An Algebraic Description of Stereochemical Correspondence ( dynamic stereochemistry / group theory )

It is shown that a stereochemical correspondence between two molecular systems can be represented by a commuting diagram of the point groups and permutation groups involved. The effect of the diagrammatic condition on the mappings of the cosets, double cosets, subgroup lattice, and double coset algebra determined for the two molecular systems by point group to permutation group homomorphisms is detailed. Chemical similarities implied by a stereochemical correspondence are indicated, and six examples are provided and discussed. The concept of stereochemical correspondence (SCC) has been put forth to describe an intrinsic similarity between molecular systems which is independent of widely different structural properties (1). In this article we will provide an algebraic description of this relationship and some further examples. The concept will be generalized somewhat from the previous definition, and the chemical and mathematical properties related by a SCC will be discussed. Briefly, the problem here is to assign an algebraic structure of some kind to a molecular system and to see if the similarity or correspondence between two molecular systems can be expressed as some mapping of the algebraic structures. The fundamental importance of this sort of procedure of assigning mathematical structures and mappings between them has been recognized by mathematicians. The study of category theory is the study of such procedures (2). Well-known examples can be found in the field of algebraic topology, in which mappings between topological spaces can be represented by homomorphisms of groups. As an example, the homeomorphism between a donut and a coffee cup can be expressed as a homomorphism between homology groups or fundamental groups (3). The rationale is that the group homomorphisms are often easier to establish and the groups themselves describe intrinsic properties (topological invariants) of the topological spaces (such as the "hole" in the donut or coffee cup). The motivation behind the work described in this article is to apply similar considerations to molecular systems in a search for intrinsic properties and relationships. Algebraic description of molecular systems A typical problem in dynamic stereochemistry is the study of the degenerate rearrangements of a molecule with skeletal symmetry group P. The set of all rearrangements also form a group G, which will often be one of the symmetric groups or a direct product of symmetric groups and may or may not include an operation of overall coordinate inversion (4, 5). Of particular importance to the present discussion is the homomorphism of P into G.