Linear carbon chain
Linear carbon chain is the simplest 1D form of carbon. According to Peierls theorem a one-dimensional equally spaced chain with one electron per ion is unstable. Imagine a lattice distortion where every other ion moves closer to one neighbour and farther away from the other. This lattice distortion becomes energetically favorable when the energy savings due to the new band gaps outweighs the elastic energy cost of rearranging the ions.
In a recent work of our group the physical properties of a linear carbon chain encapsulated within single-walled carbon nanotubes are investigated with density-functional theory using periodic boundary conditions. The dominant feature of an isolated carbon chain is the Peierls distorsion and the opening of a Peierls gap. The two weakly interacting subsystems - infinite carbon chain and nanotube - establish a common Fermi level, resulting in charge transfer (CT) which leads to a metallic combined system with a high density of states at the Fermi level. The rigid band model provides useful insights. Unusual physics arises from the effects of CT and chain-tube orbital hybridization which both tend to suppress the Peierls dimerization.