  We present an ab initio analysis with density functional theory for
superconductors (SCDFT) to understand the superconducting mechanism of doped
layered nitrides \beta-Li$_x$MNCl (M=Ti, Zr, and Hf). The current version of
SCDFT is based on the Migdal-Eliashberg theory and has been shown to reproduce
accurately experimental superconducting-transition temperatures Tc of a wide
range of phonon-mediated superconductors. In the present case, however, our
calculated Tc$\leq$4.3 K (M=Zr) and $\leq$10.5 K (M=Hf) are found to be less
than a half of the experimental Tc. In addition, Tc obtained in the present
calculation increases with the doping concentration x, opposite to that
observed in the experiment. Our results indicate that we need to consider some
elements missing in the present SCDFT based on the Migdal-Eliashberg theory.
