# The Robustness of Cluster Expansion: Assessing the Roles of Relaxation and Numerical Error

## TL;DR

This paper investigates the factors affecting the reliability of cluster expansion models in metallic alloys, introducing a heuristic to quantify atomic relaxation effects and analyzing error impacts on model accuracy.

## Contribution

It proposes a new heuristic based on atomic displacement to assess relaxation effects and evaluates how error size influences cluster expansion reliability.

## Key findings

- The heuristic effectively predicts when CE models are unreliable due to relaxation.
- Error size impacts CE accuracy more than error distribution.
- Model complexity correlates with the number of significant terms in CE models.

## Abstract

Cluster expansion (CE) is effective in modeling the stability of metallic alloys, but sometimes cluster expansions fail. Failures are often attributed to atomic relaxation in the DFT-calculated data, but there is no metric for quantifying the degree of relaxation. Additionally, numerical errors can also be responsible for slow CE convergence. We studied over one hundred different Hamiltonians and identified a heuristic, based on a normalized mean-squared displacement of atomic positions in a crystal, to determine if the effects of relaxation in CE data are too severe to build a reliable CE model. Using this heuristic, CE practitioners can determine a priori whether or not an alloy system can be reliably expanded in the cluster basis. We also examined the error distributions of the fitting data. We find no clear relationship between the type of error distribution and CE prediction ability, but there are clear correlations between CE formalism reliability, model complexity, and the number of significant terms in the model. Our results show that the \emph{size} of the errors is much more important than their distribution.