Strain Functionals: A Symmetry-adapted Set of Descriptors for Characterizing Atomic Geometries and Potential Energy Functions

We demonstrate the use of a complete and symmetry-adapted approach to characterize atomic geometries. A Gaussian kernel is used to map discrete atomic quantities to continuous fields and the local geometry is characterized using n^th order central-moments of the atomic neighborhood. The Gaussian kernel guarantees that the mapping is continuous and smooth. Rotationally invariant metrics, referred to as strain functional descriptors, are derived from these n^th order central-moments using Clebsch-Gordan coupling. Descriptors derived from a 6^th order moment expansion can distinguish between different crystal structures and identify defects such as dislocations, stacking faults, and twins at finite temperature. Combined with dimensionality reduction techniques and clustering algorithms, these descriptors are used to analyze molecular dynamics simulations of high strain-rate compression of Cu, Ta, and Ti. We will also demonstrate that the strain functional descriptors form a complete, orthogonal, and minimal basis for characterizing inter-atomic potentials. Formulation of inter-atomic potentials for Cu and Ta in the strain functional basis will be presented.