In this letter, we propose a functional on the basis of the Schwinger variational principle that accounts for the particle rearrangement by solving a projected Lippmann-Schwinger equation system. The method is tested in the static-coupled approximation for positron-H, where excellent agreements with benchmark results are found for the elastic, positronium (Ps) formation and annihilation cross sections. The effect of virtual Ps formation is evidenced through the analysis of the annihilation map. A significant increase in the electron-positron density in the vicinity of the atom is obtained.