  By applying Berry-phase theory for the effective half-filled Hubbard model,
we derive an analytical expression for the electronic polarization driven by
the relativistic spin-orbit (SO) coupling. The model itself is constructed in
the Wannier basis, using the input from the first-principles electronic
structure calculations in the local-density approximation, and then treated in
the spirit of the superexchange theory. The obtained polarization has the
following form: ${\bf P}_{ij} = \boldsymbol{\epsilon}_{ji} \boldsymbol{\cal
P}_{ij} \cdot [\boldsymbol{e}_i \times \boldsymbol{e}_j]$, where
$\boldsymbol{\epsilon}_{ji}$ is the direction of the bond $\langle ij \rangle$,
$\boldsymbol{e}_i$ and $\boldsymbol{e}_j$ are the directions of spins in this
bond, and $\boldsymbol{\cal P}_{ij}$ is the pseudovector containing all the
information about the crystallographic symmetry of the considered system. The
expression describes the ferroelectric activity in various magnets with
noncollinear but otherwise nonpolar magnetic structures, which would yield no
polarization without SO interaction, including the magnetoelectric (ME) effect,
caused by the ferromagnetic canting of spins in the external magnetic field,
and spin-spiral multiferroics. The abilities of this theory are demonstrated
for the the analysis of linear ME effect in Cr$_2$O$_3$ and BiFeO$_3$ and
properties multiferroic MnWO$_4$ and $\beta$-MnO$_2$. In all considered
examples, the theory perfectly describes the symmetry properties of the induced
polarization. However, in some cases, the values of this polarization are
underestimated, suggesting that other effects, besides the spin and electronic
ones, can also play an important role.
