The Augmented plane-wave method (APW)

The APW method was developed by Slater in 1937. Since the effective crystal potential was found to be constant in most of the open spaces between the cores, the APW method begins by assuming such a muffin-tin potential. The potential is that of a free ion at the core, and is strictly constant outside the core. The wave function for the wave vector ${\bf k}$ is now taken to be

$$\psi_{\bf k}({\bf r}) = \left\{ \begin{array}{cc} e^{i{\bf k... ... \vert{\bf r} - {\bf R}\vert< r_0 \\ \end{array}\right.$$ (242)

where $r_0$ is the core radius. Outside the core the function is a plane wave because the potential is constant there. Inside the core the function is atom-like, and is found by solving the appropriate free-atom Schrödinger equation. Also, the atomic function is chosen such that it joins continuously to the plane wave at the surface of the sphere forming the core; this is the boundary condition here.

Notice that there is no constraint relating ${\bf k}$ and $\epsilon$ for a plane-wave, since we have $\epsilon=\hbar^2k^2/2m$. It is the boundary conditions that determine the value of ${\bf k}$ for a given $\epsilon$.