A topological index for the totalπ-electron energy

Abstract

A modified topological index$\tilde Z_G$ is proposed to be defined as

$$\tilde Z_G = \sum\limits_{k = 0}^{[N/2]} {( - 1)^k } a_{2k} $$

for characterising theπ-electronic system of a conjugated hydrocarbonG withN carbon atoms, wherea _2k is the coefficient of the characteristic polynomial ofG defined as

$$P_G (X) = ( - 1)^N \det |A - XE| = \sum\limits_{k = 0}^N { a_k X^{N - k} } $$

with an adjacency matrixA and the unit matrixE.

$\tilde Z_G$ is identical toZ _G for a tree graph, or a chain hydrocarbon.Z _G increases with a (4n+2)-membered ring formation and decreases with a 4n-membered ring formation. The totalπ-electron energyE _π of the Hückel molecular orbital is shown to be related with$\tilde Z_G$ asE _π =Cln$\tilde Z_G$. With this relation generalised and extended Hückel rules for predicting the stability of an arbitrary network are proved.