  The localized Hartree-Fock potential has proven to be a computationally
efficient alternative to the optimized effective potential, preserving the
numerical accuracy of the latter and respecting the exact properties of being
self-interaction free and having the correct $-1/r$ asymptotics. In this paper
we extend the localized Hartree-Fock potential to fractional particle numbers
and observe that it yields derivative discontinuities in the energy as required
by the exact theory. The discontinuities are numerically close to those of the
computationally more demanding Hartree-Fock method. Our potential enjoys a
"direct-energy" property, whereby the energy of the system is given by the sum
of the single-particle eigenvalues multiplied by the corresponding occupation
numbers. The discontinuities $c_\uparrow$ and $c_\downarrow$ of the
spin-components of the potential at integer particle numbers $N_\uparrow$ and
$N_\downarrow$ satisfy the condition $c_\uparrow N_\uparrow+c_\downarrow
N_\downarrow=0$. Thus, joining the family of effective potentials which support
a derivative discontinuity, but being considerably easier to implement, the
localized Hartree-Fock potential becomes a powerful tool in the broad area of
applications in which the fundamental gap is an issue.
