On the orthogonalized plane wave method for calculating energy Eigen-values in a periodic potential

Abstract

The orthogonalized plane wave OPW method is used in the first step towards determination of self-consistent solutions of the Hartree-Fock equations (with Slater's free-electron simplification of the exchange terms) for electrons in a diamond-type crystal. For illustrative purposes, the techniques developed are applied to the determination of energy eigenvalues of the valence and lowest conduction states with zero wave vector in silicon crystal. The initial crystal potential is computed from the charge distribution obtained by placing the atoms forming the crystal on the points of the appropriate space lattice. The atomic charge distributions are determined from simple orthogonalized Slater functions, which can be easily constructed for all atoms, rather than from Hartree or Hartree-Fock atomic functions. A procedure for determining sufficiently good approximations to the wave functions and energy eigenvalues for the core electrons in the initial crystal potential is given. The importance for the convergence and accuracy of the OPW method of using core wave functions which are eigenfunctions of the same operator used to determine the valence and excited states is emphasized. The secular determinant of the OPW method is factored by using appropriate linear combinations of orthogonalized plane waves in the trial function for the valence and excited states. In this connection a detailed exposition is given of a method for obtaining explicit representation matrices for the group of the wave vector, which can then be used to construct basis functions for these representations from sets of orthogonalized plane waves.