  The lithium manganese oxide spinel Li$_x$Mn$_2$O$_4$, with $0\leq x\leq 2$,
is an important example for cathode materials in lithium ion batteries.
However, an accurate description of Li$_x$Mn$_2$O$_4$ by first-principles
methods like density functional theory is far from trivial due to its complex
electronic structure, with a variety of energetically close electronic and
magnetic states. It was found that the local density approximation as well as
the generalized gradient approximation (GGA) are unable to describe
Li$_x$Mn$_2$O$_4$ correctly. Here, we report an extensive benchmark for
different Li$_x$Mn$_y$O$_z$ systems using the hybrid functionals PBE0 and
HSE06, as well as the recently introduced local hybrid functional PBE0r. We
find that all of these functionals yield energetic, structural, electronic, and
magnetic properties in good agreement with experimental data. The notable
benefit of the PBE0r functional, which relies on on-site Hartree-Fock exchange
only, is a much reduced computational effort that is comparable to GGA
functionals. Furthermore, the Hartree-Fock mixing factors in PBE0r are smaller
than in PBE0, which improves the results for (lithium) manganese oxides. The
investigation of Li$_x$Mn$_2$O$_4$ shows that two Mn oxidation states, +III and
+IV, coexist. The Mn$^\text{III}$ ions are in the high-spin state and the
corresponding MnO$_6$ octahedra are Jahn-Teller distorted. The ratio between
Mn$^\text{III}$ and Mn$^\text{IV}$ and thus the electronic structure changes
with the Li content while no major structural changes occur in the range from
$x=0$ to $1$. This work demonstrates that the PBE0r functional provides an
equally accurate and efficient description of the investigated
Li$_x$Mn$_y$O$_z$ systems.
