The authors present a method for applying the Schwinger variational principle to the scattering of low-energy electrons by molecular ions. Numerical procedures for its application in the static-exchange approximation are specifically discussed. As examples, the s- and p-wave phase shifts for e-He+ are obtained. The procedure also provides the electron continuum wave functions which can be used to obtain photoionization cross sections. The resulting cross sections for photoionization of the ground and metastable states of the He are seen to be accurate.