Absence of superconductivity in iron polyhydrides at high pressures

Recently, C. M. Pepin textit{et al.} [Science textbf{357}, 382 (2017)] reported the formation of several new iron polyhydrides FeH$_x$ at pressures in the megabar range, and spotted FeH$_5$, which forms above 130 GPa, as a potential high-tc superconductor, because of an alleged layer of dense metallic hydrogen. Shortly after, two studies by A.~Majumdar textit{et al.} [Phys. Rev. B textbf{96}, 201107 (2017)] and A.~G.~Kvashnin textit{et al.} [J. Phys. Chem. C textbf{122}, 4731 (2018)] based on {em ab initio} Migdal-Eliashberg theory seemed to independently confirm such a conjecture. We conversely find, on the same theoretical-numerical basis, that neither FeH$_5$ nor its precursor, FeH$_3$, shows any conventional superconductivity and explain why this is the case. We also show that superconductivity may be attained by transition-metal polyhydrides in the FeH$_3$ structure type by adding more electrons to partially fill one of the Fe--H hybrid bands (as, e.g., in NiH$_3$). Critical temperatures, however, will remain low because the $d$--metal bonding, and not the metallic hydrogen, dominates the behavior of electrons and phonons involved in the superconducting pairing in these compounds.