Double aromaticity of neutral, planar rings of carbon atoms is demonstrated through visualisation of the induced ring currents, mapped at the ipsocentric B3LYP/6-31G(d)//B3LYP/6-31G(d) level for species C(6) to C(30), with onset of delocalised current in the in-plane pi system at C(10)/C(11). Both in-plane and conventional out-of-plane pi systems have diatropic/paratropic current in accordance with the Hückel rule, with 4 m+2 occupation of the out-of-plane pi system taking precedence, as predicted by simple nesting of Frost-Musulin diagrams. The current-density maps show characteristic double-doughnut and double-track topographies for out-of-plane and in-plane ring currents, respectively, both governed by a common framework of angular momentum rules.

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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.

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