  We have obtained an analytic expression for the $k$ dependence of excitation
energy gap for an arbitrary double $S=1/2$ spin-chain by using the nonlocal
unitary transformation and the variational method. It is checked to explain the
gap behavior of various systems, which include the Haldane system and the dimer
system in both extreme limits, and also the ladder model and the Majumdar-Ghosh
model. The string order parameter, the dimer order parameter, and the local
spin value are also calculated in the ground state. The ground-state energy
exhibits a great stabilization by an antiferromagnetic bond dimerization, which
might be realized in various new compounds. We also mention the relation of the
convergence to the Haldane state with the spin-exchange symmetry of the model.
The excited state has one domain wall of a local triplet type except in the
vicinity of the Majumdar-Ghosh point, where a local triplet is decomposed to
two $S=1/2$ free spins moving among the singlet dimers.
