  The reported new algorithm determines the exact exchange potential v_x in a
iterative way using energy and orbital shifts (ES, OS) obtained - with
finite-difference formulas - from the solutions (occupied orbitals and their
energies) of the Hartree-Fock-like equation and the Kohn-Sham-like equation,
the former used for the initial approximation to v_x and the latter - for
increments of ES and OS due to subsequent changes of v_x. Thus, solution of the
differential equations for OS, used by Kummel and Perdew (KP) [Phys. Rev. Lett.
90, 043004 (2003)], is avoided. The iterated exchange potential, expressed in
terms of ES and OS, is improved by modifying ES at odd iteration steps and OS
at even steps. The modification formulas are related to the OEP equation
(satisfied at convergence) written as the condition of vanishing density shift
(DS) - they are obtained, respectively, by enforcing its satisfaction through
corrections to approximate OS and by determining optimal ES that minimize the
DS norm. The proposed method, successfully tested for several closed-(sub)shell
atoms, from Be to Kr, within the DFT exchange-only approximation, proves highly
efficient. The calculations using pseudospectral method for representing
orbitals give iterative sequences of approximate exchange potentials (starting
with the Krieger-Li-Iafrate approximation) that rapidly approach the exact v_x
so that, for Ne, Ar and Zn, the corresponding DS norm becomes less than 10^{-6}
after 13, 13 and 9 iteration steps for a given electron density. In
self-consistent density calculations, orbital energies of 10^{-4} Hartree
accuracy are obtained for these atoms after, respectively, 9, 12 and 12 density
iteration steps, each involving just 2 steps of v_x iteration, while the
accuracy limit of 10^{-6}--10^{-7} Hartree is reached after 20 density
iterations.
