Generation of Fullerenes by Circumscribing

Abstract

An algorithm for the generation of fullerenes with isolated pentagonal rings by successively circumscribing base excised internal structures with a combination of pentagonal and hexagonal rings is detailed. The presence of adjacent bay regions is a necesary and sufficient condition for preculding this circumscribing operation for a polypent/polyhex system.

Construction of planar graphs for IPR fullerenes using 5- and 6-fold rotational symmetry: some eigenspectral analysis

“…27 A general procedure for drawing planar graphs is Tutte 's baricentric method. 28 Dias 29,30 has produced a circumscribing algorithm for construction of planar graphs (Schlegel diagrams) of fullerene polyhedra and has shown, 30 by generating all the C 84 fullerenes, that his scheme is perfectly general. To get an isolated pentagon fullerene of desired symmetry Dias's scheme can easily be used with purposeful manipulation.…”

Systematic Construction and Calculation of Electronic Properties of Fullerene Series Related by Rotational Symmetry: From Fullerenes to Bicapped Nanotubes

“…In general, Schlegel diagrams of fullerenes with rotational symmetry in this work can be generated by starting with a core cyclic C4, C5, C6 carbon ring and circumscribing it with a layer of 4, 5, 6 pentagonal rings, respectively. − This first step is done for all the series in Figures –, which gives the initial cap. The last step always involves circumscribing with a layer of pentagonal rings equal in number with those in the first step to give the final endcap.…”

“…Small fullerenes and nanotubes related to this study include work by Lu and Chen . Generating fullerenes and nanotubes by circumscribing − is simple and general and may be compared to the recently published recursive generation of IPR fullerenes . The efficient factorization of the characteristic polynomials of highly symmetrical fullerenes like C 60 Buckminsterfullerene has been described. , A highly mathematical determination of characteristic polynomials and eigenvalues has recently been reported .…”

Static third-order polarizability calculations for five isolated-pentagon-rule isomers of C78: an excellent correlation betweenlanglegammarangleand overall curvature

“…In a qualitative way, two different algorithms such as circumscribing [15] and leapfrogging [16] for the generation of the fullerenes have been proposed. Dias [17] constructed the five IPRs of C 78 by taking the circumscribing algorithm. The five C 78 IPR isomers are labelled as A(D 3h ), B(C 2v ), C(C 2v ), D(D 3 ) and E(D 3h ), and their structures are drawn in figure 1.…”

“…Due to their shell structures, the curved connections among the carbon atoms cause the misalignment of two π-orbitals (figure 2(a)) at the adjacent carbon atoms in the path of conjugation and somewhat reduce the π -orbital overlap and aromatic character. Hence, the IPR structures are more favourable in energy than those containing adjacent five-member rings because they minimize both antiaromaticity and strains [17]. Dias [17] has suggested that it is always desirable to derive a structure where strain is distributed as uniformly as possible following the principle of the 'neighbour index' of a ring [32].…”