  For decades transition-metal oxides have generated a huge interest due to the
multitude of physical phenomena they exhibit. In this class of materials, the
rare-earth nickelates, $R$NiO$_3$, stand out for their rich phase diagram
stemming from complex couplings between the lattice, electronic and magnetic
degrees of freedom. Here, we present a first-principles study of the
low-temperature phase for two members of the $R$NiO$_3$ series, with $R=$ Pr,
Y. We employ density-functional theory with Hubbard corrections accounting not
only for the on-site localizing interactions among the Ni--$3d$ electrons
($U$), but also the inter-site hybridization effects between the
transition-metals and the ligands ($V$). All the \textit{U} and \textit{V}
parameters are calculated from first-principles using density-functional
perturbation theory, resulting in a fully \emph{ab initio} methodology. Our
simulations show that the inclusion of the inter-site interaction parameters
$V$ is necessary to simultaneously capture the features well-established by
experimental characterizations of the low-temperature state: insulating
character, antiferromagnetism and bond disproportionation. On the contrary, for
some magnetic orderings the inclusion of on-site interaction parameters $U$
alone completely suppresses the breathing distortion occurring in the
low-temperature phase and produces an erroneous electronic state with a
vanishing band gap. In addition -- only when both the \textit{U} and \textit{V}
are considered -- we predict a polar phase with a magnetization-dependent
electric polarization, supporting very recent experimental observations that
suggest a possible occurrence of type-II multiferroicity for these materials.
