  We present an $\it{ab}$ $\it{initio}$ calculation of the exchange coupling
for Fe/Nb multilayers using the self-consistent full-potential linearized
augmented-plane wave (FLAPW) method. The exchange correlation potential has
been treated in the local spin density approximation (LSDA) as well as
generalized gradient approximation (GGA). We find that for the LSDA as well as
the GGA the exchange coupling oscillates with a period of 6.0 \AA of Nb spacer
thickness which is close to the experimental value in the preasymptotic region.
This is also close to the earlier calculated period (i.e. 4.5 \AA) by augmented
spherical wave (ASW) method. The LSDA shows antiferromagnetic coupling for 2
and 5 Nb monolayers (ML) but the GGA shows the ferromagnetic coupling for all
Nb spacer layers. The period of oscillation is found to be in good agreement
with the period calculated using the Ruderman-Kittel-Kasuya-Yosida (RKKY) and
quantum well (QW) models. The magnetic moment of Fe is found to be higher in
the GGA than the LSDA. Fe magnetic moment also shows strong oscillations as a
function of the spacer layer thickness, in agreement with the experimental
results. We find that the GGA results show better agreement with the experiment
than the LSDA results.
