  Predicting the compositional phase stability of strongly correlated electron
materials is an outstanding challenge in condensed matter physics. In this
work, we employ the DFT+U formalism to address the effects of local
correlations due to transition metal d electrons on compositional phase
stability in the prototype phase stable and separating materials LixCoO2 and
olivine LixFePO4, respectively. We exploit a new spectral decomposition of the
DFT+U total energy, revealing the distinct roles of the filling and ordering of
the d orbital correlated subspace. The on-site interaction U drives both of
these very different materials systems towards phase separation, stemming from
enhanced ordering of the d orbital occupancies in the x=0 and x=1 species,
whereas changes in the overall filling of the d shell contribute negligibly. We
show that DFT+U formation energies are qualitatively consistent with
experiments for phase stable LixCoO2, phase separating LixFePO4, and phase
stable LixCoPO4. However, we find that charge ordering plays a critical role in
the energetics at intermediate x, strongly dampening the tendency for the
Hubbard U to drive phase separation. Most relevantly, the phase stability of
Li1/2CoO2 within DFT+U is qualitatively incorrect without allowing charge
ordering, which is problematic given that neither charge ordering nor the band
gap that it induces are observed in experiment. We demonstrate that charge
ordering arises from the correlated subspace interaction energy as opposed to
the double counting. Additionally, we predict the Li order-disorder transition
temperature for Li1/2CoO2, demonstrating that the unphysical charge ordering
within DFT+U renders the method problematic, often producing unrealistically
large results. Our findings motivate the need for other advanced techniques,
such as DFT+DMFT, for total energies in strongly correlated materials.
