Local, Non-Redundant Molecular Coordinates from Disjoint Primitive Families: Symmetry-Adapted Construction and Applications

Internal coordinates express molecular motion in chemical terms, but primitive functions are redundant, and global delocalization mixes coordinate families. We introduce extended symmetry-adapted local coordinates (ESALCs), which organize diverse chemical primitives into disjoint families and complete, nonredundant bases. The internal-coordinate form, ESALC-I, assigns general primitives to perceived chemical domains, forms local symmetry-adapted combinations, selects exact rank through the Wilson B matrix, and retains complete molecular-symmetry subspaces within a basin of fixed topology and symmetry. When bond assignments become ambiguous, the Cartesian form, ESALC-C, supplies a full-system basis with overall translations and rotations removed. Tests on covalent, fragment, and metal-contact systems establish rank, analytic derivatives, and symmetry. Stationary-Hessian comparisons for cyclobutane and norbornane show low coupling with chemically interpretable ESALC-I families. Full geometry optimizations demonstrate use with established programs.