Electronic structure of the pyrochlore metals ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ and ${\mathrm{Cd}}_2{\mathrm{Re}}_2{\mathrm{O}}_7$

First-principles density-functional calculations within the local spin-density approximation and the generalized gradient approximation are reported for pyrochlore ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ and ${\mathrm{Cd}}_2{\mathrm{Re}}_2{\mathrm{O}}_7.$ The transition-metal $t_{2g}$ manifolds are found to be well separated in energy from the O $2p$ bands and from the higher-lying $e_g$ and Cd derived bands. The active electronic structure in the $t_{2g}$ manifold near the Fermi energy $E_F$ is found to be significantly modified by spin-orbit interactions, which we include. Both materials show semimetallic band structures, in which the $E_F$ lies in an pseudogap. The band structure of ${\mathrm{Cd}}_2{\mathrm{Os}}_2{\mathrm{O}}_7$ near $E_F$ is dominated by very heavy-hole and electron bands, though at $E_F$ the electron sections are lighter. ${\mathrm{Cd}}_2{\mathrm{Re}}_2{\mathrm{O}}_7$ has heavy-hole bands but moderate mass electron states. The results are discussed in terms of measured transport and thermodynamic properties of these compounds as well as the very different ground states of these two materials.