  Positron binding energies in the negative ions H$^-$, F$^-$, Cl$^-$ and
Br$^-$, and the closed-shell atoms Be, Mg, Zn and Ca, are calculated via a
many-body theory approach developed by the authors [J.~Hofierka \emph{et al.}
Nature~{\bf 608}, 688-693 (2022)]. Specifically, the Dyson equation is solved
using a Gaussian basis, with the positron self energy constructed from three
infinite classes of diagrams that account for the strong positron-atom
correlations that characterise the system including the positron-induced
polarization of the electron cloud, screening of the electron-positron Coulomb
interaction, virtual-positronium formation and electron-hole and positron-hole
interactions. For the negative ions, binding occurs at the static level of
theory, and the correlations are found to enhance the binding energies by
$\sim$25--50\%, yielding results in good agreement with ($\lesssim$5\% larger
than) calculations from a number of distinct methods. For the atoms, for which
binding is enabled exclusively by correlations, most notably virtual-Ps
formation, the binding energies are found to be of similar order to (but
$\sim$10--30\% larger than) relativistic coupled-cluster calculations of [C.
Harabati, V.~A.~Dzuba and V.~V. Flambaum, Phys.~Rev.~A {\bf 89}, 022517
(2014)], both of which are systematically larger than stochastic variational
calculations of [M.~Bromley and J.~Mitroy, Phys.~Rev.~A {\bf 73} (2005);
J.~Mitroy, J.~At.~Mol.~Sci.~{\bf 1}, 275 (2010)].
