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Open AccessArticle

Remarkably Efficient [4+4] Dimerization of [n]-Cyclacenes

by Ankit Somani, Divanshu Gupta and Holger F. Bettinger

Chemistry 2025, 7(2), 62; https://doi.org/10.3390/chemistry7020062 - 12 Apr 2025

Cited by 6 | Viewed by 1742

Cyclacenes with the general formula C_4nH_2n are cyclic analogs of acenes. Acenes are well-known for their high reactivity, which increases with the number of fused benzene rings. The cyclic strain, absence of a Clar sextet, and diradical or polyradical nature are expected to render cyclacenes highly reactive under ambient conditions. Their primary decomposition pathway is anticipated to involve dimerization or polymerization. We explore the reaction pathway of the [_π4_s + _π4_s] dimerization of [n]-cyclacenes for 6 ≤ n ≤ 20 by density functional theory (DFT) using spin-unrestricted and thermally-assisted-occupation (TAO) formalisms. Computational analysis predicts a stepwise reaction mechanism that starts with the formation of a van der Waals complex and proceeds through a transition state to an intermediate with a single new C–C bond and two unsaturated valences. A subsequent second transition state results in the formation of the dimerization product. However, for smaller cyclacenes (n < 10), neither the van der Waals complex nor the first transition state is involved, and the intermediate is formed without a barrier. The largest [20]-cyclacene investigated exhibits the highest barriers for these processes. However, with a barrier as low as 3.9 kcal/mol at the UB3LYP-D3(BJ)/6-31G(d) level of theory, dimerization is anticipated to occur very rapidly. Full article

(This article belongs to the Special Issue Open-Shell Systems—a Memorial Issue Dedicated to Professor Masayoshi Nakano)

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20 pages, 2292 KB

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

Cited by 1 | Viewed by 2347

Abstract

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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11 pages, 1694 KB

Open AccessArticle

The Interplay between Diradical Character and Stability in Organic Molecules

by Vaska Petakova, Miroslava Nedyalkova, Joanna Stoycheva, Alia Tadjer and Julia Romanova

Symmetry 2021, 13(8), 1448; https://doi.org/10.3390/sym13081448 - 7 Aug 2021

Cited by 2 | Viewed by 4642

Abstract

The number of scientific papers on the unique properties and the potential for various applications of compounds with a diradical character is growing constantly. The diradical character enhances and even engenders certain desired optical properties and its modulation is a modern molecular design strategy. Nowadays, molecules with a non-zero diradical character are regarded as promising materials for new-generation and highly efficient solar cells and photonics devices. What is the price, however, of the unique properties of open-shell compounds? Alongside all the benefits, the diradical character is usually associated with low stability and high reactivity—unwanted molecular qualities for practical purposes. Thus, from a fundamental and applied point of view, it is important to investigate the correlation between the diradical character and laboratory stability, which is the goal of the present paper. Here, we report a combined quantum–chemical study (conceptual DFT and spin-projected HF theory) and multivariate analysis of the diradical character of a series of o- and p-quinomethides, for the stability of which experimental data are available. Our results reveal that a compromise between the diradical character and laboratory stability of a molecule is feasible and that the relationship between these two quantities can be understood in the framework of Clar’s sextet theory. Full article

(This article belongs to the Special Issue Quantum Chemistry as Applied to Molecular Systems)

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14 pages, 1056 KB

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 2156

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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47 pages, 6519 KB

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 15 | Viewed by 3999

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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15 pages, 1942 KB

Open AccessArticle

Similarity and a Duality for Fullerenes

by Jennifer J. Edmond and Jack E. Graver

Symmetry 2015, 7(4), 2047-2061; https://doi.org/10.3390/sym7042047 - 6 Nov 2015

Viewed by 6283

Abstract

Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are partitioned into similarity classes. In this expository article, we illustrate how the values of two important fullerene parameters can be deduced for all fullerenes in a similarity class by computing the values of these parameters for just the three smallest representatives of that class. In addition, it turns out that there is a natural duality theory for similarity classes of fullerenes based on one of the most important fullerene construction techniques: leapfrog construction. The literature on fullerenes is very extensive, and since this is a general interest journal, we will summarize and illustrate the fundamental results that we will need to develop similarity and this duality. Full article

(This article belongs to the Special Issue Symmetry and Duality)

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11 pages, 286 KB

Open AccessArticle

How to Find the Fries Structures for Benzenoid Hydrocarbons

by Arkadiusz Ciesielski, Tadeusz M. Krygowski and Michał K. Cyrański

Symmetry 2010, 2(3), 1390-1400; https://doi.org/10.3390/sym2031390 - 6 Jul 2010

Cited by 25 | Viewed by 8714

Abstract

An efficient algorithm leading to the Fries canonical structure is presented for benzenoid hydrocarbons. This is a purely topological approach, which is based on adjacency matrices and the Hadamard procedure of matrix multiplication. The idea is presented for naphthalene, as an example. The Fries canonical-structures are also derived for anthracene, coronene, triphenylene, phenanthrene, benz[a]pyrene, and one large benzenoid system. The Fries concept can be convenient for obtaining Clar structures with the maximum number of sextets, which in turn effectively represent π-electron (de)localization in benzenoid hydrocarbons. Full article

(This article belongs to the Special Issue Aromaticity and Molecular Symmetry)

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