  Transition-metal compounds pose serious challenges to first-principles
calculations based on density-functional theory (DFT), due to the inability of
most approximate exchange-correlation functionals to capture the localization
of valence electrons on their $d$ states, essential for a predictive modeling
of their properties. In this work we focus on two representatives of a well
known family of cathode materials for Li-ion batteries, namely the orthorhombic
LiMPO$_4$ olivines (M = Fe, Mn). We show that extended Hubbard functionals with
on-site ($U$) and inter-site ($V$) interactions (so called DFT+U+V) can predict
the electronic structure of the mixed-valence phases, the formation energy of
the materials with intermediate Li contents, and the overall average voltage of
the battery with remarkable accuracy. We find, in particular, that the
inclusion of inter-site interactions in the corrective Hamiltonian improves
considerably the prediction of thermodynamic quantities when electronic
localization occurs in the presence of significant interatomic hybridization
(as is the case for the Mn compound), and that the self-consistent evaluation
of the effective interaction parameters as material- and ground-state-dependent
quantities allows the prediction of energy differences between different phases
and concentrations.
