Kekulé Counts, Clar Numbers, and ZZ Polynomials for All Isomers of (5,6)-Fullerenes C₅₂–C₇₀

Abstract

We report an extensive tabulation of several important topological invariants for all the isomers of carbon (5,6)-fullerenes Cn with n = 52–70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported in the form of the corresponding Zhang–Zhang (ZZ) polynomials. The ZZ polynomials appear to be distinct for each isomer cage, providing a unique label that allows for differentiation between various isomers. Several chemical applications of the computed invariants are reported. The results suggest rather weak correlation between the Kekulé count, Clar count, Clar number invariants, and isomer stability, calling into doubt the predictive power of these topological invariants in discriminating the most stable isomer of a given fullerene. The only exception is the Clar count/Kekulé count ratio, which seems to be the most important diagnostic discovered from our analysis. Stronger correlations are detected between Pauling bond orders computed from Kekulé structures (or Clar covers) and the corresponding equilibrium bond lengths determined from the optimized DFTB geometries of all 30,579 isomers of C20–C70.

Keywords

• fullerene isomers
• isomer stability
• Kekulé count
• Clar number
• Zhang–Zhang polynomial (aka ZZ polynomial or Clar covering polynomial)
• bond order