  We demonstrate that first-principles based adiabatic continuation approach is
a very powerful and efficient tool for constructing topological phase diagrams
and locating non-trivial topological insulator materials. Using this technique,
we predict that the ternary intermetallic series Li$_2M'X$ where $M'$=Cu, Ag,
Au, or Cd, and $X$=Sb, Bi, or Sn, hosts a number of topological insulators with
remarkable functional variants and tunability. We also predict that several
III-V semimetallic compounds are topologically non-trivial. We construct a
topological phase diagram in the parameter space of the atomic numbers of atoms
in Li$_2M'X$ compounds, which places a large number of topological materials
presented in this work as well as in earlier studies within a single unified
topological framework. Our results demonstrate the efficacy of adiabatic
continuation as a useful tool for exploring topologically nontrivial alloying
systems and for identifying new topological insulators even when the underlying
lattice does not possess inversion symmetry, and the approaches based on parity
analysis are not viable.
