The configuration-interaction (CI) method is applied to the study of the positronium-hydride (PsH) and positronic-beryllium (e+Be) systems. The binding energy and other properties are slowly convergent with respect to the angular momentum of the orbitals used to construct the CI basis states. The largest calculations recover 94% and 80% of the binding energy against dissociation when compared with existing calculations of PsH and e+ Be. Extrapolating using Cl convergence trends improves these results to 99% and 98%, respectively. Convergence is not so good for the electron-positron annihilation rates, but the extrapolated annihilation rates were within 10% of the best calculations. Two different schemes have been used to construct the CI basis, and it is found that it is possible to discard roughly half the CI basis with almost no degradation in the binding energy and the annihilation rate. These investigations demonstrate the feasibility of using single particle orbitals centred on the nucleus to represent positronic systems with two valence electrons.