  We present an atomic orbital based approximate scheme for self-interaction
correction (SIC) to the local density approximation of density functional
theory. The method, based on the idea of Filippetti and Spaldin [Phys. Rev. B
67, 125109 (2003)], is implemented in a code using localized numerical atomic
orbital basis sets and is now suitable for both molecules and extended solids.
After deriving the fundamental equations as a non-variational approximation of
the self-consistent SIC theory, we present results for a wide range of
molecules and insulators. In particular, we investigate the effect of
re-scaling the self-interaction correction and we establish a link with the
existing atomic-like corrective scheme LDA+U. We find that when no re-scaling
is applied, i.e. when we consider the entire atomic correction, the Kohn-Sham
HOMO eigenvalue is a rather good approximation to the experimental ionization
potential for molecules. Similarly the HOMO eigenvalues of negatively charged
molecules reproduce closely the molecular affinities. In contrast a re-scaling
of about 50% is necessary to reproduce insulator bandgaps in solids, which
otherwise are largely overestimated. The method therefore represents a
Kohn-Sham based single-particle theory and offers good prospects for
applications where the actual position of the Kohn-Sham eigenvalues is
important, such as quantum transport.
