  Design principles and novel predictions of new 3D Dirac semimetals are
presented, along with the context of currently known materials. Current
materials include those based on a topological to trivial phase transition,
such as in TlBiSe$_{2-x}$S$_x$ and Hg$_{1-x}$Cd$_x$Te, Bi$_{1-x}$Sb$_x$,
Bi$_{2-x}$In$_x$Se$_3$, and Pb$_{1-x}$Sn$_x$Se. Some more recently revealed
materials, Na$_3$Bi and Cd$_3$As$_2$, are 3D Dirac semimetals in their native
composition. The different design principles presented each yield novel
predictions for new candidates. For Case I, 3D Dirac semimetals based on charge
balanced compounds, BaAgBi, SrAgBi, YbAuSb, PtBi$_2$ and SrSn$_2$As$_2$ are
identified as candidates. For Case II, 3D Dirac semi-metals in analogy to
graphene, BaGa$_2$ is identified as a candidate, and BaPt and Li$_2$Pt are
discussed. For Case III, 3D Dirac semi-metals based on glide planes and screw
axes, TlMo$_3$Te$_3$ and the AMo$_3$X$_3$ family in general (A=K, Na, In, Tl,
X=Se,Te) as well as the Group IVb trihalides such as HfI$_3$ are identified as
candidates. Finally we discuss conventional intermetallic compounds with Dirac
cones, and identify Cr$_2$B as a potentially interesting material.
