  We investigate the conventional tight-binding model of $L$ $\pi$-electrons on
a ring-shaped mol\-e\-cule of $L$ atoms with nearest neighbor hopping. The
hopping amplitudes, $t(w)$, depend on the atomic spacings, $w$, with an
associated distortion energy $V(w)$. A Hubbard type on-site interaction as well
as nearest-neighbor repulsive potentials can also be included. We prove that
when $L=4k+2$ the minimum energy $E$ occurs either for equal spacing or for
alternating spacings (dimerization); nothing more chaotic can occur. In
particular this statement is true for the Peierls-Hubbard Hamiltonian which is
the case of linear $t(w)$ and quadratic $V(w)$, i.e., $t(w)=t_0-\alpha w$ and
$V(w)=k(w-a)^2$, but our results hold for any choice of couplings or functions
$t(w)$ and $V(w)$. When $L=4k$ we prove that more chaotic minima {\it can\/}
occur, as we show in an explicit example, but the alternating state is always
asymptotically exact in the limit $L\to\infty$. Our analysis suggests three
interesting conjectures about how dimerization stabilizes for large systems. We
also treat the spin-Peierls problem and prove that nothing more chaotic than
dimerization occurs for $L=4k+2$ {\it and\/} $L=4k$.
