Zhang–Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids Rbn1,n2,m1,m2, usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of Rbn1,n2,m1,m2: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes Ok,m,n and oblate rectangles Orm,n.

Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms

We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekuléan benzenoids that can be obtained by overlapping two parallelograms: generalized ribbons Rb, parallelograms M, vertically overlapping parallelograms MvM, horizontally overlapping parallelograms MhM, and intersecting parallelograms MxM. All formulas have the form of multiple sums over binomial coefficients. Three of the formulas are given with a proof based on the interface theory of benzenoids, while the remaining two formulas are presented as conjectures verified via extensive numerical tests. Both of the conjectured formulas have the form of a 2×2 determinant bearing close structural resemblance to analogous formulas for the number of Kekulé structures derived from the John-Sachs theory of Kekulé structures.

ZZ Polynomials for Isomers of (5,6)-Fullerenes Cn with n = 20–50

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes Cn with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of Kekulé structures K, the Clar number Cl, the first Herndon number h1, the total number of Clar covers C, and the number of Clar structures. The presented results should be useful as benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes and at generation of resonance structures for valence-bond quantum-chemical calculations.

Air stable high-spin blatter diradicals: non-Kekulé versus Kekulé structures

1,2,4-Benzotriazinyl based stable diradicals possess singlet ground states and small singlet–triplet energy gaps with a thermal accessible triplet excited state.

An examination of students' perceptions of the Kekulé resonance representation using a perceptual learning theory lens

Students in chemistry often demonstrate difficulty with the principle of resonance. Despite many attempts to mitigate this difficulty, there have been few attempts to examine the root cause of these issues. In this study, students were assessed for their perception of Kekulé structures based on perceptual learning theory, which is grounded in cognitive mechanisms of visual perception. The data from this assessment shows that students are perceiving inappropriate clues from this representation, which infers that the image itself might be an impediment to learning about resonance. Employment of a metarepresentational competence approach was used to address these misperceptions.