The multi-electron, central force problem is one that does not have an exact solution. Approximations must be applied, and some aspects of these approximations are common to all multi-particle problems, particularly those of nuclear physics. Atoms can rightly be thought of as the building blocks of our material world. Understanding how quantum mechanics describes atoms, which justifies so much that you have been taught in chemistry and modern physics courses, is the goal for the rest of this text. In this chapter, the hardest part of that broad effort will be attempted which is trying to understand the energy-level structure of an isolated, multi-electron atom. That will be what is meant by the “solution” to the problem at hand. The problem, as presented, will rapidly grow in complexity and seem to be impossible to handle. But then the complexity will shrink as so many terms of interest are shown to be equal to others or zero.