Development of bond-length alternation in very large carbon rings: LDA pseudopotential results

Carbon rings Cₙ and infinite chains C_∞ are investigated by molecular-orbital and band-structure calculations within the local-density approximation. Carbon rings C4N $(N<~8)$ exhibit a substantial first-order Jahn-Teller distortion that leads to long/short (single/triple) bond alternation decreasing with increasing $N.$ Rings C4N+2 show no alternation (i.e., aromatic behavior is retained) until very large sizes $(N>~20).$ For the infinite carbon chain uniform Brillouin-zone sampling with an even number of points Nₛ gives bond alternation. An odd number of sampling points gives no bond alternation for less than Nₛ=41. In the large Nₛ limit even and odd sampling lead to an upper and lower bound of 0.070a₀ and 0.065a₀ for bond alternation and 0.021--0.090 millihartrees/atom for condensation energy.