  The equilibrium structure and functional properties exhibited by
brownmillerite oxides, a family of perovskite-derived structures with
alternating layers of $B$O$_6$ octahedra and $B$O$_4$ tetrahedra, viz., ordered
arrangements of oxygen vacancies, is dependent on a variety of competing
crystal-chemistry factors. We use electronic structure calculations to
disentangle the complex interactions in two ferrates, Sr$_2$Fe$_2$O$_5$ and
Ca$_2$Fe$_2$O$_5$, relating the stability of the equilibrium (strain-free) and
thin film structures to both previously identified and newly herein proposed
descriptors. We show that cation size and intralayer separation of the
tetrahedral chains provide key contributions to the preferred ground state. We
show the bulk ground state structure is retained in the ferrates over a range
of strain values; however, a change in the orientation of the tetrahedral
chains, i.e., a perpendicular orientation of the vacancies relative to the
substrate, is stabilized in the compressive region. The structure stability
under strain is largely governed by maximizing the intraplane separation of the
`dipoles' generated from rotations of the FeO$_4$ tetrahedra. Lastly, we find
that the electronic band gap is strongly influenced by strain, manifesting as
an unanticipated asymmetric-vacancy alignment dependent response. This
atomistic understanding establishes a practical route for the design of novel
functional electronic materials in thin film geometries.
