  Ba2CoGe2O7, crystallizing in the noncentrosymmetric but nonpolar structure,
belongs to a special class of multiferroic materials, whose properties are
predetermined by the rotoinversion symmetry. Unlike inversion, the
rotoinversion symmetry can be easily destroyed by the magnetization. Moreover,
due to specific structural pattern, the magnetic structure of Ba2CoGe2O7 is
relatively soft. Altogether, this leads to the rich variety of multiferroic
properties, where the magnetic structure can be easily deformed by the magnetic
field, inducing the electric polarization in the direction, which depends on
the direction of the magnetic field. In this paper, we show that all these
properties can be successfully explained on the basis of realistic low-energy
model, derived from the first-principles electronic structure calculations for
the magnetically active Co 3d bands, and the Berry-phase theory of electric
polarization. Particularly, we argue that the magnetization induced electric
polarization in Ba2CoGe2O7 is essentially local and expressed via the
expectation values of some dipole matrices, calculated in the Wannier basis of
the model, and the site-diagonal density matrices of the magnetic Co sites.
Thus, the basic aspects of the behavior of Ba2CoGe2O7 can be understood already
in the atomic limit, where both magnetic anisotropy and magnetoelectric
coupling are specified by density matrix. Then, the macroscopic polarization
can be found as a superposition of electric dipoles of the individual Co sites.
We discuss the behavior of interatomic magnetic interactions, main
contributions to the magnetocrystalline anisotropy and the spin canting, as
well as the similarities and differences of the proposed picture from the
phenomenological model of spin-dependent p-d hybridization.
