  In the search for an accurate and computationally efficient approximation to
the exact exchange potential of Kohn-Sham density functional theory, we
recently compared various semilocal exchange potentials to the exact one [F.
Tran et al., Phys. Rev. B 91, 165121 (2015)]. It was concluded that the
Becke-Johnson (BJ) potential is a very good starting point, but requires the
use of empirical parameters to obtain good agreement with the exact exchange
potential. In the present work, we extend the comparison by considering the
Krieger-Li-Iafrate (KLI) approximation, which is a beyond-semilocal
approximation. It is shown that overall the KLI and BJ-based potentials are the
most reliable approximations to the exact exchange potential, however, sizeable
differences, especially for the antiferromagnetic transition-metal oxides, can
be obtained.
