  The Heusler compounds Mn$_2$TiZ (Z = Al, Ga, In, Si, Ge, Sn, P, As, Sb) are
of large interest due to their potential ferrimagnetic properties and high spin
polarization. Here, we present calculations of the structural and magnetic
properties of these materials. Their magnetic moment follows the Slater-Pauling
rule $m = N_V - 24$. None of them is actually a perfect half-metallic
ferrimagnet, but some exhibit more than 90% spin polarization and Curie
temperatures well above room temperature. The exchange interactions are
complex, direct and indirect exchange contributions are identified. The Curie
temperature scales with the total magnetic moment, and it has a positive
pressure dependence. The role of the Z element is investigated: it influences
the properties of the compounds mainly via its valence electron number and its
atomic radius, which determines the lattice parameter. Based on these results,
Mn$_2$TiSi, Mn$_2$TiGe, and Mn$_2$TiSn are proposed as candidates for
spintronic applications.
