  In this work we define single-particle potentials for a positron and a
positronium atom interacting with light atoms (H, He, Li and Be) by inverting a
single-particle Schr\"odinger equation. For this purpose, we use accurate
energies and positron densities obtained from the many-body wavefunction of the
corresponding positronic systems. The introduced potentials describe the exact
correlations for the calculated systems including the formation of a
positronium atom. We show that the scattering lengths and the low-energy s-wave
phase shifts from accurate many-body calculations are well accounted for by the
introduced potential. We also calculate self-consistent two-component
density-functional theory positron potentials and densities for the bound
positronic systems within the local density approximation. They are in a very
good agreement with the many-body results, provided that the
finite-positron-density electron-positron correlation potential is used, and
they can also describe systems comprising a positronium atom. We argue that the
introduced single-particle positron potentials defined for single molecules are
transferable to the condensed phase when the inter-molecular interactions are
weak. When this condition is fulfilled, the total positron potential can be
constructed in a good approximation as the superposition of the molecular
potentials.
