  Based on crystal chemistry considerations and quantum density functional
theory ground state calculations, rhombohedral rh-C3 and hexagonal h-C6 carbon
allotropes are proposed and energetically calculated as new stable ultra-hard
phases likewise lonsdaleite. Along the two kinds of carbon in linear C2-C1-C2
lattice, distorted tetrahedra C2(sp3) with an angle of 106.17{\deg} (smaller
than ideal 109.4{\deg}) and C1(sp)-like hybridizations are inferred from charge
density projections. The calculated elastic constants point to a strong
anisotropy of mechanical properties of rh-C3 and h-C6 with an exceptionally
large C33 values (1636 GPa and 1610 GPa, respectively), exceeding that of
lonsdaleite (1380 GPa), due to the presence of aligned tricarbon units along
the hexagonal c-axis. Both phases are characterized by large bulk moduli and
high hardness values that are slightly less than those of lonsdaleite and
diamond. Weak metallic behavior of both new phases is identified from
electronic band structure calculations.
