Identifying macroscopic variables that affect the rates and mechanisms by which surface atoms move should enhance our ability to control surface morphology. The search for new "knobs to turn" is at the root of the decade-long effort to modify thin-film growth by depositing appropriately chosen "surfactant" species. It also motivates the work reported here, in which ab-initio total energy calculations are used to understand how an externally imposed electric field should affect the mechanism and rate of adatom self-diffusion on Pt(001), a surface for which Field Ion Microscopy experiments@footnote 1@ suggest that the low energy process, concerted substitution, is supplanted by hopping when the external field is high enough. The result is that theory agrees with FIM that the barrier to concerted substitutional diffusion of a Pt atom on Pt(001) varies linearly with external electric field (slope ~0.1eÅ), increasing for fields oriented to push electrons into the surface. But, with a computed hopping barrier remaining >0.5 eV higher than that for substitution, the calculations contradict the idea that a change in FIM site visitation at fields of 1.5-2 V/Å and temperatures ~265-284K can be attributed to the onset of hopping. @FootnoteText@ @footnote *@Work supported by the U. S. Department of Energy under Contract No. DE-AC04-94AL85000. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed-Martin Company, for the U. S. Department of Energy. @footnote 1@G. L. Kellogg, Phys. Rev. Lett. 70, 1631(1993).