  While defects such as oxygen vacancies in correlated materials can modify
their electronic properties dramatically, understanding the microscopic origin
of electronic correlations in materials with defects has been elusive.
Lanthanum nickelate with oxygen vacancies, LaNiO$_{3-x}$, exhibits the
metal-to-insulator transition as the oxygen vacancy level $x$ increases from
the stoichiometric LaNiO$_3$. In particular, LaNiO$_{2.5}$ exhibits a
paramagnetic insulating phase, also stabilizing an antiferromagnetic state
below $T_N\simeq152$K. Here, we study the electronic structure and energetics
of LaNiO$_{3-x}$ using first-principles. We find that LaNiO$_{2.5}$ stabilizes
a vacancy-ordered structure with an insulating ground state and the nature of
the insulating phase is a "site-selective" paramagnetic Mott state as obtained
using density functional theory plus dynamical mean field theory (DFT+DMFT).
The Ni octahedron site develops a Mott insulating state with strong
correlations as the Ni $e_g$ orbital is half-filled while the Ni square-planar
site with apical oxygen vacancies becomes a band insulator. Our oxygen vacancy
results can not be explained by the pure change of the Ni oxidation state alone
within the rigid band shift approximation. Our DFT+DMFT density of states
explains that the peak splitting of unoccupied states in LaNiO$_{3-x}$ measured
by the experimental X-ray absorption spectra originates from two nonequivalent
Ni ions in the vacancy-ordered structure.
