Entropy in dissimilarity and chirality measures

Within the prospect of quantifying the geometrical dissimilarity of molecular models on the basis of a thermodynamical formalism, the algebra of stereogenic pairing equilibria is reviewed and applied to molecular geometry: developing Rassat's proposition, an interaction energy of two figures F and F is taken as proportional tod _H ^Emphasis>/2 (F, F), whered _H denotes the Hausdorff distance. IfG is a group of rotations in E _n the geometrical version of the general equation (E) of the chemical algebra defines a distance extensionD _p(F,F) ofd _H(F,F), which is independent of the orientations of F and F, and where the coefficientp is interpreted as the reciprocal of a temperature-like parameter:p 1/T. At K (p = ), no formal entropy contributes to the definition of the uniform distanceD . At K (p = 0), the discrimination between homo- and hetero-pairing of figures by the harmonic distance Do is averaged over orientation states. Temperature-dependent chirality measuresc _p are derived fromD _p, andc is analogous to Mislow's chirality measure. If T and oT are normalized enantiomorphic triangles with coincident centroids inE ₂,c _p(T) =D _p (T, T) is calculated forp = 0 andp = , and discussed for 0 <p < . Finally, the Hausdorff interaction model is putatively related to energy profiles versus dihedral angle inmeso- anddl-molecules.