  We investigated the electronic and structural properties of the infinite
linear carbon chain (carbyne) using density functional theory (DFT) and the
random phase approximation (RPA) to the correlation energy. The studies are
performed in vacuo and for carbyne inside a carbon nano tube (CNT). In the
vacuum, semi-local DFT and RPA predict bond length alternations of about 0.04
{\AA} and 0.13 {\AA}, respectively. The frequency of the highest optical mode
at the $\Gamma$ point is 1219 cm$^{-1}$ and about 2000 cm$^{-1}$ for DFT and
the RPA. Agreement of the RPA to previous high level quantum chemistry and
diffusion Monte-Carlo results is excellent. For the RPA we calculate the
phonon-dispersion in the full Brillouine zone and find marked quantitative
differences to DFT calculations not only at the $\Gamma$ point but also
throughout the entire Brillouine zone. To model carbyne inside a carbon
nanotube, we considered a (10,0) CNT. Here the DFT calculations are even
qualitatively sensitive to the k-points sampling. At the limes of a very dense
k-points sampling, semi-local DFT predicts no bond length alternation (BLA),
whereas in the RPA a sizeable BLA of 0.09 {\AA} prevails. The reduced BLA leads
to a significant red shift of the vibrational frequencies of about 350
cm$^{-1}$, so that they are in good agreement with experimental estimates.
Overall, the good agreement between the RPA and previously reported results
from correlated wavefunction methods and experimental Raman data suggests that
the RPA provides reliable results at moderate computational costs. It hence
presents a useful addition to the repertoire of correlated wavefunction methods
and its accuracy clearly prevails for low dimensional systems, where semi-local
density functionals struggle to yield even qualitatively correct results.
