  Recently it was claimed that nitrogen-doped lutetium hydride exhibited a
near-ambient superconducting transition with a temperature of 294 K at a
pressure of only 10 kbar, this pressure being several orders of magnitude lower
than previously demonstrated for hydrides under pressure. In this paper, we
investigate within DFT+U the electronic structure of both parent lutetium
hydride LuH$_3$ and nitrogen doped lutetium hydride LuH$_{2.75}$N$_{0.25}$. We
calculated corresponding bands, density of states and Fermi surfaces. It is
shown that in the stoichiometric system the Lu-5d states cross the Fermi level
while the H-1s states make almost no contribution at the Fermi level. However,
with nitrogen doping, the N-2p states enter the Fermi level in large quantities
and bring together a significant contribution from the H-1s states. The
presence of N-2p and H-1s states at the Fermi level in a doped compound can
facilitate the emergence of superconductivity. For instance, nitrogen doping
almost doubles the value of DOS at the Fermi level. Simple BCS analysis shows
that the nitrogen doping of LuH$_3$ can provide T$_c$ more than 100K and even
increase it with further hole doping.
