  The energetic stability of positron di-anion systems [A$^-;e^+;$A$^-$] is
studied via many-body theory, where $A^-$ includes H$^{-}$, F$^{-}$, Cl$^{-}$
and the molecular anions (CN)$^{-}$ and (NCO)$^{-}$. Specifically, the energy
of the system as a function of ionic separation is determined by solving the
Dyson equation for the positron in the field of the two anions, using a
positron-anion self energy as constructed in [J. Hofierka, B. Cunningham, C. M.
Rawlins, C. H. Patterson and D. G. Green, \emph{Nature} {\bf 606} 688 (2022)]
that accounts for correlations including polarization, screening, and
virtual-positronium formation. Calculations are performed for a positron
interacting with H$_{2}^{2-}$, F$_{2}^{2-}$, and Cl$_{2}^{2-}$, and are found
to be in good agreement with previous theory. In particular, we confirm the
presence of two minima in the potential energy of the [H$^-;e^+$;H$^-$] system
with respect to ionic separation: one a positronically-bonded [H$^-;e^+$;H$^-$]
local minimum at ionic separations $r\sim3.4$~\AA\phantom{}, and a global
minimum at smaller ionic separations $r\lesssim1.6$~\AA\phantom{} that gives
overall instability of the system with respect to dissociation into a H$_2$
molecule and a positronium negative ion, Ps$^-$. The first predictions are made
for positronic bonding in dianions consisting of molecular anionic fragments,
specifically for (CN)$_{2}^{2-}$, and (NCO)$_{2}^{2-}$. In all cases we find
that the molecules formed by the creation of a positronic bond are stable
relative to dissociation into A$^-$ and $e^+$A$^-$ (positron bound to a single
anion), with bond energies on the order of 1~eV and bond lengths on the order
of several \r angstroms.
