  First-principles electronic structure calculations have been performed for
the double perovskite Bi$_2$CoMnO$_6$ in its non-centrosymmetric polar state
using generalized gradient approximation plus Hubbard U approach. We find that
while Co is in a high spin state, Mn is in an intermediate spin state. The
calculated dynamical charge tensors are anisotropic reflecting a low-symmetry
structure of the compound. Magnetic structure dependent phonon frequencies
indicate the presence of spin-phonon coupling. Using Berry phase method, we
obtain a spontaneous electronic polarization of 5.88 ${\mu}C/cm^2$ which is
close to the experimental value observed for a similar compound,
Bi$_2$NiMnO$_6$.
