Permutation-invariant distance between atomic configurations

TL;DR

This paper introduces a permutation-invariant distance metric for atomic configurations that directly compares structures without reducing dimensionality, accounting for permutations and rotations efficiently, and is useful for structural analysis and fingerprint evaluation.

Contribution

A novel, direct, and efficient permutation-invariant distance metric for atomic configurations that avoids complex minimizations and is applicable to structural analysis and fingerprint validation.

Findings

The proposed distance is a true metric on atomic configurations.

It effectively discriminates and classifies local atomic structures.

The method simplifies structural comparison by avoiding fingerprint reduction.

Abstract

We present a permutation-invariant distance between atomic configurations, defined through a functional representation of atomic positions. This distance enables to directly compare different atomic environments with an arbitrary number of particles, without going through a space of reduced dimensionality (i.e. fingerprints) as an intermediate step. Moreover, this distance is naturally invariant through permutations of atoms, avoiding the time consuming associated minimization required by other common criteria (like the Root Mean Square Distance). Finally, the invariance through global rotations is accounted for by a minimization procedure in the space of rotations solved by Monte Carlo simulated annealing. A formal framework is also introduced, showing that the distance we propose verifies the property of a metric on the space of atomic configurations. Two examples of applications are proposed. The first one consists in evaluating faithfulness of some fingerprints (or descriptors), i.e. their capacity to represent the structural information of a configuration. The second application concerns structural analysis, where our distance proves to be efficient in discriminating different local structures and even classifying their degree of similarity.