Witek Henryk A
Department of Applied Chemistry, Institute of Molecular Science, National Yang Ming Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300092, Taiwan.
Center for Emergent Functional Matter Science, National Yang Ming Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300092, Taiwan.
Molecules. 2021 Apr 26;26(9):2524. doi: 10.3390/molecules26092524.
Multiple zigzag chains Zm,n of length and width 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 encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains Zm,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 Zm,n multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size m2×m2 consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zkm,n, i.e., derivatives of Zm,n with a single attached polyacene chain of length . 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 Zm,n and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Zm,n.
长度为 且宽度为 的多个锯齿链Zm,n构成了一类重要的矩形规则石墨烯薄片。这些基本并缩苯类化合物的物理和化学性质可以与其各种拓扑不变量相关联，这些拓扑不变量可方便地编码为组合多项式的系数，通常称为多个锯齿链Zm,n的ZZ多项式。当前的研究报告了一种基于对约翰 - 萨克斯定理的假设扩展来确定这些ZZ多项式的新方法，该定理先前用于枚举各种苯类烃的凯库勒结构。我们表明，Zm,n多个锯齿链的ZZ多项式可以方便地表示为由简单超几何多项式组成的大小为m2×m2的托普利兹（或几乎托普利兹）矩阵的行列式。所提出的分析可以扩展到广义多个锯齿链Zkm,n，即Zm,n的衍生物，带有一个长度为 的单连接并苯链。所有给出的公式都伴有形式证明。所开发的理论机制用于预测大型和无限多个锯齿链Zm,n中的芳香性分布模式，以及计算有限多个锯齿链Zm,n的双自由基态中的自旋密度分布。
