On representing chemical environments

TLDR

The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. The study reviews recent methods for representing atomic neighborhood environments and proposes the Smooth Overlap of Atomic Positions approach to overcome limitations in fidelity and convergence. The authors evaluate existing descriptors and introduce the Smooth Overlap of Atomic Positions method, then test these representations by fitting potential energy surfaces of small silicon clusters and bulk crystal. The analysis reveals that many popular descriptors are special cases of a general basis‑function expansion, that higher angular wave numbers are required for faithful representation as neighbor count rises, and that the proposed Smooth Overlap of Atomic Positions method offers a more efficient similarity measure closely related to invariant descriptors.

Abstract

We review some recently published methods to represent atomic neighborhood environments, and analyze their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighborhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbors increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether different approach, called Smooth Overlap of Atomic Positions, that sidesteps these difficulties by directly defining the similarity between any two neighborhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.