Topological index as applied to π-electronic systems. III. Mathematical relations among various bond orders

Mathematical relations among the Coulson (pCl), Ham and Ruedenberg (pHRl), Pauling (pPl), and topological (pTl) bond orders have been derived by combining the enumeration technique of the graph theory and the complex integral developed by Coulson and Longuet-Higgins. If one defines a function FG,l(y) for bond l of graph G as FG,l(y) =Δr,s(iy)/Δ (iy)(l=rs¯) the following relations are proved for alternant hydrocarbons: pCl=2/πℱ∞0FG,l(y) dy, pHRl=pPl=FG,l(0), pTl=FG,l(1) for tree graphs, pTl=FG,l(1) for nontree graphs. Linear relationship pCl∝pTl+ApPl (A≪1) is also proved. Abnormal bond orders for 4n-membered ring systems are discussed. Graphical methods for decomposing determinants and adjuncts are proposed.