Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties

Abstract

This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modification of Pólya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperfine patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperfine patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.

New Insights into Aromaticity through Novel Delta Polynomials and Delta Aromatic Indices

“…Combinatorics and graph theory have been applied to the enumeration of conjugated circuits, isomers of polycyclic aromatics and their derivatives, spectral polynomials, matching polynomials, distance polynomials and a number of polycyclic aromatics and fullerene cages . An intriguing feature of such applications is that some of these symmetry-based techniques involve such novel group theoretical techniques, such as Euler's totient functions, Polya's theory of enumeration, etc., to predict their polysubstituted isomers and spectra [77]. Many of these polysubstituted aromatics, macrocyclic arenes, heteropolycyclic arenes and related halocarbons have been studied owing to their significant importance as environmental pollutants, carcinogens, hepatotoxins, industrial chemicals and petroleum products.…”

Density functional and graph theory computations of vibrational, electronic, and topological properties of porous nanographenes

“…The success of the density functional methods and graph‐theoretical techniques [ 17–68 ] is evident in that one can handle such large polycyclic aromatics, fullerenes, nanospheres, and holey nanographenes with these techniques. While the DFT technique has considerably reduced the computational cost, making such computations tractable, graph‐theoretical techniques have provided powerful means to gain new insights into the underlying connectivity through techniques such as the conjugated circuit theory, [ 45,46 ] Clar's aromatic sextets [ 34 ] and their generalizations, [ 46 ] topological and bond resonance theories, have provided insights into relative stabilities, magnetic and electronic properties of these species.…”

Topological and Spectral Properties of Wavy Zigzag Nanoribbons

“…Topological descriptors are structurebased numeric entities that capture the underlying network connectivity through graph theory. These techniques together with energy spectra, delocalization energies, bond resonance energies, aromatic sextets, etc., provide considerable insight into their stabilities, reactivities, aromaticity, ring currents, and so forth [46][47][48][49][50][51][52][53]. As these topological descriptors are invariant to labelings, they play a vital role in the quantitative analysis of structural activity, property, and toxicity relationships (QSAR/QSPR/QSTR) [54].…”