Application of self-consistent $\alpha$ method to improve the performance of model exchange potentials

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 $\alpha$).