Graph Theory of Ring-Current Diamagnetism

Abstract

Various magnetic criteria of aromaticity have been proposed so far, because they are easily calculated and applicable to a variety of cyclic π-systems. Many researchers, however, are reluctant to accept all or some of them. Our graph theory of ring-current diamagnetism revealed serious flaws in familiar magnetic criteria of aromaticity. Physically meaningful information on aromaticity can nevertheless be extracted from the ring-current diamagnetism. In particular, magnetic resonance energy (MRE), derived from the ring-current diamagnetic susceptibility, is interpretable as a kind of aromatic stabilization energy (ASE) and helps in consistently interpreting energetic and magnetic criteria of aromaticity. MRE and related quantities can be determined without reference to any hypothetical polyene-like structure but with a reasonable assumption that, when a magnetic field is applied to a cyclic π-system, a current is induced in each circuit in proportion to the ASE arising from the circuit. Unlike other conventional magnetic indexes, they are independent of the size and shape of the π-system.

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]…”