  In this work, we present a comparative first principles study of mechanical
properties, electronic structure, phonon dispersion relation, electron-phonon
coupling and magnetism in three isostructural superconductors, namely, ThFeAsN,
ThCoAsN and ThNiAsN. Experimentally, ThFeAsN and ThNiAsN show superconducting
properties, while ThCoAsN has not been synthesized. Our calculated elastic
constants show that all these systems are mechanically stable. Significant
differences in the electronic structures of these three compounds in terms of
density of states, band structures and Fermi surfaces, are found. Our phonon
calculations reveal that all the systems including ThCoAsN, are dynamically
stable. Phonon dispersion relations indicate that the optical modes of all the
three systems are almost the same while there are significant variations in the
low frequency manifold consisting of mixed modes. The electron-phonon coupling
constants and superconducting transition temperatures calculated based on the
Eliashberg formalism, predict a rather high $T_c$ of 6.4 K for ThCoAsN and also
a $T_c$ of 3.4 K for ThNiAsN which agrees well with the experimental value of
4.3 K. Nevertheless, we find a $T_c$ of 0.05 K for ThFeAsN, which is much
smaller than the experimental $T_c$ of $\sim$30 K. However, a simple analysis
considering the amplifying effects of spin density wave order and out-of-plane
soft phonon modes suggests that the $T_c$ could be increased considerably to
$\sim$10 K. Finally, we also discuss the effect of anion As height on the
electronic structures and study possible magnetic states in these three
compounds.
