A theoretical treatment of the exchange contributions in Bethe’s theory of the stopping of electrons is presented. The effects of the binding of the electron within the atom are systematically evaluated. Starting with the first Born approximation, an expansion in powers of the ratio of the electronic energy of the target atom to the incident electron’s kinetic energy is obtained, and the leading contribution recovers Bethe’s results. The approach involves the use of sum rules that are generalizations of the Bethe sum rule, specifically dealing with the contributions from electron exchange. Though the discussion is restricted to the stopping power here, the approach is quite general and may be applied to the calculation of the exchange contributions to other quantities related to the cross sections of electron-atom scattering. Our results indicate that there may be significant corrections when one accounts for the binding of the target electrons. ©2001 American Institute of Physics.