Bond alternation in ring-shaped molecules: The stability of the peierls instability

We investigate the phenomenon of bond alternation in ring molecules of the type (CH)_2n, which occurs due to the Peierls instability. We prove that the energy-minimizing configuration of bond lengths always has period two when n is odd. When n is even, a new instability may destroy the periodicity two as long as n is not too large. We also analyze the corresponding problem for the Heisenberg antiferromagnetic "spin-Peierls" system and prove that instabilities other than period two never occur there.