An Augmented Plane Wave Method for the Periodic Potential Problem

A new method is proposed for solving a periodic potential problem in which the potential can be approximated as a constant outside spheres surrounding the atoms, spherically symmetrical within the spheres. We set up unperturbed functions consisting of a plane wave outside the spheres, joined continuously and with continuous derivative to functions derived from the spherical problem within the spheres. These spherical solutions are linear combinations of eigenfunctions of Schr\"odinger's equation within the spheres, subject to the boundary conditions that the logarithmic derivative of the function of each l value at the surface equals the logarithmic derivative of the corresponding Bessel function in the expansion of the plane wave, thereby insuring continuity of the derivative of the wave function over the sphere if the function itself is continuous. The coefficients in the expansion within the spheres are determined by demanding that the expectation value of the energy of the wave function be stationary when the coefficients are varied. The secular equation connected with this variation problem can be solved exactly, leading to wave functions having the general character of orthogonalized plane waves. A linear combination of such functions is then used to build up an approximate solution of Schr\"odinger's equation. It is shown that the tightly-bound states are handled quite differently from the conduction band, and that the treatment of the conduction band can well resemble the free-electron approximation, thereby perhaps explaining the empirical success of the free-electron approximation for the conduction electrons in metals. The method can be extended to a case where the potential does not have the simple behavior postulated, by treating the difference between the actual potential and the postulated form as a perturbation.