  We study the locations of the gapless points which occur for quantum spin
chains of finite length (with a twisted boundary condition) at particular
values of the nearest neighbor dimerization, as a function of the spin S and
the number of sites. For strong dimerization and large values of S, a tunneling
calculation reproduces the same results as those obtained from more involved
field theoretic methods using the non-linear sigma-model approach. A different
analytical calculation of the matrix element between the two Neel states gives
a set of gapless points; for strong dimerization, these differ significantly
from the tunneling values. Finally, the exact diagonalization method for a
finite number of sites yields a set of gapless points which are in good
agreement with the Neel state calculations for all values of the dimerization,
but the agreement with the tunneling values is not very good even for large S.
This raises questions about possible corrections to the tunneling results.
