Ring-Current Maps for Benzenoids: Comparisons, Contradictions, and a Versatile Combinatorial Model

Abstract

As a key diagnostic property of benzenoids and other polycyclic hydrocarbons, induced ring current has inspired diverse approaches for calculation, modeling, and interpretation. Grid-based methods include the ipsocentric ab initio calculation of current maps, and its surrogate, the pseudo-π model. Graph-based models include a family of conjugated-circuit (CC) models and the molecular-orbital Huckel-London (HL) model. To assess competing claims for physical relevance of derived current maps for benzenoids, a protocol for graph-reduction and comparison was devised. Graph reduction of pseudo-π grid maps highlights their overall similarity to HL maps, but also reveals systematic differences. These are ascribed to unavoidable pseudo-π proximity limitations for benzenoids with short nonbonded distances, and to poor continuity of pseudo-π current for classes of benzenoids with fixed bonds, where single-reference methods can be unreliable. Comparison between graph-based approaches shows that the published CC models all shadow HL maps reasonably well for most benzenoids (as judged by L 1 -, L 2 -, and L ∞ -error norms on scaled bond currents), though all exhibit physically implausible currents for systems with fixed bonds. These comparisons inspire a new combinatorial model (Model W) based on cycle decomposition of current, taking into account the two terms of lowest order that occur in the characteristic polynomial. This improves on all pure-CC models within their range of applicability, giving excellent adherence to HL maps for all Kekulean benzenoids, including those with fixed bonds (halving the rms discrepancy against scaled HL bond currents, from 11% in the best CC model, to 5% for the set of 18 360 Kekulean benzenoids on up to 10 hexagonal rings). Model W also has excellent performance for open-shell systems, where currents cannot be described at all by pure CC models (4% rms discrepancy against scaled HL bond currents for the 20112 non-Kekulean benzenoids on up to 10 hexagonal rings). Consideration of largest and next-to-largest matchings is a useful strategy for modeling and interpretation of currents in Kekulean and non-Kekulean benzenoids (nanographenes).

Postscript on Viable Ground-States for Calculating Topological π-Electron Ring-Currents Using the Hückel–London–Pople–McWeeny Model

“…In this process, each orbital in a shell of p orbitals that has a total occupation of q electrons is assigned an occupation number, q/p, which is possibly fractional; on this model, the resulting electronic occupation in the shell in question could, in some sense, be thought of as being, thereby, 'smeared' or 'averaged' over all the orbitals (p in number) in that shell. [42] This invites future investigation.…”

“…[11], arise because the McWeeny formalism [40,41] -the equations of which are directly incorporated into the HLPM approach [1][2][3] -explicitly assumes that closed-shell systems are being dealt with. [40,41] We point out, however, that Fowler et al [39] have recently documented a scheme in which a calculation based on the Hückel [24] -London [43] method may be carried out in what they call [42] a 'fractional-occupation approximation'. In this process, each orbital in a shell of p orbitals that has a total occupation of q electrons is assigned an occupation number, q/p, which is possibly fractional; on this model, the resulting electronic occupation in the shell in question could, in some sense, be thought of as being, thereby, 'smeared' or 'averaged' over all the orbitals (p in number) in that shell.…”

Partitioning Hückel–London Currents into Cycle Contributions

“…The dual nature of HL theory as a graph theoretical method based on molecular-orbital theory, makes it interesting to compare HL results with conjugated-circuit models on the one hand, and with more sophisticated wavefunction and density functional approaches to electronic structure on the other. The relevance of the present graph-theoretical investigation to ab initio calculation is that HL currents already typically mimic pseudo-π currents [43], which in turn are usually excellent mimics for current maps derived from full ab initio and density functional calculations. Some systematic exceptions to this statement are discussed in [43].…”

“…Conjugated-circuit models have an attractive simplicity, but have crucial drawbacks for non-Kekulean systems, where they predict zero current, and for Kekulean systems with fixed bonds, where they predict 'dead zones' of vanishing current [43][44][45]. The current maps from conjugated-circuit models can be seen as approximate versions of HL current maps in which only certain 'important' cycles have been selected and given model-dependent weightings.…”

“…Comparison of HL and CC currents in benzenoids by cycle size has allowed us to evaluate these selection and weighting schemes, and to propose a new model, also based on matchings, that gives an approximation to HL currents for both Kekulean and non-Kekulean benzenoids that is better than any of the published CC models [43]. The dual nature of HL theory as a graph theoretical method based on molecular-orbital theory, makes it interesting to compare HL results with conjugated-circuit models on the one hand, and with more sophisticated wavefunction and density functional approaches to electronic structure on the other.…”