Variational methods in the computation of energy bands

The formulation of the augmented-plane-wave (APW) method of solving the energy-band problem is reexamined from a fully variational point of view, and various ways of increasing the power and efficiency of the method are suggested. Classes of variational expressions for the energy are derived which are suitable for trial wave functions discontinuous on a separation surface in the crystal unit cell. New trial functions are constructed which permit better matching across the (spherical) separation surface; one of these permits both the function and its derivative to be continuous and is still simple. The matrix elements of a secular equation for the energy are obtained for two classes of variational expressions and three kinds of trial function. Comparison with standard APW matrix elements shows certain differences arising from the fully variational approach; these are discussed along with relative merits of the different secular equations, and the best choice of numerous parameters appearing in the formulation, but no numerical results are given.