  Motivated by recent experiments on chemically synthesized magnetic molecular
chains we investigate the lowest lying energy band of short spin-$s$
antiferromagnetic Heisenberg chains focusing on effects of open boundaries. By
numerical diagonalization we find that the Land\'e pattern in the energy
levels, i.e. E(S) \propto S(S+1) for total spin S, known from e.g. ring-shaped
nanomagnets, can be recovered in odd-membered chains while strong deviations
are found for the lowest excitations in chains with an even number of sites.
This particular even-odd effect in the short Heisenberg chains cannot be
explained by simple effective Hamiltonians and symmetry arguments. We go beyond
these approaches, taking into account quantum fluctuations by means of a path
integral description and the valence bond basis, but the resulting quantum
edge-spin picture which is known to work well for long chains does not agree
with the numerical results for short chains and cannot explain the even-odd
effect. Instead, by analyzing also the classical chain model, we show that
spatial fluctuations dominate the physical behavior in short chains, with
length N < exp(\pi s), for any spin s. Such short chains are found to display a
unique behavior, which is not related to the thermodynamic limit and cannot be
described well by theories developed for this regime.
