  Using density functional theory plus self-consistent Hubbard $U$
(DFT$+U_{sc}$) calculations, we have investigated the structural and electronic
properties of the rare-earth cobaltites \textit{R}CoO$_3$ (\textit{R} = Pr --
Lu). Our calculations show the evolution of crystal and electronic structure of
the insulating low-spin (LS) \textit{R}CoO$_3$ with increasing rare-earth
atomic number (decreasing ionic radius), including the invariance of the Co-O
bond distance ($d_{Co-O}$), the decrease of the Co-O-Co bond angle ($\Theta$),
and the increase of the crystal field splitting ($\Delta_{CF}$) and band gap
energy ($E_g$). Agreement with experiment for the latter improves considerably
with the use of DFT$+U_{sc}$ and all trends are in good agreement with
experimental data. These trends enable a direct test of prior rationalizations
of the trend in spin-gap associated with the spin crossover in this series,
which is found to expose significant issues with simple band based arguments.
We also examine the effect of placing the rare-earth \textit{f}-electrons in
the core region of the pseudopotential. The effect on lattice parameters and
band structure is found to be small, but distinct for the special case of
\textit{Pr}CoO$_3$ where some \textit{f}-states populate the middle of the gap,
consistent with recent reports of unique behavior in Pr-containing cobaltites.
Overall, this study establishes a foundation for future predictive studies of
thermally induced spin excitations in rare-earth cobaltites and similar
systems.
