  By exploiting freedoms in the definitions of 'correlation', 'exchange' and
'Hartree' physics in ensemble systems we better generalise the notion of 'exact
exchange' (EXX) to systems with fractional occupations functions of the
frontier orbitals, arising in the dissociation limit of some molecules. We
introduce the Linear EXX ("LEXX") theory whose pair distribution and energy are
explicitly \emph{piecewise linear} in the occupations $f^{\sigma}_{i}$. {\hi}We
provide explicit expressions for these functions for frontier $s$ and $p$
shells. Used in an optimised effective potential (OEP) approach it yields
energies bounded by the piecewise linear 'ensemble EXX' (EEXX) energy and
standard fractional optimised EXX energy: $E^{EEXX}\leq E^{LEXX} \leq E^{EXX}$.
Analysis of the LEXX explains the success of standard OEP methods for diatoms
at large spacing, and why they can fail when both spins are allowed to be
non-integer so that "ghost" Hartree interactions appear between \emph{opposite}
spin electrons in the usual formula. The energy $E^{LEXX}$ contains a
cancellation term for the spin ghost case. It is evaluated for H, Li and Na
fractional ions with clear derivative discontinuities for all cases. The
$p$-shell form reproduces accurate correlation-free energies of B-F and Al-Cl.
We further test LEXX plus correlation energy calculations on fractional ions of
C and F and again shows both derivative discontinuities and good agreement with
exact results.
