Geometrical and algebraic approach to central molecular chirality: a chirality index and an Aufbau description of tetrahedral molecules

On the basis of empirical Fischer projections, we develop an algebraic approach to the central molecular chirality of tetrahedral molecules. The elements of such an algebra are obtained from the 24 projections which a single chiral tetrahedron can generate in S and R absolute configurations. They constitute a matrix representation of the O(4) orthogonal group. According to this representation, given a molecule with n chiral centres, it is possible to define an "index of chirality chi = {n, p}", where n is the number of stereogenic centres of the molecule and p the number of permutations observed under rotations and superimpositions of the tetrahedral molecule to its mirror image. The chirality index not only assigns the global chirality of a given tetrahedral chain, but indicates also a way to predict the same property for new compounds, which can be built up consistently.