An Augmented Plane-Wave Method for the Periodic Potential Problem. II

It is shown that the augmented plane-wave method recently proposed can be given an alternative interpretation which leads to a much simpler analytical formulation. We join a plane wave of energy E₀ outside the spherical atoms continuously, but with a derivative which is discontinuous, to spherical solutions of Schr\"odinger's equation inside the spherical atoms, corresponding to an energy E, to be determined. We compute the expectation value of the energy for this combined wave function, consisting of contributions from the plane-wave region, the spherical atoms, and also a surface contribution from the surface of the sphere, since the discontinuous derivative is equivalent to an infinite Laplacian which integrates to a finite contribution over the sphere. We now regard E as a parameter, and vary it to make the expectation value of energy stationary. The resulting wave function is proved to be identical with that set up in Part (I). Furthermore, the energy E inside the spheres proves to be identical with the expectation value of the energy, so that our functions are exact solutions of Schr\"odinger's equation inside the sphere, but not outside the sphere, since the energy of the plane wave E₀ is different from E. However this discrepancy is just canceled in the expectation value of energy by the surface integral. The resulting formulas for energy and wave function are much more convenient to use then those in Part (I).