The conventional Hartree and Hartree-Fock approaches for treating many-electron bound systems have been extended recently to positive energy scattering problems, in which both the bound and continuum orbitals are determined in a fully self-consistent way. In the present study, this generalized self-consistent field (SCF) theory is been tested for electron-hydrogen and positron-hydrogen scattering at low energies, where the target wave function is assumed unknown and solved self-consistently, together with the scattering function. Our results show that the SCF theory converges to the correct amplitudes and to the exact boundary conditions as more configurations are added. A quasi-bound property on the phase shift is observed. This justifies a posteriori the use of the amputated functions and the weak asymptotic condition (WAC), upon which the generalized SCF theory is based. We then apply the theory to the more complex positron-helium and electron-helium scattering systems, where the helium target function is again calculated simultaneously with the scattering function.