Isomer numbers of nonrigid molecules. The cyclohexane case

A generalized wreath product method for the enumeration of stereo and position isomers of polysubstituted organic compounds

“…The group of permutations for the enumeration of isomers at room temperature should contain the point group operations and the permutations induced by any additional internal degrees of freedom. Leonard [6,7] enumerated the isomers of nonrigid cyclohexane molecules by including the permutations induced by a ring-flip operator in the permutation group of the rigid molecule. Davidson [8] investigated the three-fold rotors in alkanes, two-fold rotors in polyphenyls and certain higher-fold rotors in metal complexes from the standpoint of their cycle indices with what is known as the wreath product also referred to as the Kranz group.…”

Isomer counting for fluctional molecules

“…Consequently, the full covering group is a subgroup of Dqh-It can be considered to be either Dq or Cgv (these are isomorphic to each other). For isomer counting, the Dg group should be used since it is the maximal rotational subgroup of D6h-This is also the covering group for a hypothetical allplanar cyclohexane structure (with the hydrogens and carbons all lying in the same plane) (5). De is both the full covering group and the proper rotation group for both of these structures.…”

“…The rotation of the CH3 groups relative to one another introduces an isodynamic <?%. The nonrigid group is thus (6) S -Sc = G3 A DSd (5) and has an order of 36. This is the full covering group, since the protons do not all lie on a line or a plane.…”

Pseudo-point-group approach to stereochemistry and stereoisomerism of cyclohexane derivatives

“…Leonard et a1.2~'3~ have reported an enumeration of cyclohexanes by combining the Polya-Redfield theorem with extended point groups. An alternative demonstration reported by Flurry4),5) is based on an isodynamic operator (or the corresponding group F), which was presented by Altmann.6~ Although these two methods have assigned the flipping cyclohexane itself to the group D3dR6 X2), 3) or to the group F x D3d4) in place of the conventional D6h, they have not identified the symmetries of cyclohexane derivatives (Item 1).…”

“…[3] or [4] D6h(/G) t D3d = 2D3d(/G), [5] where G is an anisoenergetic subgroup. Isoenergetic cases have been characterized by two isoenergetic conformers (Q for Type II; A for Type I; or Q and Q for Type I'), as shown in Table I.…”