Hydrocarbons and nanostructures, consisting of trigonally coordinated C-atoms, represent a huge family of functional materials. Depending on the configuration and the size, they may be highreactivity intermediates, raw materials or catalyzers of organic reactions, or semiconducting polymers with variable bandgap. Small molecules have been intensively studied by chemists, and crystalline polymers by physicist, but the knowledge on the medium-size oligomers is rather incomplete. For a better understanding of such materials, the size-dependence of several structural properties of linear polyenes and infinite polymers were calculated at the same level of density functional theory. Linear polyenes are the simplest derivatives of trigonal carbon and their study is instructive for the understanding of more complicated series. The analysis of heats of formation and bondlength distributions reveals a singlet-triplet transition in the trans-cisoid polyenes at the carbon number of n = 28. An efficient method is suggested for the common treatment of the molecular orbitals of polyenes and the energy bands of polyacetylene: the mapping of the energy levels of polyenes to the bands of polyacetylene. The excellent agreement of the levels of molecular orbitals and the polymer bands makes possible the reconstruction of the bands in the knowledge of molecular orbitals and vice versa. A future extension of the suggested mapping to 2 dimensions may be suitable for a uniform treatment of trigonal carbon systems from methyl radical through polycyclic aromatic hydrocarbons to graphene.
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