Phase Separation in Li$_x$FePO$_4$ Induced by Correlation Effects
  We report on a significant failure of LDA and GGA to reproduce the phase stability and thermodynamics of mixed-valence Li$_x$FePO$_4$ compounds. Experimentally, Li$_x$FePO$_4$ compositions ($0 \leq x \leq 1$) are known to be unstable and phase separate into Li FePO$_4$ and FePO$_4$. However, first-principles calculations with LDA/GGA yield energetically favorable intermediate compounds an d hence no phase separation. This qualitative failure of LDA/GGA seems to have its origin in the LDA/GGA self-interaction which de localizes charge over the mixed-valence Fe ions, and is corrected by explicitly considering correlation effects in this material. This is demonstrated with LDA+U calculations which correctly predict phase separation in Li$_x$FePO$_4$ for $U-J \gtrsim 3.5$eV. T he origin of the destabilization of intermediate compounds is identified as electron localization and charge ordering at different iron sites. Introduction of correlation also yields more accurate electrochemical reaction energies between FePO$_4$/Li$_x$FePO$_ 4$ and Li/Li$^+$ electrodes. 
F. Zhou, C. A. Marianetti, M. Cococcioni, D. Morgan, G. Ceder