  Fractional-spin errors are inherent in all current approximate density
functionals, including Hartree-Fock theory, and their origin has been related
to strong static correlation effects. The conventional way to encode
fractional-spin calculations is to construct an ensemble density that scales
between the high-spin and low-spin densities. In this article, we explore the
variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in
one-electron systems using restricted and unrestricted ensemble densities, and
the exact generalized Hartree-Fock representation. By considering the hydrogen
atom and H$_2^+$ cation, we analyze how the unrestricted and generalized
Hartree-Fock schemes minimize this error by localizing the electrons or
rotating the spin coordinates. We also reveal a clear similarity between the
Coulomb hole of He-like ions and the density depletion near the nucleus induced
by the fractional-spin error in the unpolarized hydrogen atom. Finally, we
analyze the effect of the fractional-spin error on the M{\o}ller-Plesset
adiabatic connection, excited states, and functional- and density-driven
errors.
