In this chapter we return to the goal of robust first-principles methods that can treat materials and phenomena, such as the examples of Ch. 2: from covalent semiconductors like silicon to lanthanides with partially filled f shells like cerium; from optical properties in ionic materials like LiF to metal–insulator transitions in transition metal compounds, and much more. A natural approach is Green's function methods; however, it is very difficult to have one method that treats all cases. In fact, this is not the goal. We seek understanding as well as results. We seek combinations of methods that are each firmly rooted in fundamental theory, with the flexibility to take advantage of different capabilities and provide robust, illuminating predictions for the properties of materials.

This is an active area with on-going developments; the topics of this chapter are not meant to be exhaustive, but rather to formulate basic principles and indicators of possible paths for future work. The approach developed here can be used with any method that partitions a system into parts, treats the parts using techniques that may be different for the different parts, and then constructs the solution for the entire system. The formulation in terms of Green's functions, self-energies, and screened interactions is already given in Ch. 7; here we put this together with the practical methods developed in Chs. 9–20. If the procedures are judiciously chosen, the results have the potential to be more efficient, more accurate, and more instructive than a comparable calculation with a single method.