  The few reported members of the antiperovskite structure class $Ae_3Pn_APn_B$
of alkaline earth ($Ae$ = Ca,Sr,Ba) pnictides ($Pn$ = N,P,As,Sb,Bi) compounds
are all based on the B-site anion $Pn_B$=N. All fit can be categorized as
narrow gap semiconductors, making them of interest for several reasons. Because
chemical reasoning suggests that more members of this class may be stable, we
provide here a density functional theory (DFT) based survey of this entire
class of $3\times5\times5$ compounds. We determine first the relative energetic
stability of the distribution of pairs of $Pn$ ions in the A and B sites of the
structure, finding that the $B$ site always favors the small pnictogen anion.
The trends of the calculated energy gaps with $Ae$ cation and $Pn$ anions are
determined, and we study effects of spin-orbit coupling as well as two types of
gap corrections to the conventional DFT electronic spectrum. Because there have
been suggestions that this class harbors topological insulating phases, we have
given this possibility attention and found that energy gap corrections indicate
the cubic structures will provide at most a few topological insulators.
Structural instability is addressed by calculating phonon dispersion curves for
a few compounds, with one outcome being that distorted structures should be
investigated further for thermoelectric and topological character. Examples of
the interplay between spin-orbit coupling and strain on the topological nature
are provided. A case study of $\mathrm{Ca_3BiP}$ including the effect of strain
illustrates how a topological semimetal can be transformed into topological
insulator and Dirac semimetal.
