  In the constrained minimization method of Gidopoulos and Lathiotakis (J.
Chem. Phys. 136, 224109), the Hartree exchange and correlation Kohn-Sham
potential of a finite $N$-electron system is replaced by the electrostatic
potential of an effective charge density that is everywhere positive and
integrates to a charge of $N-1$ electrons. The optimal effective charge density
(electron repulsion density, $\rho_{\rm rep}$) and the corresponding optimal
effective potential (electron repulsion potential $v_{\rm rep}$) are obtained
by minimizing the electronic total energy in any density functional
approximation. The two constraints are sufficient to remove the
self-interaction errors from $v_{\rm rep}$, correcting its asymptotic behavior
at large distances from the system. In the present work, we describe, in
complete detail, the constrained minimization method, including recent
refinements. We also assess its performance in removing the self-interaction
errors for three popular density functional approximations, namely LDA, PBE,
and B3LYP, by comparing the obtained ionization energies to their experimental
values for an extended set of molecules. We show that the results of the
constrained minimizations are almost independent of the specific approximation
with average percentage errors 15%, 14%, 13% for the above DFAs respectively.
These errors are substantially smaller than the corresponding errors of the
plain (unconstrained) Kohn-Sham calculations at 38%, 39%, and 27% respectively.
Finally, we showed that this method correctly predicts negative values for the
HOMO energies of several anions.
