An expansion method is described which provides a simple and rapid means of calculating for any atom the expectation values of sums of one-electron operators. All the equations which arise can be solved analytically and the results are obtained as functions of the nuclear charge. For inner shells the accuracy is comparable with that of the Hartree-Fock approximation. The method gives a quantitative description of the effects of direct and exchange interactions between electron shells. Results are given for all members of the helium and beryllium iso-electronic sequences.