Nonhanded chirality in octahedral metal complexes

Chiral molecules can either be handed (i.e., "shoes") or nonhanded ("potatoes"). The only chiral ligand partition for tetrahedral metal complexes (or for a tetrahedral carbon atom such as that found in amino acids and other chiral biological molecules) is the fully unsymmetrical degree 6 partition (1(4)), which leads to handed metal complexes of the type MABCD with a lowest-degree chirality polynomial consisting of the product of all six possible linear factors of the type (s(i)-s(j)) where 1 < or = i,j < or = 4. The lowest-degree chiral ligand partitions for octahedral metal complexes are the degree 6 partitions (31(3)) and (2(3)) leading to handed chiral metal complexes of the types fac-MA(3)BCD and cis-MA(2)B(2)C(2). The form of the lowest-degree chirality polynomial for the (31(3)) chiral ligand partition of the octahedron resembles that of the (1(4)) chiral ligand partition of the tetrahedron, likewise with four different ligands. However, the form of the lowest-degree chirality polynomial for the (2(3)) chiral ligand partition of the octahedron corresponds to the square of the chirality polynomial of the (1(3)) chiral ligand partition of the polarized triangle, which likewise has three different ligands. Ligand partitions for octahedral metal complexes such as (2(2)1(2)), (21(4)), and (1(6)), which are less symmetrical than the lowest-degree chiral ligand partitions (31(3)) and (2(3)), lead to chiral octahedral metal complexes which are nonhanded. In such complexes, pairs of enantiomers can be interconverted by simple ligand interchanges without ever going through an achiral intermediate.