  The random phase approximation (RPA) to the correlation energy is extended to
fractional occupations and its performance examined for exact conditions on
fractional charges and fractional spins. RPA satisfies the constancy condition
for fractional spins that leads to correct bond dissociation and no static
correlation error for H$_2$ but massively fails for fractional charges, with an
enormous delocalization error even for a one-electron system such as H$_2^+$.
Other methods such as range-separated RPA can reduce this delocalization error
but only at the cost of increasing the static correlation error. None of the
RPA methods seem to have the discontinuous nature required to satisfy both
exact conditions and the full unified condition, emphasizing the need to go
further than just smooth functionals of the orbitals.
