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We investigate the electronic structure and lattice stability of pristine andfunctionalized (with either hydrogen or oxygen) $\alpha$-graphyne systems. Weidentify lattice instabilities due to soft-phonon modes, and describe twomechanisms leading to gap opening in the Dirac-fermion electronic spectrum ofthese systems: symmetry breaking, connected with the lattice instabilities, andpartial incorporation of an $sp^3$-hybrid character in the covalent-bondingnetwork of a buckled hydrogenated $\alpha$-graphyne lattice that retains thesymmetries of the parent pristine $\alpha$-graphyne. In the case of anoxygen-functionalized $\alpha$-graphyne structure, each O atom bindsasymmetrically to two twofold-coordinated C atoms, breaking inversion andmirror symmetries, and leading to the opening of a sizeable gap of 0.22 eV atthe Dirac point. Generally, mirror symmetries are found to suffice for theoccurrence of gapless Dirac cones in these $\alpha$-graphyne systems, even inthe absence of inversion symmetry centers. Moreover, we analyze the gapless andgapped Dirac cones of pristine and functionalized $\alpha$-graphynes from theperspective of the dispersion relations for massless and massive free Diracfermions. We find that mirror-symmetry breaking mimics a Dirac-fermionmass-generation mechanism in the oxygen-functionalized $\alpha$-graphyne,leading to gap opening and to isotropic electronic dispersions with a rathersmall electron-hole asymmetry. In the hydrogen-functionalized case, we findthat carriers show a remarkable anisotropy, behaving as massless fermions alongthe M-K line in the Brillouin zone and as massive fermions along the $\Gamma$-Kline.

Theoretical chemistry of $\alpha$-graphyne: functionalization, symmetry breaking, and generation of Dirac-fermion mass