This dissertation report's calculations of ionization-excitation cross sections of helium and, for the first time, of the negative hydrogen ion, differential in the energy transfer and total, by Compton scattering. They are compared to the Compton cross section by a free electron. The electronic initial state is the ground state, used with two limits, uncorrelated and fully correlated. The final state is selected as uncorrelated. We find the H-- cross section, when the bound electron remains in the ground state, to be 20% smaller than He. This was not expected since H -- geometrical cross section is about 16 times larger than He. However when the sum over the bound states of the second electron is taken into account both cross sections are comparable with two times the Compton cross section by a free electron. A general analysis based on the use of sum rules and the quasi-elastic scattering is applied to both shake and correlated calculations of the cross sections