Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem
Abstract
:1. Introduction
2. Preliminaries
- The determinant of the John–Sachs matrix is equal to the number of Kekulé structures as stipulated by Equation (8). Similarly, we request that the determinant of the generalized John–Sachs matrix is equal to the ZZ polynomial of
- As the Kekulé structures of are simply the Clar covers of of order 0, the number of Kekulé structures can be obtained from the ZZ polynomial of by evaluating it at ,
3. Discovery of the Determinantal Formulas
- The matrices are Toeplitz matrices with diagonals: 1 subdiagonal , 1 diagonal , and n consecutive superdiagonals .
- The value on the subdiagonal, , does not depend on n.
- The value of the diagonal element is equal to , given explicitly byThis value is consistent with the natural extension of the John–Sachs matrix diagonal elements, i.e., with replacing by .
- The value on the l-th superdiagonal is a product of a multiplicative factor , a numerical factor , and a polynomial of degree 2, 1, or 0.
- The polynomials always start from 1 and contain coefficients, which factorize into small primes. This suggests that they are hypergeometric polynomials, i.e., hypergeometric functions with at least one of the upper indices being a negative integer. As the formula for the diagonal elements in Equation (12) contains a hypergeometric function , it is natural to seek also in this form. The lower parameter is suggested by the denominator in the linear term of the polynomials , and it is equal to 3 for , 5 for , 7 for , etc. Therefore, in a general case, we can expect for a value of . Another observation is that the coefficient in the linear term of the polynomial, equal to , is always positive, which, taking into account that at least one of the upper indices should be negative, shows that actually both and are negative. One of these numbers, say , is immediately recognizable as , because of the constant degree of the polynomials for . The other index, , is a function , which for should be 0 (because of the degree 0 of the polynomial for ) and for should be 1 (because of the degree 0 of the polynomial for ). All these facts suggest that . A straightforward verification with Maple [78] shows that indeed the hypergeometric function reproduces all the reported polynomials in the matrix elements given above. Note that this polynomial reproduces somewhat fortuitously also the diagonal entry .
- The last remaining task is the identification of the two-dimensional sequence of numbers , which numerical values for small n and l are given by the following triangle:This task can be readily performed by typing, for example, the last rows of this triangle in the The On-Line Encyclopedia of Integer Sequences [79], which recognizes it as the sequence A085478 generated by
4. Formal Proof of Determinantal Formulas for and
4.1. with Even m
4.2. with Odd m
- Decompose the determinant into a sum of two determinants differing only by the last column [80]
- Rewrite the first determinant in Equation (40) by subtracting its last column from its last-but-one column obtaining
- Using these relations simplifies the determinant in Equation (41) to the following form:
4.3. with Even m
- We rewrite the last column of the determinant in Equation (44) as , with and .
- We decompose the determinant in Equation (44) as a sum of two determinants, the first one having in the last column and the second one having in the last column.
- We identify the second determinant with the help of Figure 2 as .
- In the first determinant, we subtract the last column from the last-but-one column obtaining
- Laplace expansion of the determinant in Equation (46) with respect to the last row gives
4.4. with Odd m
5. Chemical Applications
5.1. Local ZZ Aromaticity Indicators for Multiple Zigzag Chains
5.2. Spin Densities in Biradical Multiple Zigzag Chains
6. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Sample Availability
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Witek, H.A. Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem. Molecules 2021, 26, 2524. https://doi.org/10.3390/molecules26092524
Witek HA. Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem. Molecules. 2021; 26(9):2524. https://doi.org/10.3390/molecules26092524
Chicago/Turabian StyleWitek, Henryk A. 2021. "Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem" Molecules 26, no. 9: 2524. https://doi.org/10.3390/molecules26092524