A model for combinatorial organic chemistry

Abstract

The set of coset representations, CR's, of a group G, [G(/G1), G(/G2), ..., G(/Gs)], where G1 = [I], Gs = G, the marks, m(ij) of subgroup Gj on a given G(/Gi), 1 < or = i < or = s, and the subduction of G(/Gi) by Gj, j < or = i, G(/Gi) precipitates to Gj, are essential tools for the enumeration of stereoisomers and their classification according to their subgroup symmetry (Fujita, S. Symmetry and Combinatorial Enumeration in Chemistry; Springer-Verlag: Berlin, 1991). In this paper, each G(/Gi) is modeled by a set of colored equivalent configurations (called homomers), H = h1, h2, ..., hr, r = G/Gi, such that a given homomer, h(k), remains invariant only under all g epsilon Gi, where g is an element of symmetry. The resulting homomers generate the corresponding set of marks almost by inspection. The symmetry relations among a set H can be conveniently stored in a Cayley-like diagram (Chartrand, G. Graphs as Mathematical Models; Prindle, Weber and Schmidt Incorporated: Boston, MA, 1977; Chapter 10), which is a complete digraph on r vertices so that an arc from vertex v(i) to vertex v(j) is colored with the set Sij of symmetry elements such that h(i)[g(if)]-->h(j),g(ij) epsilon Sij. In addition, each vertex, v(i), is associated with a loop that is colored with a set Sii so that g(ii) epsilon Sii stabilizes h(i). A Cayley-like diagram of a given CR, G[G(/Gi)], leads to graphical generation of G(/Gi) precipitates to Gj for all values of j and also to all m(ij)'s. Several group-theoretical results are rederived and/or became more envisagable through this modeling. The approach is examplified using C2, C3, D2, T, and D3 point groups and is applied to trishomocubane, a molecule that belongs to the D3 point group.