  Recently, Lu-N-H materials were reported to have room-temperature
superconductivity, and the Hubbard U correction on the Lu's
$\textit{f}$-electrons is necessary, and a constant U = 5.5 eV was applied to
different Lu-N-H configurations (Nature 615, 244 (2023)). Following simulations
indicate that the superconducting transition temperature (T$_{c}$) of LuH$_{3}$
with U = 0 eV is 50 ~ 60 K, while the N-doped LuH$_{3}$ is below 30 K. Quite
recently, calculations with U = 5 eV shows that the T$_{c}$ of N-doped
LuH$_{3}$ exceeds 100 K. The properties of Lu-N-H are sensitive to the applied
U values. Here, the structural and electronic Lu-N-H properties at
high-pressure (0 ~ 10 GPa) are systematically investigated based on the hybrid
functional. We show that different Lu-N-H configurations should possess
different U values varying from 6.4 eV to 7.4 eV. Furthermore, at pressure
ranging from 0 GPa to 1 GPa, the $\textit{f}$ and $\textit{d}$ band centers of
N-doped LuH$_{3}$ show oscillation or even plateau, and the band gap of
insulators also shows a platform near this pressure, this is consistent with
the pressure range where room-temperature superconductivity appeared in Lu-N-H.
Our work provides insights into the understanding of Lu-N-H materials and other
hydrogen-rich superconductors based on the rare-earth elements.
