  The spinel-structured lithium manganese oxide (LiMn$_2$O$_4$) is a material
currently used as cathode for secondary lithium-ion batteries, but whose
properties are not yet fully understood. Here, we report a computational
investigation of the inversion thermodynamics and electronic behaviour of
LiMn$_2$O$_4$ derived from spin-polarised density functional theory
calculations with a Hubbard Hamiltonian and long-range dispersion corrections
(DFT+$U-$D3). Based on the analysis of the configurational free energy, we have
elucidated a partially inverse equilibrium cation distribution for the
LiMn$_2$O$_4$ spinel. This equilibrium degree of inversion is rationalised in
terms of the crystal field stabilisation effects and the difference between the
size of the cations. We compare the atomic charges with the oxidation numbers
for each degree of inversion. We found segregation of the Mn charge once these
ions occupy the tetrahedral and octahedral sites of the spinel. We have
obtained the atomic projections of the electronic band structure and density of
states, showing that the normal LiMn$_2$O$_4$ has half-metallic properties,
while the fully inverse spinel is an insulator. This material is in the
ferrimagnetic state for the inverse and partially inverse cation arrangement.
The optimised lattice and oxygen parameters, as well as the equilibrium degree
of inversion, are in agreement with the available experimental data. The
partially inverse equilibrium degree of inversion is important in the
interpretation of the lithium ion migration and surface properties of the
LiMn$_2$O$_4$ spinel.
