  We calculate the properties of the 4$d$ ferromagnet SrRuO$_3$ in bulk and
thin film form with the aim of understanding the experimentally observed metal
to insulator transition at reduced thickness. Although the spatial extent of
the 4$d$ orbitals is quite large, many experimental results have suggested that
electron-electron correlations play an important role in determining this
material's electronic structure. In order to investigate the importance of
correlation, we use two approaches which go beyond the conventional local
density approximation to density functional theory (DFT): the local spin
density approximation + Hubbard $U$ (LSDA+$U$) and the pseudopotential
self-interaction correction (pseudo-SIC) methods. We find that the details of
the electronic structure predicted with the LSDA do not agree with the
experimental spectroscopic data for bulk and thin film SrRuO$_3$. Improvement
is found by including electron-electron correlations, and we suggest that bulk
orthorhombic SrRuO$_3$ is a {\it weakly strongly-correlated} ferromagnet whose
electronic structure is best described by a 0.6 eV on-site Hubbard term, or
equivalently with corrections for the self-interaction error. We also perform
{\it ab initio} transport calculations that confirm that SrRuO$_3$ has a
negative spin polarization at the Fermi level, due to the position of the
minority Ru 4$d$ band center. Even with correlations included in our
calculations we are unable to reproduce the experimentally observed
metal-insulator transition, suggesting that the electronic behavior of
SrRuO$_3$ ultra-thin films might be dominated by extrinsic factors such as
surface disorder and defects.
