  The exchange-driven contribution to the magnetoelectric susceptibility
$\hat\alpha$ is formulated using a microscopic model Hamiltonian coupling the
spin degrees of freedom to lattice displacements and electric field, which may
be constructed from first-principles data. Electronic and ionic contributions
are sorted out, and the latter is resolved into a sum of contributions from
different normal modes. If intrasublattice spin correlations can be neglected,
the longitudinal component $\alpha_\parallel$ becomes proportional to the
product of magnetic susceptibility and sublattice magnetization in accordance
with Rado's phenomenological model. As an illustration, the method is applied
to analyze the temperature dependence of the longitudinal magnetoelectric
susceptibility of Cr$_2$O$_3$ using first-principles calculations and the
quantum pair cluster approximation for magnetic thermodynamics. A substantial
electronic contribution is found, which is opposite to the ionic part. The
sensitivity of the results to the Hubbard $U$ parameter and the sources of
error are studied. It is also found that non-Heisenberg interactions are too
weak to account for the sign change of $\alpha_\parallel$ in Cr$_2$O$_3$.
