A quantitative approach to calculating properties of inhomogeneous two-component Coulomb-Fermi systems is presented. As an application, the ground-state electronic structure of a jellium vacancy containing a trapped positron is calculated self-consistently. While the resulting density profiles and energetics are quite different from those obtained neglecting cross correlations, the conventional estimates for the annihilation rates are shown to remain valid, due to canceling effects of the increase in the mean electron density and the decrease in short-range screening.
Nieminen, R. M., Boronski, E., & Lantto, L. J. (1985). Two-component density-functional theory: Application to positron states. Physical Review B, 32(2), 1377-1379. https://doi.org/10.1103/PhysRevB.32.1377