We describe a new grid-based (or localized orbital-based) method for treating the effects of exchange and correlation on electronic transmission through a molecular target where there are initially other bound electrons. Our algorithm combines the approaches of (i) solid-state grid-based algorithms using self-energies and (ii) the complex Kohn method from electron-molecule scattering. For the general problem of a molecular target with n -electrons, our algorithm should ideally solve for electronic transmission with a computational cost scaling as n2, although the present implementation is limited to one-dimensional problems. In this paper, we implement our algorithm to solve three one-dimensional model problems involving two electrons: (i) Single-channel resonant transmission through a double-barrier well (DBW), where the target already contains one bound-state electron [Rejec, Phys. Rev. B 67, 075311 (2003)]; (ii) multichannel resonant transmission through a DBW, where the incoming electron can exchange energy with the bound electron; (iii) transmission through a triple-barrier well (TBW), where the incoming electron can knock forward the bound electron, yielding a physical model of electron-assisted electron transfer. This article offers some insight about the role and size of exchange and correlation effects in molecular conduction, where few such rigorous calculations have yet been made. Such multibody effects have already been experimentally identified in mesoscopic electron transport, giving rise to the "0.7 anomaly," whereby electrons traveling through a narrow channel pair up as singlets and triplets. We expect the effect of electronic correlation to be even more visible for conduction through molecules, where electrons should partially localize into bonding and antibonding orbitals.