  Intriguing physical property of materials stems from their chemical
constituent whereas the connection between them is often not clear. Here, we
uncover a general chemical classification for the two quantum phases in the
honeycomb ABX structure--topological insulator (TI) and topological Dirac
semimetal (TDSM). First, we find among the 816 (existing as well as
hypothetical) calculated compounds, 160 TI's (none were noted before), 96
TDSM's, 282 normal insulators (NI's), and 278 metals. Second, based on this
classification, we have distilled a simple chemical regularity based on
compound formulae for the selectivity between TI and TDSM: The ABX compounds
that are TDSM have B atoms (part of the BX honeycomb layers) that come from the
Periodic Table columns XI (Cu, Ag, Au) or XII (Zn, Cd, Hg), or Mg (group II),
whereas the ABX compounds whose B atoms come from columns I (Li, Na, K, Rb, Cs)
or II (Ca, Sr, Ba) are TI's. Third, focusing on the ABX Bismide compounds that
are thermodynamically stable, we find a structural motif that delivers
topological insulation and stability at the same time. This study opens the way
to simultaneously design new topological materials based on the compositional
rules indicated here.
