Characteristic Polynomial Expression and Integral Expression for Imaginary Bond–Bond Polarizability. Rules for London Susceptibility and for Ring Current

Abstract

The McWeeny theory for calculating magnetic properties of conjugated hydrocarbons is studied from a graph-theoretical point of view. Imaginary bond–bond polarizability is expressed in terms of characteristic polynomial. The expression clarifies the dependence of imaginary bond–bond polarizability on the structure. It is shown that the expression enables one easily to derive the equations for the quantity proved in two previous papers. The characteristic polynomial expression for imaginary bond–bond polarizability is rewritten in Coulson’s integral form. The rules for London susceptibility and for ring current are proved from the integral expression.

Some Observations on Triplet Ground-States in the Context of ‘Topological’ (HLPM) Ring-Currents in Conjugated Systems

“…Despite its ostensibly infrequent occurrence, we nevertheless emphasise that this 'triplet ground-state problem' does represent an intrinsic limitation of the HLPM approach. It is, furthermore, also a limitation that is necessarily inherent in other equivalent (though ostensibly different) methods [13][14][15][33][34][35][36][37][38] of calculating magnetic properties due to π-electron ring-currents -methods that are likewise founded on the Hückel molecular-orbital conventions. [11,12,39] When a triplet ground-state arises, topological ring-currents and bond-currents cannot be calculated by the HLPM method, [9,10,[13][14][15][16][17][18][19] and its equivalents.…”

“…[11,12,39] When a triplet ground-state arises, topological ring-currents and bond-currents cannot be calculated by the HLPM method, [9,10,[13][14][15][16][17][18][19] and its equivalents. [33][34][35][36][37][38] Infinite paramagnetism is formally to be predicted in such situations. [11,44,45] We conclude by drawing attention to an observation that Rouvray and one of the present authors (RBM) made in Refs.…”

“…This case arises when the last two π-electrons to be distributed in the course of applying the Aufbau process [20][21][22][23] have to be assigned to a pair of doubly degenerate molecular-orbitals (that is, a pair of repeated eigenvalues of the molecular graph [8,17,21] ) -thereby necessarily resulting in two unpaired spins and thus a triplet groundstate. This is where there is a problem with the HLPM approach -a problem that is shared with other ostensibly different formalisms [33][34][35][36][37][38] that also rely on the simple Hückel model. [11,12,39] How this comes about is explained in the next sub-section.…”

“…Triplet ground-states are, accordingly, not catered for in the HLPM method, nor are they by other methods [33][34][35][36][37][38] that are based on the Hückel approach. As a very special case of the more-general scenario outlined here, the first explicit encounter with this phenomenon was in the context of the [4n]-annulenes without 'bond-alternation', studied by Pople and Untch [44] in their classic paper of more than fifty years ago.…”

“…The overall aim of the present communication is to describe the problems that a triplet ground-state brings about, to record some specific examples of this problem's occurrence, and to emphasise that it represents an intrinsic limitation of the HLPM approach. It is, furthermore, also a limitation that is necessarily inherent in other equivalent (though ostensibly different) methods [13][14][15][33][34][35][36][37][38] of calculating magnetic properties due to π-electron ring-currents -methods that are likewise founded on the Hückel molecular-orbital conventions. [11,12,39]…”