  Self interaction error remains an impotrant problem in density functional
theory. A number of approximations to exact exchange aimed to correct for this
error while retainining computational efficiency had been suggested recently.
We present a critical comparison between model exchange potentials generated
through the application of the asymptotically-adjusted self-consistent
$\alpha$, AASC$\alpha$, method and BJ effective exchange potential advanced in
[A.D. Becke and E.R. Johnson, J. Chem. Phys. 124, 221101 (2006)] and [V.N.
Staroverov, J. Chem. Phys. 129, 134103 (2008)]. In particular we discuss their
compliance with coordinate-scaling, virial and functional derivative
conditions. We discuss the application of the AASC$\alpha$ method to generate
the AA-BJ potential. A numerical comparison is carried out through the
implementation of a fully-numerical diatomic molecule code yielding molecular
virial energies and ionization potentials approximated by the energies of the
HOMO orbitals. It is shown that some of the shortcomings of these model
potentials, such as the non-compliance with the Levy-Perdew virial relation,
may be eliminated by multiplying the response term by an orbital-dependent
functional $\alpha$, which can be simplified to a constant determined during
the self-consistent procedure (self-consistent $\alpha$).
