A topological approach to predicting properties of infinite polymers.: Part VI.: Rational formulas for the normalized Wiener index and a comparison with index J

A general formula W-infinity = d/[3(v + mu (0))] is derived for the normalized Wiener index of infinite polymers, W-infinity. It makes possible the calculation of this important polymer descriptor directly from simple structural information: the number of atoms (v) and rings (mu (0)) in the repeating polymer cell, and the topological distance d (the number of bonds along the shortest path) between the corresponding pairs of equivalent atoms in two neighboring monomer units. In view of the previously shown ([1] D. Bonchev, O. Mekenyan, A topological approach to the calculation of the pi -electron energy and energy gap of infinite conjugated polymers, Z. Naturforsch. 35a (1980) 739-747; [2] D. Bonchev, O. Mekenyan, O.E. Polansky, A topological approach to the predicting of the electron energy characteristics of conjugated infinite polymers. II. PPP-calculations. Z. Naturforsch. 36a (1981) 643-646; [3] D. Bonchev, O. Mekenyan, O.E. Polansky, A topological approach to the predicting of the electron energy characteristics of conjugated infinite polymers. III. The influence of some structural modifications of polymers, Z. Naturforsch. 36a(1981) 647-650; [4] O. Mekenyan, S. Dimitrov, D. Bonchev, Graph-theoretical approach to the calculation of physicohemical properties of polymers, fur. Polym. J. 19 (1963) 1185-1193, [5] D. Bonchev, O. Mekenyan, V. Kamenska, A topological approach to the modeling of polymer properties (the TEMPO method), J. Math. Chem. 11 (1992) 107-132) high correlation of W-infinity with the total pi -electron energy and physicochemical properties of polymers, this result might be regarded as a step towards the design of polymers with tailored properties. The approach is illustrated with examples of acyclic polymers, and of polymers with isolated rings, with cata-condensed or peri-condensed rings. The reciprocal relationship with the similar index J(infinity) is pointed out and an approximate hyperbolic dependence is presented between these two indices. (C) 2001 Elsevier Science B.V. All rights reserved.