Fig. 2: Symmetry, local atomic geometry, and geometric structure comparisons in AFLOW-XtalFinder.

a Crystal isometries are calculated internally with AFLOW-SYM. The space groups and occupied Wyckoff positions are compared, revealing isopointal structures. b The local least-frequently occurring atom (LFA) geometries are computed and compared between structures. An example local LFA geometry (2-D projection) is shown for the quaternary Heusler structure (ABCD_cF16_216_c_d_b_a)^17,18, highlighting the closest neighbors (via solid lines) for each LFA type to the central Mg atom (purple). Shaded concentric circles indicate the tolerance threshold for capturing atoms in the coordination shell with a thickness of 10% of the distance from the central and connected atom. Local geometry vectors are compared against local geometries in other structures to determine mapping potential. c Two structures (\({{\mathbb{X}}}_{{\rm{ref}}}\) and \({{\mathbb{X}}}_{{\rm{test}}}\)) are mapped onto one another by expanding \({{\mathbb{X}}}_{{\rm{test}}}\) into a supercell and exploring commensurate lattice and origin choices with respect to \({{\mathbb{X}}}_{{\rm{ref}}}\). The yellow lattice (highlighted by the green box) is a potential match with \({{\mathbb{X}}}_{{\rm{ref}}}\). \({{\mathbb{X}}}_{{\rm{test}}}\) is transformed into the new representation (\({\widetilde{{\mathbb{X}}}}_{{\rm{test}}}\)), and the structures are quantitatively compared via the misfit criteria. The structures are evaluated via their lattice deviation (ϵ_latt), coordinate displacement (ϵ_coord), and figure of failure (ϵ_fail). Distances between mapped atoms (d^map) that are less than half the atom’s nearest neighbor (d_nn/2) are accounted for in the coordinate displacement (green dashed lines and arrows), while larger distances are described in the figure of failure (red dashed lines and arrows).