Computational Physics

6.5 Augmented plane wave methods 135

Figure 6.8. The muffin tin approximation.

In the next sections, we consider the augmented plane wave (APW) method and
its linearised version, and construct a program for calculating the band structure of
copper. In Section 6.7, the pseudopotential method will be considered with a few
applications.

6.5 Augmented plane wave methods

6.5.1 Plane waves and augmentation

As we have just indicated, the problem in constructing basis sets starting from plane
waves lies in the core region. Close to the nucleus, the potential is approximately
spherically symmetric and this symmetry should be exploited in constructing the
basis functions. In the augmented plane wave (APW) [ 11 , 12 ] method, the nuclei
are surrounded by spheres in which the potential is considered to be spherically
symmetric and outside which the potential is constant, and usually taken to be zero.
From a two-dimensional picture (Figure 6.8) it is clear why this approximation is
known as the ‘muffin tin’ approximation.
Outside the muffin tin spheres, the basis functions are simple plane waves eiq·r.
Inside the spheres, they are linear combinations of the solutions to the Schrödinger
equation, evaluated at a predefined energyE. These linear combinations can be
written as

∑∞

l= 0

∑l

m=−l

AlmRl(r)Yml(θ,φ) (6.26)