The mechanism for the 3 x 3 distortion of Sn/ge (111)

We show that two distinct $3\times 3$ ground states, one nonmagnetic, metallic, and distorted, the other magnetic, semimetallic (or insulating) and undistorted, compete in $\alpha$-phase adsorbates on semiconductor (111) surfaces. In Sn/Ge(111), LSDA/GGA calculations indicate, in agreement with experiment, that the distorted metallic ground state prevails. The reason for stability of this state is analysed, and is traced to a sort of bond density wave, specifically a modulation of the antibonding state filling between the adatom and a Ge-Ge bond directly underneath.