Journal of Phase Equilibria and
Diffusion 38 (2017) 238-251.

font-family:" calibri fr-ca>[2] J. M. Sanchez, Phys. Rev. B 95 (2017) 216202.

[3] A. H. Nguyen, C. W.
Rosenbrock, C. S. Reese, G. L. W. Hart, Phys. Rev. B 96 (2017) 014107.

[4] K. Baumann, Trends Anal.
Chem. 22 (2003) 395-406.

[5] J. Bray, G. Collinge, C.
Stampfl, Y. Wang, J.-S. McEwen, Top. Catal. 61 (2018) 763-775.

Figure 1. The ECIs found for the two optimum
LG CEs fitted to the adsorption energies of the O/Fe(100) system using the RPBE
exchange correlation functional and where Fe(100) surface and the oxygen
adatoms in panels (A-B) were allowed to fully relax during optimization (color
coded red), and where the Fe(100) surface was kept fixed in its optimum clean
surface configuration and the oxygen adatoms were kept fixed in their
isolated-oxygen optimum position on this surface in panels (C-D) (color coded
blue). The corresponding arrangement of the oxygen adatoms for the given
cluster IDs is shown below where the brown circles are Fe atoms and the red
circles are O adatoms. The LMO CV score and RMSR are shown as an inset and the
errors determined using our method are shown as error bars. Note that the ECIs
for Cluster ID #1 (the adsorption energy of an isolated O atom) and its
associated error are given in text as an inset due to magnitude discrepancy.