The Lippmann-Schwinger equation for the scattering of electrons by atoms and molecules is investigated from the perspective of Brillouin-Wigner perturbation theory. It is shown that the solution of the Lippmann-Schwinger can be obtained from many-body Brillouin-Wigner methods for bound-state problems. In particular, the equations of many-body Brillouin-Wigner coupled cluster theory for bound-state systems can be shown to lead directly to equations for the amplitudes for electron-molecule scattering processes.