  We present a Hamiltonian in real space which is well suited to study
numerically the behavior of holes introduced in III-V semiconductors by Mn
doping when the III$^{3+}$ ion is replaced by Mn$^{2+}$. We consider the actual
lattice with the diamond structure. Since the focus is on light doping by
acceptors, a bonding combination of III and V p-orbitals is considered since
the top of the valence band, located at the $\Gamma$ point, has p character in
these materials. As a result, an effective model in which the holes hop between
the sites of an fcc lattice is obtained. We show that around the $\Gamma$ point
in momentum space the Hamiltonian for the undoped case is identical to the one
for the Luttinger-Kohn model. The spin-orbit interaction is included as well as
the on-site interaction between the spin of the magnetic impurity and the spin
of the doped holes. The effect of Coulomb interactions between Mn$^{2+}$ and
holes, as well as Mn$^{3+}$ doping are discussed. Through large-scale Monte
Carlo simulations on a Cray XT3 supercomputer, we show that this model
reproduces many experimental results for ${\rm Ga_{\it 1-x}Mn_{\it x}As}$ and
${\rm Ga_{\it 1-x}Mn_{\it x}Sb}$, and that the Curie temperature does not
increase monotonically with $x$. The cases of Mn doped GaP and GaN, in which
Mn$^{3+}$ is believed to play a role, are also studied.
