  We investigate the impact of electron-lattice coupling on the stability of
various magnetic orders in rare-earth nickelates. We use the Hartree-Fock
approximation, at zero temperature, to study an effective, two-band model with
correlations characterized by a Hubbard $U$ and a Hund's $J$. This is coupled
to breathing-mode distortions of the octahedral oxygen cages, described
semi-classically, with a Holstein term. We analyze the effect of the various
parameters on the resulting phase diagram, in particular on the charge
disproportionation and on the magnetic order. We confirm that the coupling to
the lattice cooperates with Hund's coupling and thus encourages charge
disproportionation. We also find that it favors the fully disproportionated,
4-site periodic magnetic order of type $\Uparrow 0 \Downarrow 0$. Other
convergent magnetic phases, such as the collinear
$\uparrow\uparrow\downarrow\downarrow$ and non-collinear
$\uparrow\rightarrow\downarrow\leftarrow$ states, do not couple to the lattice
because of their lack of charge disproportionation. Novel phases, e.g. with
charge disproportionation but no magnetic order, are also found to be
stabilized in specific conditions.
