  We have carried out an ab-initio study of $\alpha$-Sn (111), with the aim of
predicting and understanding its structure, reconstructions, and electronic
states. We consider a variety of structural possibilities, and optimize them by
moving atoms according to Hellmann-Feynman forces. Our results indicate that
the unreconstructed surface is highly unstable, while a variety of
reconstructions compete for the true ground state. Extrapolated trends from
diamond to Si to Ge are well borne out, with a $2\times 1 \pi$-bonded chain
reconstruction prevailing in the absence of adatoms, and a $c(4\times 2)$ or
$(2\times 2)$ basic adatom-restatom unit reconstruction otherwise. Accompanying
surface bucklings are in both cases larger than in Si and Ge, with consequently
large ionic charge transfers predicted. Search for a $\beta$-Sn-like metallic
state of the surface turned out to be inconclusive.
