  We have used a many-body model Hamiltonian to study the nature of the
magnetic ground state of hetero-binuclear complexes involving rare-earth and
copper ions. We have taken into account all diagonal repulsions involving the
rare-earth 4f and 5d orbitals and the copper 3d orbital. Besides, we have
included direct exchange interaction, crystal field splitting of the rare-earth
atomic levels and spin-orbit interaction in the 4f orbitals. We have identified
the inter-orbital $4f$ repulsion, U$_{ff}$ and crystal field parameter,
$\Delta_f$ as the key parameters involved in controlling the type of exchange
interaction between the rare earth $4f$ and copper 3d spins. We have explored
the nature of the ground state in the parameter space of U$_{ff}$, $\Delta_f$,
spin-orbit interaction strength $\lambda$ and the $4f$ filling n$_f$. We find
that these systems show low-spin or high-spin ground state depending on the
filling of the $4f$ levels of the rare-earth ion and ground state spin is
critically dependent on U$_{ff}$ and $\Delta_f$. In case of half-filling
(Gd(III)) we find a reentrant low-spin state as U$_{ff}$ is increased, for
small values of $\Delta_f$, which explains the recently reported apparent
anomalous anti-ferromagnetic behaviour of Gd(III)-radical complexes. By varying
U$_{ff}$ we also observe a switch over in the ground state spin for other
fillings . We have introduced a spin-orbit coupling scheme which goes beyond
L-S or j-j coupling scheme and we find that spin-orbit coupling does not
significantly alter the basic picture.
