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20 pages, 2292 KiB

Open AccessArticle

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

by Henryk A. Witek and Rafał Podeszwa

Molecules 2024, 29(17), 4013; https://doi.org/10.3390/molecules29174013 - 24 Aug 2024

Viewed by 1651

We report an extensive tabulation of several important topological invariants for all the isomers of carbon

(5, 6)

-fullerenes

C_{n}

with n = 52–70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported [...] Read more.

We report an extensive tabulation of several important topological invariants for all the isomers of carbon

(5, 6)

-fullerenes

C_{n}

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. Full article

(This article belongs to the Special Issue Multiconfigurational and DFT Methods Applied to Chemical Systems—2nd Edition)

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27 pages, 3539 KiB

Open AccessArticle

Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

by Henryk A. Witek

Molecules 2021, 26(9), 2524; https://doi.org/10.3390/molecules26092524 - 26 Apr 2021

Cited by 8 | Viewed by 2794

Abstract

Multiple zigzag chains

Z (m, n)

of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently [...] Read more.

Multiple zigzag chains

Z (m, n)

Z (m, n)

. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the

Z (m, n)

multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size

⌈\frac{m}{2}⌉ \times ⌈\frac{m}{2}⌉

consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains

Z_{k} (m, n)

, i.e., derivatives of

Z (m, n)

with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains

Z (m, n)

and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains

Z (m, n)

. Full article

(This article belongs to the Special Issue Molecular Modeling: Advancements and Applications)

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14 pages, 1056 KiB

Open AccessFeature PaperArticle

Zhang–Zhang Polynomials of Ribbons

by Bing-Hau He, Chien-Pin Chou, Johanna Langner and Henryk A. Witek

Symmetry 2020, 12(12), 2060; https://doi.org/10.3390/sym12122060 - 11 Dec 2020

Cited by 12 | Viewed by 1845

Abstract

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

R b (n_{1}, n_{2}, m_{1}, m_{2})

, usually referred to [...] Read more.

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

R b (n_{1}, n_{2}, m_{1}, m_{2})

, 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

R b (n_{1}, n_{2}, m_{1}, m_{2})

: 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

O (k, m, n)

and oblate rectangles

O r (m, n)

. Full article

(This article belongs to the Special Issue Symmetry on the Genealogy of Conjugated Acyclic Polyenes ‑ Dedicated to the Two Active Mathematical Chemists, Diudea and Aihara)

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20 pages, 3110 KiB

Open AccessEditor’s ChoiceArticle

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

by Henryk A. Witek and Johanna Langner

Symmetry 2020, 12(10), 1599; https://doi.org/10.3390/sym12101599 - 25 Sep 2020

Cited by 10 | Viewed by 2238

Abstract

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

R b

, parallelograms M, vertically overlapping parallelograms

M v M

, horizontally [...] Read more.

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

R b

, parallelograms M, vertically overlapping parallelograms

M v M

, horizontally overlapping parallelograms

M h M

, and intersecting parallelograms

M x M

. 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 \times 2

determinant bearing close structural resemblance to analogous formulas for the number of Kekulé structures derived from the John-Sachs theory of Kekulé structures. Full article

(This article belongs to the Special Issue Symmetry on the Genealogy of Conjugated Acyclic Polyenes ‑ Dedicated to the Two Active Mathematical Chemists, Diudea and Aihara)

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47 pages, 6519 KiB

Open AccessArticle

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

by Henryk A. Witek and Jin-Su Kang

Symmetry 2020, 12(9), 1483; https://doi.org/10.3390/sym12091483 - 9 Sep 2020

Cited by 13 | Viewed by 2944

Abstract

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes C

_{n}

with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of [...] Read more.

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes C

_{n}

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

C l

, the first Herndon number

h_{1}

, 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. Full article

(This article belongs to the Section Chemistry: Symmetry/Asymmetry)

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