A multiple scattering perturbation theory for electron densities, originally discussed by N. H. March and A. M. Murray, is derived in a new way. The new derivation does not depend on special choices for the origin of the energy scale. Therefore, it clarifies the point that the result of the perturbation calculation, the electron density, should be independent of the energy origin chosen for the purposes of the calculation even though the individual terms of the series do depend on that choice. This point is not evident in the previous derivations. Appreciation of this point permits adoption of an origin of the energy scale which varies with position without any change in form of the perturbation expansion. This extends the original theory beyond the significant limitation that the zero-order result is the uniform density of the free-electron gas. In particular, the energy origin can be chosen so that the zero-order contribution reproduces any physical model electron density. The theory then gives successive corrections and allows investigation of the usefulness of physical models by an analysis of the low-order corrections. These ideas permit a compact new derivation of the Thomas-Fermi theory, a derivation which also produces explicit formulae for corrections of all orders. An especially useful choice for the energy origin yields the optimized Thomas-Fermi theory as the zero-order contribution. Therefore, new results of this development are explicit formulae for the corrections to that simple theory.