  We use first-principles density functional theory-based calculations to
investigate structural instabilities in the high symmetry cubic perovskite
structure of rare-earth (R $=$ La, Y, Lu) and Bi-based biferroic chromites,
focusing on $\Gamma$ and $R$ point phonons of states with para-, ferro-, and
antiferromagnetic ordering. We find that (a) the structure with G-type
antiferromagnetic ordering is most stable, (b) the most dominant structural
instabilities in these oxides are the ones associated with rotations of oxygen
octahedra, and (c) structural instabilities involving changes in Cr-O-Cr bond
angle depend sensitively on the changes in magnetic ordering. The dependence of
structural instabilities on magnetic ordering can be understood in terms of how
super-exchange interactions depend on the Cr-O-Cr bond angles and Cr-O bond
lengths. We demonstrate how adequate buckling of Cr-O-Cr chains can favour
ferromagnetism. Born effective charges (BEC) calculated using the Berry phase
expression are found to be anomalously large for the A-cations, indicating
their chemical relevance to ferroelectric distortions.
