Some graph-theoretical aspects of simple ring current calculations on conjugated systems

Abstract

In this paper we examine the simple theory of π -electron ring currents in conjugated systems (devised originally by London (1937) and extended by People (1958) and McWeeny (1958)), with particular reference to the topological or graph-theoretical aspects of it (all the necessary graph-theoretical ideas and terminology are explained in the text). There is a close correspondence between the adjacency matrix of the graph representing the σ-bond skeleton of the carbon atoms comprising a given conjugated system, and the secular equations which arise in the theory (a relation now well known to be common to all formalisms based on Hückel ‘topological’ molecular orbitals), but in addition we here emphasize that several other graph-theoretical ideas–notably those concerning circuits and spanning trees–specifically underlie the ring current concept. In this connexion, the question of whether any given molecular graph is semi-Hamiltonian or non-Hamiltonian is of prime importance, and it is pointed out that a unitary transformation originally proposed by McWeeny applies to semi-Hamiltonian molecular graphs, whereas one recently devised by Gayoso & Boucekkine can be applied to any simple, connected graph–as also can an explicit ring current formula (based on the London–McWeeny theory) just published by the present author. These ideas are illustrated by some simple numerical calculations, and an example is given of a conjugated system (decacyclene) whose molecular graph is apparently non-Hamiltonian. It is emphasized that although much graph theory is inherent in the ring current concept, the ring current index itself is not a completely topological quantity–even when a purely topological wavefunction (such as the simple Hückel one) has been used to calculate it.