  Applying the approximate DFT-1/2 quasiparticle scheme, band structure
unfolding, and generalized quasichemical approximation to describe chemical and
structural disorder, we investigate the electronic structure of
Bi$_{1-x}$Sb$_x$ alloys from first principles. We calculate the important
energy levels near the Fermi energy versus the Sb concentration $x$ where the
trivial-topological (TT) and semimetal-semiconductor (SMSC) transitions occur.
We demonstrate that the energy variation of the relevant states deviates
significantly from linear behavior and that the bowings are important to
correctly describe the critical compositions. The influence of strain on the
energy levels is briefly discussed. It is concluded that the type or sign of
strain applied on antimony atoms during the growth of the alloy should be
heavily dependent on its composition.
