Interactions with a single-atom tip become important a few angstroms prior to adhesive contact to a metallic sample. The collapse of the barrier produces a connecting channel; directed atomiclike electronic states form at its ends, resulting in an overall attraction. The wave functions of states near the Fermi level nevertheless remain evanescent. These effects are found to be strongly site dependent even on a very flat metal surface, and may explain the atomic resolution achieved in scanning tunneling microscopy of such surfaces. Deformation of the tip cannot explain such observations. © 1990 The American Physical Society.