Rigid Molecule Packing Similarity Index Based on Dual Quaternions

Mapping comparisons between periodic 3D rigid-molecule (or fixed molecular conformer) arrangements to a scalar value, typically referred to as a similarity index, is essential in fields such as crystal structure prediction (CSP) and molecular engineering. Evaluating crystal lattice similarity is crucial for filtering random structure pools and differentiating polymorphs. Here, we introduce a configurational similarity index based on dual quaternions to simultaneously capture relative rigid-molecule (fixed molecular-conformer) positions and orientations in crystal lattices. Within this framework, pairs of molecular configurations with known atomic correspondence are compared using a metric derived from the product of their dual quaternions. After validating the metric on water dimer configurations, we employ it to contrast clusters of molecules representing different molecular packing motifs. This pairwise comparison is condensed into a single index, Drt, which measures total crystal packing similarity. Using Drt to screen pools of randomly generated structures for oxalic acid, maleic hydrazide, and formamide, we successfully retrieve packing motifs similar to the experimental references. Upon geometry optimization, these identified structures converge to the corresponding optimized experimental configurations, highlighting the reliability of this metric for integration into CSP methodologies.