Structure comparison algorithms

Tasks 9.1.1-9.1.3. References: pymatgen StructureMatcher and analysis.structure_matcher; Niggli (1928); Kabsch (1976) and Horn (1987).

Matcher

1. Both structures are expanded to P1 and their lattices are Niggli reduced (moyo through reduce_lattice). The fractional coordinates are moved by the Cartesian mapping, so the reduction never changes geometry.
2. An integer matrix M is searched with M * L2 ~ L1: every row of L1 is expressed in the basis of L2, rounded to the nearest integer vector, and accepted when the fractional error is at most ltol. All combinations with the required determinant are kept (up to 32, ordered by a length and angle score).
3. For each candidate, the second cell is mapped into the first and a translation is searched by anchoring the first four atoms of the first structure on every species-compatible atom of the second. Atoms are then assigned greedily to the nearest unmatched partner within stol; the assignment must cover every atom.
4. The best RMS displacement over all candidates is returned. Supercell matching uses a determinant equal to the atom-count ratio (and the volume ratio must agree), and the smaller structure is expanded before matching.

Tolerances default to ltol = 0.2, stol = 0.3 Angstrom and angle_tol = 5 degrees. Species are compared exactly, by canonical element, or ignored.

RMSD and alignment

The periodic RMSD uses the minimum image of the first cell for the fractional difference of each mapped pair. Kabsch alignment uses Horn's quaternion method: the 4x4 matrix built from the cross-covariance is diagonalized with Jacobi rotations and its largest eigenvector gives the rotation. The result is deterministic and needs no external linear-algebra routine.

Equality classes

Structures are visited in input order and joined to the first class whose representative matches; this gives a deterministic partition independent of hash iteration order.