Chemical algebra III: Thermochemical approach to completely G-invariant distances

Abstract

The search for a definition of distances over sets of skeletal analogs (identified to G-Hilbert spaces of vector ligand parameters) is initiated from the algebraic formulation of the constant of stereogenic pairing equilibria (pairing product). A basic definition equation is devised from thermodynamical speculations. The equation is proved to have always a single potential distance solution D_p as soon as the pairing product is discriminating. The equation of D_p is constructed in order to satisfy three consistency requirements: completeG-invariance (arbitrary orientations selected to describe skeletal analogs do not affect the value of D_p); extension properties (D_p coincides with two standard completelyG-invariant distances or with the Euclidean distance in borderline cases); all the distance properties except, perhaps, the triangular inequality. The latter point remains challenging in general, and is computationally verified in some examples.