  Global optimization of transition paths in complex atomic scale systems is
addressed in the context of misfit dislocation formation in a strained Ge film
on Si(001). Such paths contain multiple intermediate minima connected by
minimum energy paths on the energy surface emerging from the atomic
interactions in the system. The challenge is to find which intermediate states
to include and to construct a path going through these intermediates in such a
way that the overall activation energy for the transition is minimal. In the
numerical approach presented here, intermediate minima are constructed by
heredity transformations of known minimum energy structures and by identifying
local minima in minimum energy paths calculated using a modified version of the
nudged elastic band method. Several mechanisms for the formation of a 90{\deg}
misfit dislocation at the Ge-Si interface are identified when this method is
used to construct transition paths connecting a homogeneously strained Ge film
and a film containing a misfit dislocation. One of these mechanisms which has
not been reported in the literature is detailed. The activation energy for this
path is calculated to be 26% smaller than the activation energy for half loop
formation of a full, isolated 60{\deg} dislocation. An extension of the common
neighbor analysis method involving characterization of the geometrical
arrangement of second nearest neighbors is used to identify and visualize the
dislocations and stacking faults.
