Overview of Chapters 12–17

There are three basic approaches to the calculation of independent-particle electronic states in materials. There are no fundamental disagreements: all agree when applied carefully and taken to convergence. Indeed, each of the approaches leads to instructive, complementary ways to understand electronic structure and each can be developed into a general framework for accurate calculations.

• Each method has its advantages: each is most appropriate for a range of problems and can provide particularly insightful information in its realm of application.

• Each method has its pitfalls: the user beware. It is all too easy to make glaring errors or over-interpret results if the user does not understand the basics of the methods.

The three types of methods and their characteristic pedagogical values are:

1. Plane wave and grid methods provide general approaches for solution of differential equations, including the Schrödinger and Poisson equations. At first sight, plane waves and grids are very different, but in fact each is an effective way of representing smooth functions. Furthermore, grids are involved in modern efficient plane wave calculations that use fast Fourier transforms.

Chapter 12 is devoted to the basic concepts and methods of electronic structure. Plane waves are presented first because of their simplicity and because Fourier transforms provide a simple derivation of the Bloch theorem.