  We investigate the electronic structure of Sr$_3$NiRhO$_6$, a
quasi-one-dimensional spin chain system using \emph{ab initio} band structure
calculations. Spin polarized calculations within GGA reveal that Ni and Rh have
finite moments and they are antiferromagnetically coupled along the chain axis
in the ground state. While these results obtained within the local spin density
approximations provide remarkable representation of the magnetic phase, the
experimentally observed insulating behavior could not be captured within this
method. GGA+$U$ calculations show that opening up of an insulating gap requires
on-site Coulomb interaction among the Rh 4$d$ electrons, $U_{dd}^{Rh}$
$\approx$ 2.5 eV and the correlation among Ni 3$d$ electrons, $U_{dd}^{Ni}$
$\approx$ 4.5 eV suggesting this system to be a Mott insulator. Electron
correlation among $d$ electrons leads to significant enhancement of the O 2$p$
character in the energy bands in the vicinity of the Fermi level and the $d$
bands appear at lower energies. Energy gap in the up spin density of states
appears to be significantly small ($\sim$ 0.12 eV) while it is $>$ 2 eV in the
down spin density of states suggesting possibility of spin polarized conduction
in the semiconducting phase.
