Second-order Jahn-Teller effect on carbon4N+2member ring clusters

Occurrence of bond-length alternation of monocyclic carbon $4N+2$ member rings is examined based on the hybrid density-functional calculations. In the case that $N<~4,$ the bond-length nonalternant cumulenic structure $\left[D_{\mathrm{}(2N+1)h}\right]$ is found to be the most stable and this structure becomes unstable when $N>~5.$ Then the bond-length alternant structure $\left[C_{\mathrm{}(2N+1)h}\right]$ becomes the most stable among the ring-shape clusters. We observe that the bond-length alternation mode of the stable cumulenic structures is considerably lowered as the size becomes large. This softening is analogous to the Kohn anomaly in crystal systems and is the origin of the Peierls-like (bond-length alternation) distortion of large clusters. These features are discussed in terms of the second-order Jahn-Teller effect.