| This work is divided into three sections: (1) In the first section we have investigated the basis dependence of calculated energy eigenvalues of gap states, introduced by ideal vacancies and surfaces, in the tight binding (orbital removal) method by employing a Green's function analysis. We find that (a) if Wannier functions are employed there are no ideal vancancy or surface gap states; (b) if atomic orbitals are used, no ideal vacancy gap states exist in the limit as the number of orbitals on the atom to be removed approaches infinity, although (c) spurious solutions for finite band models can exist. We find the reason for these results is that the orbital removal method is not equivalent to removing the potential of the removed atom. Thus, if the size of the basis is allowed to increase without limit, the orbital removal method yields the same energy eigenvalues as the Hamiltonian of the unperturbed system. This result is independent of the method used to solve the eigenvalue equation. (2) In the second section we develop a generalized effective mass equation to deal with rapidly varying perturbations. (Our analysis is limited to a single non-degenerate band). We apply this formalism to calculate the binding energy of an impurity and then a vacancy within the exactly soluble Kronig-Penney model. In each case our formalism shows a considerable improvement over the results of a standard EMT calculation. In the case of the impurity, for a small but rapidly varying perturbation, the generalized form yields the exact result, while EMT never achieves more than 25% of the correct energy. For the vacancy as well the results of our formalism are clearly superior although here it is clear that for large perturbations the one band formalism is inadequate. (3) In the third section we present an exactly soluble model system of two interacting electrons attracted to a common force center by a harmonic oscillator potential. This model is used to test the appropriateness of various approximation schemes for the v(,xc){lcub}n(r){rcub} and E(,xc){lcub}n(r){rcub}. In addition, we show analytically that the exact Kohn-Sham eigenvalue for the highest occupied orbital equals the difference between the ground states of the two and one particle systems. |
