  Extracting relevant information from atomistic simulations relies on a
complete and accurate characterization of atomistic configurations. We present
a framework for characterizing atomistic configurations in terms of a complete
and symmetry-adapted basis, referred to as strain functionals. In this approach
a Gaussian kernel is used to map discrete atomic quantities, such as number
density, velocities, and forces, to continuous fields. The local atomic
configurations are then characterized using nth order central moments of the
local number density. The initial Cartesian moments are recast unitarily into a
Solid Harmonic Polynomial basis using SO(3) decompositions. Rotationally
invariant metrics, referred to as Strain Functional Descriptors (SFDs), are
constructed from the terms in the SO(3) decomposition using Clebsch-Gordan
coupling. A key distinction compared to related methods is that a minimal but
complete set of descriptors is identified. These descriptors characterize the
local geometries numerically in terms of shape, size, and orientation
descriptors that recognize n-fold symmetry axes and net shapes such as
trigonal, cubic, hexagonal, etc. They can easily distinguish between most
different crystal symmetries using n = 4, identify defects (such as
dislocations and stacking faults), measure local deformation, and can be used
in conjunction with machine learning techniques for in situ analysis of finite
temperature atomistic simulation data and quantification of defect dynamics.
