Our calculations show that alkali metals at 1 2 a monolayer coverage adsorb at the quasi-hexagonal (H) pedestal sites on Si(001)-2 × 1 and form linear chains along the  dimer rows. To arrive at this conclusion, it is essential to incorporate the substrate relaxation because only then the H site is pulled down in energy relative to the other competing sites. The main relaxation consists of opening of the Si dimer bond length at the surface. The H site occupancy and the dimer bond opening are consistent with the conclusions of a recent LEED study. But a cluster model calculation has suggested that alkali metal chains at the H site are unstable towards a zig-zag Peierls type of distortion. It is shown that such a Peierls instability is not expected in the linear alkali metal chains on Si(001)-2 × 1. A theorem leading to gapless Peierls transitions due to zig-zag distortions is explicitly given along with an elementary proof. © 1991.