  We investigate the electronic structure and lattice stability of pristine and
functionalized (with either hydrogen or oxygen) $\alpha$-graphyne systems. We
identify lattice instabilities due to soft-phonon modes, and describe two
mechanisms leading to gap opening in the Dirac-fermion electronic spectrum of
these systems: symmetry breaking, connected with the lattice instabilities, and
partial incorporation of an $sp^3$-hybrid character in the covalent-bonding
network of a buckled hydrogenated $\alpha$-graphyne lattice that retains the
symmetries of the parent pristine $\alpha$-graphyne. In the case of an
oxygen-functionalized $\alpha$-graphyne structure, each O atom binds
asymmetrically to two twofold-coordinated C atoms, breaking inversion and
mirror symmetries, and leading to the opening of a sizeable gap of 0.22 eV at
the Dirac point. Generally, mirror symmetries are found to suffice for the
occurrence of gapless Dirac cones in these $\alpha$-graphyne systems, even in
the absence of inversion symmetry centers. Moreover, we analyze the gapless and
gapped Dirac cones of pristine and functionalized $\alpha$-graphynes from the
perspective of the dispersion relations for massless and massive free Dirac
fermions. We find that mirror-symmetry breaking mimics a Dirac-fermion
mass-generation mechanism in the oxygen-functionalized $\alpha$-graphyne,
leading to gap opening and to isotropic electronic dispersions with a rather
small electron-hole asymmetry. In the hydrogen-functionalized case, we find
that carriers show a remarkable anisotropy, behaving as massless fermions along
the M-K line in the Brillouin zone and as massive fermions along the $\Gamma$-K
line.
