  The magnetoelectric behavior of BiFeO$_3$ has been explored on the basis of
accurate density functional calculations. The structural, electronic, magnetic,
and ferroelectric properties of BiFeO$_3$ are predicted correctly without
including strong correlation effect in the calculation. Moreover, the
experimentally-observed elongation of cubic perovskite-like lattice along the
[111] direction is correctly reproduced. At high pressure we predicted a
pressure-induced structural transition and the total energy calculations at
expanded lattice show two lower energy ferroelectric phases, closer in energy
to the ground state phase. Band-structure calculations show that BiFeO$_3$ will
be an insulator in A- and G-type antiferromagnetic phases and a metal in other
magnetic configurations. Chemical bonding in BiFeO$_3$ has been analyzed using
various tools and electron localization function analysis shows that
stereochemically active lone-pair electrons at the Bi sites are responsible for
displacements of the Bi atoms from the centro-symmetric to the
noncentrosymmetric structure and hence the ferroelectricity. A large
ferroelectric polarization (88.7 $\mu$C/cm$^{2}$) is predicted in accordance
with recent experimental findings. The net polarization is found to mainly ($>$
98%) originate from Bi atoms. Moreover the large scatter in experimentally
reported polarization values is due to the large anisotropy in the spontaneous
polarization.
