  We present a detailed first principles study of Fe-pnictides with particular
emphasis on competing magnetic interactions, structural phase transition, giant
magneto-elastic coupling and its effect on phonons. The exchange interactions
$J_{i,j}(R)$ are calculated up to $\approx 12 $\AA $. We find that $J_{i,j}(R)$
has an oscillatory character with an envelop decaying as $1/R^3$ along the
stripe-direction while it is very short range along the diagonal direction and
antiferromagnetic. A brief discussion of the neutron scattering determination
of these exchange constants from a single crystal sample with orthorhombic
twinning is given. The lattice parameter dependence of the exchange constants,
$dJ_{i,j}/da$ are calculated for a simple spin-Peierls like model to explain
the fine details of the tetragonal-orthorhombic phase transition. We then
discuss giant magneto-elastic effects in these systems. We show that when the
Fe-spin is turned off the optimized c-values are shorter than experimetnal
values by 1.4 \AA $ $ for CaFe$_2$As$_2$, by 0.4 \AA $ $ for BaFe$_2$As$_2$,
and by 0.13 \AA $ $ for LaOFeAs. Finally, we show that Fe-spin is also required
to obtain the right phonon energies, in particular As c-polarized and Fe-Fe
in-plane modes. Since treating iron as magnetic ion always gives much better
results than non-magnetic ones and since there is no large c-axis reduction
during the normal to superconducting phase transition, the iron magnetic moment
should be present in Fe-pnictides at all times. We discuss the implications of
our results on the mechanism of superconductivity in these fascinating
Fe-pnictide systems.
