  Accurate first-principles predictions of the structural, electronic,
magnetic, and electrochemical properties of cathode materials can be key in the
design of novel efficient Li-ion batteries. Spinel-type cathode materials
Li$_x$Mn$_2$O$_4$ and Li$_x$Mn$_{1.5}$Ni$_{0.5}$O$_4$ are promising candidates
for Li-ion battery technologies, but they present serious challenges when it
comes to their first-principles modeling. Here, we use density-functional
theory with extended Hubbard functionals - DFT+$U$+$V$ with on-site $U$ and
inter-site $V$ Hubbard interactions - to study the properties of these
transition-metal oxides. The Hubbard parameters are computed from
first-principles using density-functional perturbation theory. We show that
while $U$ is crucial to obtain the right trends in properties of these
materials, $V$ is essential for a quantitative description of the structural
and electronic properties, as well as the Li-intercalation voltages. This work
paves the way for reliable first-principles studies of other families of
cathode materials without relying on empirical fitting or calibration
procedures.
