  First-principles calculations of the electronic structure, magnetism and
structural stability of inverse-Heusler compounds with the chemical formula
\textit{X$_2$YZ} are presented and discussed with a goal of identifying
compounds of interest for spintronics. Compounds for which the number of
electrons per atom for \textit{Y} exceed that for \textit{X} and for which
\textit{X} is one of Sc, Ti, V, Cr, Mn, Fe, Co, Ni, or Cu; \textit{Y} is one of
Ti, V, Cr, Mn, Fe, Co, Ni, Cu, or Zn; and \textit{Z} is one of Al, Ga, In, Si,
Ge, Sn, P, As or Sb were considered. The formation energy per atom of each
compound was calculated. By comparing our calculated formation energies to
those calculated for phases in the Inorganic Crystal Structure Database (ICSD)
of observed phases, we estimate that inverse-Heuslers with formation energies
within 0.052 eV/atom of the calculated convex hull are reasonably likely to be
synthesizable in equilibrium. The observed trends in the formation energy and
relative structural stability as the \textit{X}, \textit{Y} and \textit{Z}
elements vary are described. In addition to the Slater-Pauling gap after 12
states per formula unit in one of the spin channels, inverse-Heusler phases
often have gaps after 9 states or 14 states. We describe the origin and
occurrence of these gaps. We identify 14 inverse-Heusler semiconductors, 51
half-metals and 50 near half-metals with negative formation energy. In
addition, our calculations predict 4 half-metals and 6 near half-metals to lie
close to the respective convex hull of stable phases, and thus may be
experimentally realized under suitable synthesis conditions, resulting in
potential candidates for future spintronics applications.
