  We study the origin of the gap and the role of chemical composition in the
half-ferromagnetic Heusler alloys using the full-potential screened KKR method.
In the paramagnetic phase the C1_b compounds, like NiMnSb, present a gap.
Systems with 18 valence electrons, Z_t, per unit cell, like CoTiSb, are
semiconductors, but when Z_t > 18 antibonding states are also populated, thus
the paramagnetic phase becomes unstable and the half-ferromagnetic one is
stabilized. The minority occupied bands accommodate a total of nine electrons
and the total magnetic moment per unit cell in mu_B is just the difference
between Z_t and $2 \times 9$. While the substitution of the transition metal
atoms may preserve the half-ferromagnetic character, substituting the $sp$ atom
results in a practically rigid shift of the bands and the loss of
half-metallicity. Finally we show that expanding or contracting the lattice
parameter by 2% preserves the minority-spin gap.
