  We revisit the problem of unphysical charge density
delocalization/fractionalization induced by the self-interaction error of
common approximate Kohn-Sham Density Functional Theory functionals on
simulation of small to medium-size proteins in vacuum. Aside from producing
unphysical electron densities and total energies, the vanishing of the
HOMO-LUMO gap associated with the unphysical charge delocalization leads to an
unphysical low-energy spectrum and catastrophic failure of most popular solvers
for the Kohn-Sham (KS) self-consistent field (SCF) problem. We apply a robust
quasi-Newton SCF solver [Phys. Chem. Chem. Phys. 26, 6557 (2024)] to obtain
solutions for some of these difficult cases. The anatomy of the charge
delocalization is revealed by the natural deformation orbitals obtained from
the density matrix difference between the Hartree-Fock and KS solutions; the
charge delocalization can occur not only between charged fragments (such as in
zwitterionic polypeptides) but also may involve neutral fragments. The
vanishing-gap phenomenon and troublesome SCF convergence are both attributed to
the unphysical KS Fock operator eigenspectra of molecular fragments (e.g.,
amino acids or their side chains). Analysis of amino acid pairs suggests that
the unphysical charge delocalization can be partially ameliorated by the use of
{\em some} range-separated hybrid functionals, but not by semilocal or standard
hybrid functionals. Last, we demonstrate that solutions without the unphysical
charge delocalization can be located even for semilocal KS functionals highly
prone to such defects, but such solutions have non-Aufbau character and are
unstable with respect to mixing of the non-overlapping "frontier" orbitals.
