  We present results of an {\it ab initio} density functional theory study of
three bismuth-based multiferroics, BiFeO$_{3}$, Bi$_{2}$FeCrO$_{6}$, and
BiCrO$_{3}$. We disuss differences in the crystal and electronic structure of
the three systems, and we show that the application of the LDA+$U$ method is
essential to obtain realistic structural parameters for Bi$_{2}$FeCrO$_{6}$. We
calculate the magnetic nearest neighbor coupling constants for all three
systems and show how Anderson's theory of superexchange can be applied to
explain the signs and relative magnitudes of these coupling constants. From the
coupling constants we then obtain a mean-field approximation for the magnetic
ordering temperatures. Guided by our comparison of these three systems, we
discuss the possibilities for designing a multiferroic material with large
magnetization above room temperature.
