We have performed extensive first-principles calculations to study the bind
ing and diffusion of a Ge ad-dimer on the Si(001) surface. The diffusion of
a Ge ad-dimer along the lowest-energy-barrier pathway is found to be piece
wise but with strong correlation, and its energy barrier is 0.77 eV. Such a
correlated piecewise diffusion pathway is also found to be favored by a Si
ad-dimer on Si(001), with an energy barrier of 1.02 eV in excellent agreem
ent with experimental measurements.